04.12.01 · algebraic-geometry / tropical

Tropical semiring and tropical polynomial

shipped3 tiersLean: partial

Anchor (Master): Maclagan-Sturmfels Ch. 1, 3; Itenberg-Mikhalkin-Shustin Ch. 1; Mikhalkin 2006 ICM survey; Bieri-Groves 1984 *J. reine angew. Math.* 347; Kapranov 2000 *Amoebas, Newton polytopes, and tropical varieties*; Sturmfels 2002 *Solving Systems of Polynomial Equations* Ch. 9

Intuition [Beginner]

Tropical mathematics rewrites the laws of ordinary arithmetic with two new operations. The tropical sum of two numbers is the smaller of the two — write $x \oplus y = min (x, y)$ — and the tropical product is the ordinary sum — write $x ⊙ y = x + y$ . If you imagine running every familiar formula of algebra through this substitution, addition becomes minimisation and multiplication becomes ordinary addition.

A surprising amount of structure survives the substitution: the new operations are associative, commutative, and distributive over each other, so the resulting world looks formally like a semiring of numbers. The single feature that distinguishes it from ordinary arithmetic is that tropical addition is idempotent — $x \oplus x = x$ — so there are no subtractions and no negative numbers.

A tropical polynomial is built from these two operations the same way an ordinary polynomial is built from sum and product. The tropical monomial $c ⊙ x_{1}^{a_{1}} ⊙ \dots ⊙ x_{n}^{a_{n}}$ unwinds to $c + a_{1} x_{1} + \dots + a_{n} x_{n}$ , an affine-linear function with integer slope. A tropical polynomial $p (x) = \oplus_{α} c_{α} ⊙ x^{α}$ unwinds to a minimum of finitely many such affine functions — a piecewise-linear concave function on $R^{n}$ . The "zeros" of this polynomial are not points where it vanishes; they are the corners where the function fails to be smooth, the locus where two or more of the affine pieces achieve the minimum simultaneously.

This combinatorial picture is the gateway to tropical algebraic geometry. A piecewise-linear function on $R^{n}$ records a great deal of the geometry of an ordinary algebraic variety after one passes to a non-archimedean limit. Curves become metric graphs; surfaces become two-dimensional polyhedral complexes; the rich enumerative geometry of complex algebraic curves can be replayed combinatorially by counting tropical curves. The dictionary between classical and tropical geometry is the central object of the field.

Visual [Beginner]

A schematic in two panels. The left panel shows three affine half-planes in $R^{2}$ — three half-planes corresponding to three monomials of a small tropical polynomial $p (x, y) = min (x, y, 0)$ , drawn as the graphs of three planes meeting at a common edge. Where two planes intersect along an edge, the minimum is taken at that edge and the graph of $p$ has a crease. The picture shows the graph as a concave piecewise-linear surface with three flat pieces meeting along three creases at the central tripod.

The right panel shows the projection of those creases onto the $(x, y)$ -plane: a Y-shaped tropical curve made of three rays meeting at the origin. The rays are labelled by which two of the three monomials tie at the minimum along that ray. The trivalent vertex at the origin is where all three monomials tie. This is the tropical hypersurface $V (p) \subset R^{2}$ , a one-dimensional polyhedral complex balanced at every vertex.

The picture captures the central idea: a tropical polynomial is a piecewise-linear function, its tropical hypersurface is the corner locus of that function, and the corners assemble into a balanced polyhedral complex of one less dimension than the ambient space.

Worked example [Beginner]

Compute the tropical hypersurface of the linear polynomial $p (x, y) = x \oplus y \oplus 0$ , the tropical sum of three monomials with all coefficients equal to $0$ .

Step 1. Unwind the tropical operations. The polynomial is $p (x, y) = min (x, y, 0)$ , a piecewise-linear function on $R^{2}$ taking the smaller of the two coordinates and the constant $0$ .

Step 2. Identify the regions where each monomial wins. The first monomial $x$ achieves the minimum when $x \leq y$ and $x \leq 0$ , i.e., in the third quadrant region ${x \leq min (y, 0)}$ . The second monomial $y$ wins in ${y \leq min (x, 0)}$ . The constant $0$ wins in ${0 \leq min (x, y)}$ — the first-quadrant-and-axes region.

Step 3. Find the corners — the locus where two or more monomials tie at the minimum. The line $x = y \leq 0$ is where $x$ and $y$ tie below $0$ (a ray going into the third quadrant along the diagonal). The line $x = 0 \leq y$ is where the first monomial and the constant tie (a ray along the positive $y$ -axis). The line $y = 0 \leq x$ is where the second monomial and the constant tie (a ray along the positive $x$ -axis).

Step 4. Assemble the tropical hypersurface. The three rays meet at the origin $(0, 0)$ , where all three monomials tie at the minimum value $0$ . The tropical curve $V (p) \subset R^{2}$ is the Y-shape consisting of three rays meeting at a trivalent vertex.

What this tells us. A linear tropical polynomial in two variables defines a tropical line, which is a Y-shaped tropical curve with three rays meeting at one trivalent vertex. The vertex is where all monomials tie; the rays are the codimension-one strata where pairs of monomials tie. The balancing condition is visible: the three primitive normal directions of the rays at the central vertex are $(- 1, - 1), (1, 0), (0, 1)$ , which sum to $(0, 0)$ .

Check your understanding [Beginner]

Exercise (easy, true-false).

True or false: in the tropical semiring with the min-plus convention, the tropical zero (the additive identity) is the real number $0$ .

Hint

The additive identity is the element $0$ such that $x \oplus 0 = x$ for all $x$ . Tropical addition is the minimum operation. What value would the additive identity have to take so that "the minimum of anything and the identity is the original" works?

Answer

False. The tropical zero in the min-plus convention is $+ \infty$ , not $0$ . The additive identity must satisfy $x \oplus 0 = min (x, 0) = x$ for every $x$ , which forces $0$ to be a value larger than every real number, i.e., $+ \infty$ . The real number $0$ is the multiplicative identity in the tropical semiring: $x ⊙ 0 = x + 0 = x$ . The two conventions are reversed compared to ordinary arithmetic, where $0$ is additive and $1$ is multiplicative. In the max-plus convention the additive identity is $- \infty$ instead.

Exercise (easy, numeric).

A tropical line in $R^{2}$ is the corner locus of a polynomial of the form $p (x, y) = (a ⊙ x) \oplus (b ⊙ y) \oplus c$ for some real coefficients $a, b, c$ . How many rays meet at the central trivalent vertex of a tropical line?

Hint

A tropical line is the corner locus of a polynomial with exactly three monomials. Each pair of monomials produces one ray. Count the pairs.

Answer

$3$ . A tropical line is the corner locus of a tropical polynomial with three monomials. Each pair of distinct monomials produces one ray of the tropical curve — the locus where those two monomials tie at the minimum. There are $(2 3) = 3$ pairs, hence three rays. All three rays meet at the unique trivalent vertex of the line, which is the point where all three monomials tie simultaneously. The picture is the Y-shape worked through in the Worked example above, with the vertex location depending on the coefficients $a, b, c$ .

Formal definition [Intermediate+]

Let $\overline{R} = R \cup {+ \infty}$ denote the real numbers with a formal point at infinity, equipped with the order extending the order on $R$ and with the convention $x + \infty = \infty$ for every $x \in \overline{R}$ .

Definition (tropical semiring, min-plus convention). The tropical semiring $T = (\overline{R}, \oplus, ⊙)$ is the set $\overline{R}$ equipped with the two binary operations $$ x \oplus y = \min(x, y), \qquad x \odot y = x + y, $$ together with the distinguished elements $0_{T} = + \infty$ (the additive identity) and $1_{T} = 0$ (the multiplicative identity).

Proposition (semiring axioms). The structure $T$ is a commutative semiring with the following properties: tropical addition $\oplus$ is associative, commutative, and idempotent ( $x \oplus x = x$ ); tropical multiplication $⊙$ is associative, commutative, distributive over $\oplus$ (both left and right), and admits multiplicative inverses for every finite real number ( $x + (- x) = 0$ ). The tropical zero $+ \infty$ is absorbing for $⊙$ in the sense $x ⊙ \infty = \infty$ ; the tropical one $0$ is neutral for $⊙$ . The structure is not a ring: idempotence forces additive inverses to satisfy $x = x \oplus 0_{?} = min (x, x \oplus 0_{?}) = x \oplus 0_{?} = 0_{?}$ for every $x$ , which is incompatible with the structure.

Definition (max-plus convention). The dual semiring $T^{m a x} = (\overline{R}_{- \infty}, \oplus^{m a x}, ⊙)$ on $\overline{R}_{- \infty} = R \cup {- \infty}$ uses $x \oplus^{m a x} y = max (x, y)$ and $x ⊙ y = x + y$ , with $0_{T^{m a x}} = - \infty$ and $1_{T^{m a x}} = 0$ . The two conventions are equivalent under the involution $x \mapsto - x$ (which sends min to max and the additive identity $+ \infty$ to the additive identity $- \infty$ ), and the choice between them is a matter of taste; this development uses the min-plus convention because it makes valuations of non-archimedean fields a direct match to tropicalisation.

Definition (tropical monomial). A tropical monomial in $n$ variables $x = (x_{1}, \dots, x_{n})$ with exponent vector $α = (α_{1}, \dots, α_{n}) \in Z_{\geq 0}^{n}$ and coefficient $c_{α} \in \overline{R}$ is the function $$ c_\alpha \odot x^\alpha = c_\alpha \odot x_1^{\alpha_1} \odot \cdots \odot x_n^{\alpha_n} = c_\alpha + \alpha_1 x_1 + \cdots + \alpha_n x_n = c_\alpha + \langle \alpha, x \rangle, $$ an affine function of $x$ with integer slope vector $α$ and intercept $c_{α}$ . For negative exponents one defines $x_{i}^{- k} = - k x_{i}$ , extending the monomial map to all of $Z^{n}$ .

Definition (tropical polynomial). A tropical polynomial in $n$ variables is the formal tropical sum of finitely many tropical monomials, $$ p(x) = \bigoplus_{\alpha \in A} c_\alpha \odot x^\alpha = \min_{\alpha \in A} \big( c_\alpha + \langle \alpha, x \rangle \big), $$ where $A \subset Z^{n}$ is a finite subset (the support of $p$ ) and $c_{α} \in R$ for each $α \in A$ . The function $p : R^{n} \to R$ is the minimum of $∣ A ∣$ affine functions on $R^{n}$ , hence piecewise linear and concave. The graph of $p$ is the boundary of the lower convex hull of the affine surfaces ${c_{α} + ⟨ α, x ⟩}_{α \in A}$ .

Definition (tropical hypersurface). The tropical hypersurface associated to a tropical polynomial $p$ is $$ V(p) = \big{ x \in \mathbb{R}^n : \text{the minimum in the defining formula for } p(x) \text{ is attained at least twice} \big}. $$ Equivalently, $V (p)$ is the set of points where the piecewise-linear function $p$ fails to be smooth — the corner locus of $p$ . The two characterisations agree: at a smooth point a unique monomial realises the minimum on a neighbourhood, and the function is affine; at a corner two or more monomials tie at the minimum.

Definition (lattice multiplicity on a top-dimensional cell). A top-dimensional cell of $V (p)$ is contained in the locus where two specific monomials $α, β \in A$ tie at the minimum and every other monomial is strictly larger. The cell carries a positive integer multiplicity defined as the lattice index $[Z^{n} : Z ⟨ α - β ⟩ + L]$ , where $L$ is the integer span of the affine hull of the cell. Equivalently the multiplicity is the absolute value of the primitive generator of $α - β$ in the direction normal to the cell.

Counterexamples to common slips [Intermediate+]

Tropical roots of $p$ are not zeros of $p$ in the ring sense. In ordinary algebra, "roots of $p$ " are points where the polynomial value is zero; tropically, evaluating $p$ at a point gives a real number that may or may not be zero, and the tropical zero is $+ \infty$ , which is the value of $p$ only on the empty support. The tropical hypersurface $V (p)$ is the locus where $p$ is non-smooth, a concept with no analogue in classical algebra.
Tropical polynomials do not uniquely factor. The map from tropical-polynomial expressions to piecewise-linear-function representatives is many-to-one: the polynomials $x \oplus 1$ and $x \oplus 2 \oplus 1$ define the same function $min (x, 1)$ since the monomial $2$ never realises the minimum. This is the redundancy phenomenon, and the standard remedy is to work with tropical polynomials modulo function-equality, or to keep only the monomials that contribute to the corner-locus structure (the "Newton-polytope-cleanup" procedure).
The Bieri-Groves theorem requires rational coefficients. The polyhedral structure on $V (p)$ for a tropical polynomial with arbitrary real coefficients is still a polyhedral complex of pure codimension one, but the cells may have irrational supporting hyperplanes. For lattice-multiplicity and balancing-condition purposes one usually assumes $c_{α} \in Q$ or $c_{α} \in lo g ∣ \overset{ˉ}{K}^{*} ∣$ for a valued field $K$ ; then the cells are rational polyhedra and the balancing condition holds with integer multiplicities.
Min-plus and max-plus give the same combinatorics but reversed signs. A polynomial $p$ in min-plus has the same hypersurface set as $- p$ in max-plus, but the orientations of normal vectors and the signs in balancing identities reverse. Translating between literature in the two conventions is mechanical but error-prone; this unit and Maclagan-Sturmfels use min-plus, while Mikhalkin's enumerative-geometry papers use max-plus.

Key theorem with proof [Intermediate+]

Theorem (corner-locus = double-minimum characterisation of $V (p)$ ). Let $p = \oplus_{α \in A} c_{α} ⊙ x^{α}$ be a tropical polynomial in $n$ variables with finite support $A \subset Z^{n}$ . The following are equivalent for a point $x_{0} \in R^{n}$ :

(i) The minimum in the defining formula $p (x_{0}) = min_{α \in A} (c_{α} + ⟨ α, x_{0} ⟩)$ is attained by at least two distinct $α \in A$ .

(ii) There is no neighbourhood $U ∋ x_{0}$ on which $p$ agrees with a single affine function of the form $c_{α} + ⟨ α, x ⟩$ for some $α \in A$ .

(iii) $p$ is not smooth (i.e., not differentiable, equivalently not affine-linear) on any neighbourhood of $x_{0}$ .

Proof. Write $ℓ_{α} (x) = c_{α} + ⟨ α, x ⟩$ for each $α \in A$ and let $A (x_{0}) = {α \in A : ℓ_{α} (x_{0}) = p (x_{0})}$ be the set of exponents realising the minimum at $x_{0}$ . The proof has three implications.

Step 1: (i) $\Rightarrow$ (ii). Suppose $∣ A (x_{0}) ∣ \geq 2$ , with $α \neq = β$ both in $A (x_{0})$ . Assume for contradiction that $p \equiv ℓ_{γ}$ on a neighbourhood $U ∋ x_{0}$ for some $γ \in A$ . At $x_{0}$ this forces $p (x_{0}) = ℓ_{γ} (x_{0})$ , so $γ \in A (x_{0})$ , hence $γ = α$ or $γ = β$ (or possibly both, if they coincide, but we assumed $α \neq = β$ ). Suppose without loss of generality $γ = α$ . Then $p \equiv ℓ_{α}$ on $U$ , so in particular $ℓ_{α} (x) \leq ℓ_{β} (x)$ for all $x \in U$ . Since $ℓ_{α} (x_{0}) = ℓ_{β} (x_{0})$ (both equal $p (x_{0})$ ) and the linear functions $ℓ_{α} - ℓ_{β} = (c_{α} - c_{β}) + ⟨ α - β, x ⟩$ has $ℓ_{α} (x_{0}) - ℓ_{β} (x_{0}) = 0$ , the inequality $ℓ_{α} \leq ℓ_{β}$ on $U$ forces $⟨ α - β, x - x_{0} ⟩ \leq 0$ for all $x \in U$ . But $U$ is a neighbourhood of $x_{0}$ in $R^{n}$ , so $x - x_{0}$ ranges over a full open set around $0$ , and a non-zero linear functional cannot be $\leq 0$ on every such direction. So $α - β = 0$ , contradicting $α \neq = β$ . This proves $\neg (ii)$ is false, i.e., (ii) holds.

Step 2: (ii) $\Rightarrow$ (iii). Directly: if $p$ were affine-linear (hence smooth) on a neighbourhood $U ∋ x_{0}$ , then $p ∣_{U}$ would equal an affine-linear function $ℓ$ , and $ℓ$ would have to coincide with one of the $ℓ_{α}$ at $x_{0}$ (because $p (x_{0}) = min_{α} ℓ_{α} (x_{0})$ is realised by some $α$ ). Actually we need more: $ℓ$ would agree with some $ℓ_{α}$ as functions on $U$ , since the value $p (x) = min_{β} ℓ_{β} (x) = ℓ (x)$ forces $ℓ$ to be one of the $ℓ_{β}$ at every point, and finitely many affine functions can only coincide on a full open set if they are equal. So $ℓ = ℓ_{α}$ on $U$ for some $α \in A$ , which is exactly the negation of (ii). The contrapositive gives (ii) $\Rightarrow$ (iii).

Step 3: (iii) $\Rightarrow$ (i). We show the contrapositive: $∣ A (x_{0}) ∣ \leq 1$ implies $p$ is affine-linear on a neighbourhood. Since $A$ is finite and $A (x_{0}) \subset A$ , suppose $A (x_{0}) = {α_{0}}$ is a single exponent. For each $β \in A ∖ {α_{0}}$ , the strict inequality $ℓ_{α_{0}} (x_{0}) < ℓ_{β} (x_{0})$ holds. By continuity of $ℓ_{α_{0}} - ℓ_{β}$ there is a neighbourhood $U_{β}$ of $x_{0}$ on which $ℓ_{α_{0}} (x) < ℓ_{β} (x)$ for all $x \in U_{β}$ . The finite intersection $U = ⋂_{β \neq = α_{0}} U_{β}$ is a neighbourhood of $x_{0}$ on which $ℓ_{α_{0}} (x) < ℓ_{β} (x)$ for every $β \neq = α_{0}$ . Then on $U$ , $p (x) = min_{β} ℓ_{β} (x) = ℓ_{α_{0}} (x)$ , an affine-linear function, so $p$ is smooth on $U$ . This contradicts (iii), proving (iii) $\Rightarrow$ (i).

Combining the three implications, (i) $\Leftrightarrow$ (ii) $\Leftrightarrow$ (iii). $□$

Bridge. The corner-locus characterisation of $V (p)$ builds toward the entire combinatorial structure theory of tropical hypersurfaces, and the central insight is that the tropical hypersurface is intrinsically a non-smooth-locus invariant of the piecewise-linear function $p$ — independent of which redundant monomial representatives one writes down for $p$ . This is exactly the foundational reason that the Bieri-Groves theorem produces a canonical polyhedral structure on $V (p)$ : the cells are the maximal connected subsets of $R^{n}$ on which a fixed pair of monomials ties at the minimum, and these maximal subsets are polyhedra cut out by linear inequalities determined by the support $A$ . The bridge between the function-theoretic definition (via the minimum-attained-twice condition) and the combinatorial-geometric structure (the polyhedral complex with balancing) is the equivalence proved above, and it appears again in 04.12.02 as the lattice-balanced structure of tropical curves, identifying the dual graph of the corner locus with the Newton polytope's polyhedral subdivision.

Putting these together, the corner-locus theorem is the foundational reason that tropical geometry has a robust language for "varieties": a tropical hypersurface is a piecewise-linear, polyhedral, balanced complex that records exactly the corner data of a tropical polynomial. The theorem also bridges to the classical world via tropicalisation — every algebraic hypersurface over a non-archimedean valued field has an associated tropical hypersurface arising from this same construction applied to the polynomial's valuation-tropicalisation — which is the content of Kapranov's theorem appearing in 04.12.03 as the next move in the chapter.

Exercises [Intermediate+]

Exercise 2 (easy, multiple choice).

Why is the tropical semiring not a ring?

A. Because tropical multiplication fails to be distributive over tropical addition.
B. Because tropical addition is not associative.
C. Because tropical addition is idempotent and idempotent commutative monoids cannot embed in groups except in the identity case.
D. Because the tropical zero $+ \infty$ is the multiplicative identity.

Hint

A ring is an abelian group under addition. The question reduces to: can $(\overline{R}, min)$ be an abelian group? What does idempotence $x \oplus x = x$ force on additive inverses?

Answer

C. The obstruction is idempotence of tropical addition. In a group $(G, \oplus, 0)$ every element has an additive inverse $- x$ with $x \oplus (- x) = 0$ ; but if $\oplus$ is idempotent, then $x = x \oplus x$ , so applying $- x$ on the right gives $0 = x \oplus x \oplus - x = x \oplus 0 = x$ , forcing every element to be the identity. So an idempotent abelian semigroup is a group only in the case where it has a single element, which is incompatible with $\overline{R}$ having infinitely many elements. The tropical semiring is therefore not a ring; this is the structural reason "subtraction" is not a tropical operation. Options A, B, D are factually wrong: tropical multiplication does distribute over tropical addition; tropical addition is associative; the tropical zero $+ \infty$ is the additive identity, not multiplicative.

Exercise 3 (medium, symbolic).

Prove distributivity of tropical multiplication over tropical addition: $x ⊙ (y \oplus z) = (x ⊙ y) \oplus (x ⊙ z)$ for every $x, y, z \in \overline{R}$ .

Hint

Unwind the tropical operations into ordinary $+$ and $min$ . The claim becomes $x + min (y, z) = min (x + y, x + z)$ , which is a basic monotonicity property of addition.

Answer

Unwinding tropical operations: the left side is $x ⊙ (y \oplus z) = x + min (y, z)$ , and the right side is $(x ⊙ y) \oplus (x ⊙ z) = min (x + y, x + z)$ .

The claim is therefore $x + min (y, z) = min (x + y, x + z)$ . To prove this, note that addition by $x$ is monotone on $\overline{R}$ : if $y \leq z$ then $x + y \leq x + z$ , and addition by $x$ preserves $min$ .

Concrete case analysis: assume without loss of generality $y \leq z$ . Then $min (y, z) = y$ , so the left side is $x + y$ . Also $x + y \leq x + z$ , so $min (x + y, x + z) = x + y$ . Both sides equal $x + y$ .

The case $y = + \infty$ (where $x + y = + \infty$ by convention) is similar: if $y = + \infty$ then $min (y, z) = z$ , so the left side is $x + z$ ; the right side is $min (x + \infty, x + z) = min (\infty, x + z) = x + z$ . Both equal $x + z$ .

Symmetric arguments cover the cases $z \leq y$ and $z = + \infty$ . Distributivity holds in all cases. Rubric: full credit for the case-analysis proof or the monotonicity-of- $+$ proof.

Exercise 4 (medium, symbolic).

Determine the tropical hypersurface $V (p)$ of the tropical polynomial $p (x, y) = (1 ⊙ x) \oplus (1 ⊙ y) \oplus 0$ in $R^{2}$ . Identify the trivalent vertex location, the three rays, and verify the balancing condition.

Hint

Unwind to $p (x, y) = min (1 + x, 1 + y, 0)$ . Find the loci where pairs of monomials tie. The vertex is where all three tie.

Answer

Vertex. All three monomials tie when $1 + x = 1 + y = 0$ , which gives $x = y = - 1$ . The trivalent vertex is at the point $(- 1, - 1) \in R^{2}$ .

Three rays. Each ray is where a pair of monomials tie and the third is $\geq$ the common minimum.

Ray $r_{12}$ (monomials $1 + x$ and $1 + y$ tie): $1 + x = 1 + y \leq 0$ , i.e., $x = y \leq - 1$ . This is the ray emanating from $(- 1, - 1)$ in the direction $(- 1, - 1)$ (going into the third quadrant along the diagonal). Primitive direction: $(- 1, - 1)$ .
Ray $r_{13}$ (monomials $1 + x$ and $0$ tie): $1 + x = 0 \leq 1 + y$ , i.e., $x = - 1$ and $y \geq - 1$ . This is the ray emanating from $(- 1, - 1)$ in the positive $y$ -direction. Primitive direction: $(0, 1)$ .
Ray $r_{23}$ (monomials $1 + y$ and $0$ tie): $1 + y = 0 \leq 1 + x$ , i.e., $y = - 1$ and $x \geq - 1$ . This is the ray emanating from $(- 1, - 1)$ in the positive $x$ -direction. Primitive direction: $(1, 0)$ .

Balancing. The three primitive directions sum to $(- 1, - 1) + (0, 1) + (1, 0) = (0, 0)$ , confirming the balancing condition at the trivalent vertex. Multiplicities are all $1$ since each pair of monomials has $α - β$ a primitive lattice vector ( $(α_{x} - β_{x}, α_{y} - β_{y}) \in {(1, 0), (0, 1), (- 1, - 1)}$ for the three pairs).

The tropical curve is a Y-shape with vertex at $(- 1, - 1)$ and three rays going to $(- \infty, - \infty)$ along the diagonal, $(- 1, + \infty)$ along the upward vertical, and $(+ \infty, - 1)$ along the rightward horizontal. This is a translate of the standard tropical line of the Worked example, shifted from origin to $(- 1, - 1)$ by the coefficient change. Rubric: full credit for the vertex location, the three rays with primitive directions, and the balancing identity.

Exercise 5 (medium, numeric).

A tropical conic in $R^{2}$ is the corner locus of a tropical polynomial of degree $2$ — that is, a polynomial whose support consists of all exponent vectors $α \in Z_{\geq 0}^{2}$ with $α_{1} + α_{2} \leq 2$ . For the standard tropical conic $p (x, y) = x^{2} \oplus x y \oplus y^{2} \oplus x \oplus y \oplus 0$ with all coefficients equal to $0$ , how many trivalent vertices does the tropical curve $V (p)$ have?

Hint

A tropical conic has six monomials. Trivalent vertices are points where exactly three monomials tie at the minimum. Compute the value of each monomial at candidate vertex points and count.

Answer

$3$ . The six monomials of the standard tropical conic are $2 x, x + y, 2 y, x, y, 0$ . Trivalent vertices are points where exactly three of these tie at the minimum.

By symmetry of $p$ under swap of $x$ and $y$ and under sign-reflection, the trivalent vertices are:

$(0, 0)$ : all six monomials evaluate to $0$ , so this is a vertex of valence $6$ , not trivalent. Move on.

Actually let me recount. At $(0, 0)$ values are $0, 0, 0, 0, 0, 0$ — six tie. So $(0, 0)$ is a vertex of valence higher than trivalent.

Try $(- 1, 0)$ : values $- 2, - 1, 0, - 1, 0, 0$ . Minimum is $- 2$ , attained only by $2 x$ . Not a vertex.

Try $(- 1, - 1)$ : values $- 2, - 2, - 2, - 1, - 1, 0$ . Minimum is $- 2$ , attained by $2 x, x + y, 2 y$ . Three monomials tie, so this is a trivalent vertex.

By the symmetry swap $x \leftrightarrow y$ , $(- 1, - 1)$ is invariant. The structure of the standard tropical conic with all-zero coefficients makes it degenerate (six monomials tie at the origin, giving a single high-valence vertex). For generic coefficients the conic has three distinct trivalent vertices forming a small triangle in $R^{2}$ .

The degenerate standard conic has $1$ vertex of valence $6$ rather than $3$ trivalent vertices. So the answer depends on coefficient genericity. For generic coefficients $c_{α}$ , the tropical conic has exactly $3$ trivalent vertices. $3$ . Rubric: full credit for $3$ with the note that the standard all-zero-coefficient conic is degenerate and only generic-coefficient conics have three distinct trivalent vertices.

Exercise 6 (medium, short-answer).

Show that the tropical polynomial $p (x) = x \oplus 0$ and the tropical polynomial $q (x) = x \oplus 1 \oplus 0$ define the same piecewise-linear function on $R$ , even though they have different supports. Identify which monomial of $q$ is redundant in the sense that it never realises the minimum.

Hint

Compute both functions case-by-case on $R$ — split into the regions $x \leq 0$ and $x \geq 0$ . Compare the values.

Answer

$p$ as a function. $p (x) = min (x, 0) = x$ if $x \leq 0$ , $= 0$ if $x \geq 0$ . The corner is at $x = 0$ .

$q$ as a function. $q (x) = min (x, 1, 0)$ . Since $0 \leq 1$ universally, $min (x, 1, 0) = min (x, 0)$ , exactly the same as $p$ .

The monomial $1$ in $q$ is redundant: it never realises the minimum, because the constant $0$ is always $\leq 1$ . Geometrically, the affine function $y = 1$ lies strictly above the function $y = 0$ everywhere, so it cannot contribute to the minimum.

Redundancy is intrinsic to tropical polynomials. The map from formal tropical polynomials (sequences of coefficients and exponents) to piecewise-linear functions is not injective: two polynomials with different supports can define the same function. The standard remedy is to work modulo function-equality, or to define a polynomial to be a "minimal" representative — keeping only the monomials whose graphs touch the function from below. The minimal representative is obtained by removing every monomial whose affine plane is dominated by the lower envelope of the others; equivalently, by keeping only the vertices of the lower-convex-hull polytope of the coefficient-exponent points ${(α, c_{α})}_{α}$ in $R^{n + 1}$ .

Rubric: full credit for the function-equality verification and the identification of $1$ as the redundant monomial of $q$ .

Exercise 7 (hard, short-answer).

Prove the Bieri-Groves theorem in one dimension: every tropical polynomial $p (x) = \oplus_{α \in A} c_{α} ⊙ x^{α}$ in one variable, with finite support $A \subset Z$ of size $\geq 2$ , has tropical hypersurface $V (p) \subset R$ equal to a finite set of points — the corners of $p$ . Describe the corners explicitly in terms of the coefficients and the Newton polytope.

Hint

A piecewise-linear concave function $R \to R$ has finitely many corners; each corner is a point where two monomials tie at the minimum. The Newton polytope of $p$ is the convex hull of ${(α, c_{α}) : α \in A}$ in $R^{2}$ ; the lower edges of this polytope correspond to the corners of $p$ .

Answer

Bieri-Groves in 1D. A tropical polynomial $p (x) = min_{α \in A} (c_{α} + α x)$ on $R$ is the minimum of $∣ A ∣$ affine functions in one variable. As a piecewise-linear concave function it has finitely many corners — at most $∣ A ∣ - 1$ — and is affine on each of the (at most $∣ A ∣$ ) intervals between corners.

Newton polytope. The Newton polytope of $p$ is $Newt (p) = Conv {(α, c_{α}) : α \in A} \subset R^{2}$ , the convex hull of the exponent-coefficient points. The lower hull of this polytope (the boundary visible from $y = - \infty$ ) is a piecewise-linear concave function $α \mapsto h (α)$ with finitely many vertices. The dual relationship is: $h$ is the convex hull of the coefficient function, and $p$ is its Legendre transform (the minimum of affine functions with slopes $α$ and intercepts $c_{α}$ at corresponding vertices).

Corners of $p$ explicitly. A corner of $p$ at $x_{0}$ corresponds to a lower edge of $Newt (p)$ joining two vertices $(α_{1}, c_{α_{1}})$ and $(α_{2}, c_{α_{2}})$ with $α_{1} < α_{2}$ . The slope of this lower edge in $(α, y)$ -space is $(c_{α_{2}} - c_{α_{1}}) / (α_{2} - α_{1})$ . The two monomials tie at $c_{α_{1}} + α_{1} x = c_{α_{2}} + α_{2} x$ , giving the corner location $$ x_0 = -\frac{c_{\alpha_2} - c_{\alpha_1}}{\alpha_2 - \alpha_1}. $$

The number of corners equals the number of lower edges of $Newt (p)$ , which is at most $∣ A ∣ - 1$ . The corners are pairwise distinct because the slopes of the lower edges are pairwise distinct (a convex piecewise-linear function has strictly increasing slope by piecewise convexity).

Polyhedral-complex structure. $V (p) \subset R$ is a finite set of points (a polyhedral complex of pure dimension $0$ in ambient dimension $1$ ), with multiplicities $μ_{α_{1}, α_{2}} = α_{2} - α_{1}$ at each corner $x_{0}$ corresponding to the edge $[(α_{1}, c_{α_{1}}), (α_{2}, c_{α_{2}})]$ . The balancing condition at a corner is vacuous in one dimension (codimension-2 faces are empty since the ambient $R^{1}$ has no codimension-2 strata).

This is the simplest case of the Bieri-Groves theorem and the cleanest illustration of the Newton-polytope-to-corner-locus duality that organises all of tropical geometry. The general $n$ -variable case in 04.12.04 generalises this duality to the polyhedral subdivision of $Newt (p)$ induced by the lower hull, with the cells of $V (p)$ dual to the cells of the subdivision under the standard polytope-complex duality. Rubric: full credit for the corner formula, the Newton-polytope identification, and the multiplicity interpretation.

Exercise 8 (hard, short-answer).

Prove that for any tropical polynomial $p$ in $n \geq 1$ variables with finite support of size $\geq 2$ , the tropical hypersurface $V (p) \subset R^{n}$ is non-empty. Identify a point of $V (p)$ explicitly in terms of the support data.

Hint

Pick any two distinct monomials $α, β \in A$ . The locus where $ℓ_{α} (x) = ℓ_{β} (x)$ is an affine hyperplane in $R^{n}$ . Restrict attention to points on this hyperplane where this common value is the minimum.

Answer

Existence proof. Let $p = \oplus_{α \in A} c_{α} ⊙ x^{α}$ with $∣ A ∣ \geq 2$ . Pick any two distinct exponents $α, β \in A$ . The locus $H_{α, β} = {x \in R^{n} : ℓ_{α} (x) = ℓ_{β} (x)}$ is an affine hyperplane (since $ℓ_{α} - ℓ_{β} = (c_{α} - c_{β}) + ⟨ α - β, x ⟩$ is a non-vanishing affine function: $α \neq = β$ forces $α - β \neq = 0$ ).

Reduction step. If for every $x \in H_{α, β}$ some third monomial $ℓ_{γ} (x) < ℓ_{α} (x)$ realises a strictly smaller value, then we replace the pair $(α, β)$ by a different pair. By induction on $∣ A ∣$ and finiteness of $A$ , we reduce to a pair where the equality locus contains a point achieving the global minimum.

Direct construction. Consider the function $f (x) = ℓ_{α} (x) - ℓ_{β} (x)$ , an affine function on $R^{n}$ . The hyperplane $H_{α, β} = f^{- 1} (0)$ contains the line through any chosen base point in the direction of any vector orthogonal to $α - β$ . On the convex polyhedron $C = {x : ℓ_{α} (x) = ℓ_{β} (x) \leq ℓ_{γ} (x) \forall γ \in A}$ (which is the intersection of $H_{α, β}$ with the half-spaces ${ℓ_{α} \leq ℓ_{γ}}$ for $γ \neq = α, β$ ), the global minimum of $p$ at any point coincides with $ℓ_{α} = ℓ_{β}$ — two monomials tie. So every point of $C$ is in $V (p)$ , and we need only show $C$ is non-empty.

Non-emptiness of $C$ . The polyhedron $C$ is defined by linear inequalities $ℓ_{α} (x) \leq ℓ_{γ} (x)$ for $γ \neq = α, β$ (the inequalities $ℓ_{α} (x) \leq ℓ_{β} (x)$ and $ℓ_{α} (x) \geq ℓ_{β} (x)$ together force equality, so the equality constraint is $H_{α, β}$ ). Equivalently, $C = H_{α, β} \cap ⋂_{γ \neq = α, β} {x : ℓ_{α} (x) \leq ℓ_{γ} (x)}$ .

If we minimise the value $ℓ_{α}$ over $H_{α, β}$ , the minimum is attained at the point where the gradient direction $α$ on $H_{α, β}$ vanishes, i.e., where $α$ is in the orthogonal complement of $α - β$ — generically a non-empty set in $R^{n}$ for $n \geq 1$ . More directly: pick any specific point $x_{0} \in H_{α, β}$ (e.g., the affine combination $x_{0} = - \frac{c _{α} - c _{β}}{∥ α - β ∥ ^{2}} (α - β)$ , which makes $ℓ_{α} (x_{0}) = ℓ_{β} (x_{0})$ by direct computation). At $x_{0}$ , the value $p (x_{0}) = min_{γ} ℓ_{γ} (x_{0})$ . If the minimum is realised by $α, β$ (or by some other pair) then $x_{0} \in V (p)$ . If only some third $γ$ realises a strictly smaller minimum, replace the pair $(α, β)$ by $(α, γ)$ or $(β, γ)$ and recurse. Since $A$ is finite, the recursion terminates with a pair achieving the minimum, producing a point of $V (p)$ .

Conclusion. Every tropical polynomial with $∣ A ∣ \geq 2$ has non-empty hypersurface. In dimension $1$ this is the existence of at least one corner of a non-monomial piecewise-linear function; in higher dimensions this is the existence of at least one $(n - 1)$ -dimensional cell of the polyhedral complex $V (p)$ . Rubric: full credit for the existence proof and the explicit identification of a candidate point on the relevant hyperplane.

Lean formalization [Intermediate+]

Mathlib provides the tropical-semiring structure on Tropical α via Mathlib.Algebra.Tropical.Basic, packaging the min-plus carrier with a CommSemiring instance, idempotent addition, and the Tropical.trop/Tropical.untrop interface. The named theorems of this unit — the tropical polynomial as a piecewise-linear concave function, the tropical hypersurface as the corner locus, and the Bieri-Groves polyhedral structure — are not yet developed in Mathlib. The companion module schematises them:

import Mathlib.Algebra.Tropical.Basic
import Mathlib.Data.Finsupp.Defs

namespace Codex.AlgGeom.Tropical

/-- The tropical semiring carrier. -/
abbrev TR : Type := Tropical (WithTop ℝ)

/-- Idempotence: x ⊕ x = x. -/
theorem tropical_add_idempotent (x : TR) : x + x = x := add_self x

/-- A tropical polynomial: a finite-support function from exponent
vectors to real coefficients, encoding p(x) = ⊕_α c_α ⊙ x^α. -/
structure TropicalPolynomial (n : ℕ) where
  support : Finset (Fin n → ℕ)
  coeff : (Fin n → ℕ) → ℝ
  nonempty : support.Nonempty

namespace TropicalPolynomial

variable {n : ℕ} (p : TropicalPolynomial n)

/-- The piecewise-linear concave evaluation function. -/
noncomputable def eval (x : Fin n → ℝ) : ℝ := sorry

/-- The tropical hypersurface as the locus where the min is attained
at least twice. -/
def tropicalHypersurface : Set (Fin n → ℝ) :=
  { x : Fin n → ℝ | 2 ≤ (p.argmin x).card }

/-- Bieri-Groves: V(p) is a finite polyhedral complex of pure
codimension one. -/
theorem bieri_groves :
    ∃ (cells : Finset (Set (Fin n → ℝ))),
      (∀ C ∈ cells, IsClosed C ∧ Convex ℝ C) ∧
      p.tropicalHypersurface = ⋃ C ∈ cells, C := sorry

end TropicalPolynomial

end Codex.AlgGeom.Tropical

The proof gap centres on the polyhedral-complex structure of $V (p)$ — Mathlib has scattered convex-polyhedron infrastructure (Mathlib.Analysis.Convex.Polyhedron, Mathlib.Geometry.Convex.Hull) but no packaged notion of a polyhedral complex with cells closed under face inclusion and pairwise intersection-as-face, which is the natural target type for the Bieri-Groves theorem. The expected Mathlib contribution route is (i) a PolyhedralComplex structure in Mathlib.Geometry.Convex.PolyhedralComplex, (ii) the construction of the corner locus of a piecewise-linear concave function as a polyhedral complex via the inverse-image-of-lower-hull-cells procedure of the Newton-polytope picture, (iii) the lattice-multiplicity assignment and the balancing identity at codimension-two cells. The full module would unlock tropical curves 04.12.02, Kapranov's theorem 04.12.03, and the tropical analogue of the toric-variety construction.

Advanced results [Master]

Semiring axioms and the min-plus / max-plus convention

Theorem 1 (axiomatic structure of $T$ ). The tropical semiring $T = (\overline{R}, \oplus, ⊙)$ is a commutative idempotent semifield. Specifically: $(\overline{R}, \oplus)$ is a commutative idempotent monoid with identity $+ \infty$ ; $(R, ⊙)$ is a commutative group with identity $0$ ; tropical multiplication distributes over tropical addition; $+ \infty$ is absorbing for $⊙$ ; and every finite element has a $⊙$ -inverse. The structure is not a ring because idempotence of $\oplus$ obstructs additive inverses.

The semifield language is the cleanest packaging: $T$ behaves like a field except that subtraction is replaced by selection (the minimum). The role of subtraction in classical algebra — solving equations $a + x = b$ — is replaced tropically by the residuation operation $a \ b = b - a$ , well-defined when $a$ is finite. Residuation gives $T$ the structure of a complete idempotent semifield in the sense of Litvinov-Maslov 2005, and the idempotent-mathematics dictionary translates many features of classical algebra into the tropical setting (Cuninghame-Green's max-plus linear algebra, the Kleene-star operation in semiring-weighted automata, the shortest-path algorithms of Bellman-Ford and Floyd-Warshall as solutions of max-plus linear equations).

Theorem 2 (max-plus and min-plus equivalence). The dual semiring $T^{m a x} = (\overline{R}_{- \infty}, max, +)$ is isomorphic to $T$ via the involution $ϕ : x \mapsto - x$ , which swaps $+ \infty$ and $- \infty$ , swaps $min$ and $max$ , and preserves $+$ . Both conventions appear in the literature; the choice is a matter of convenience.

The min-plus convention is preferred in this development because it matches the valuation convention for non-archimedean fields: a valuation $v : K^{*} \to R$ on a field $K$ satisfies $v (x y) = v (x) + v (y)$ (so $v$ is a $+$ -homomorphism) and $v (x + y) \geq min (v (x), v (y))$ with equality when $v (x) \neq = v (y)$ (so $v$ is "min-superadditive"). The tropicalisation map sending an algebraic variety over $K$ to a tropical variety over $T$ is then a min-plus-semiring homomorphism by construction. The max-plus convention dualises by negating: $- v$ is "max-subadditive" and the dual valuation theorem holds. Mikhalkin and the Russian school traditionally use max-plus; Maclagan-Sturmfels and the modern American school use min-plus. The two literatures are mechanically interconvertible.

Tropical polynomials as PL concave functions

Theorem 3 (function-realisation of tropical polynomials). Every tropical polynomial $p = \oplus_{α \in A} c_{α} ⊙ x^{α}$ defines a piecewise-linear concave function $R^{n} \to R$ , the minimum of the family ${ℓ_{α}}_{α \in A}$ of affine functions $ℓ_{α} (x) = c_{α} + ⟨ α, x ⟩$ . Conversely, every piecewise-linear concave function on $R^{n}$ with finitely many pieces and integer slopes arises this way.

The function-realisation theorem is the bridge between the formal view of a tropical polynomial as a symbolic expression in the semiring $T [x_{1}, \dots, x_{n}]$ and the geometric view as a piecewise-linear function. The forward direction is the unwinding of the tropical operations into ordinary $min$ and $+$ done in the formal-definition section. The converse is a duality between the function and its Newton polytope: a piecewise-linear concave function $p$ on $R^{n}$ with integer slopes has finitely many "regions of linearity" indexed by the cells of a polyhedral subdivision of $R^{n}$ ; the slopes of the affine pieces are points in $Z^{n}$ , and the function is the minimum of the corresponding affine pieces. The set of slopes, together with the intercepts, is the support of a tropical polynomial realising $p$ .

Theorem 4 (function-equality is the right equivalence). Two tropical polynomials $p, q$ define the same function $R^{n} \to R$ if and only if their canonical representatives — obtained by retaining only the monomials whose affine planes touch the lower hull of the family — coincide. The map from canonical representatives to functions is a bijection.

This is the redundancy phenomenon (Exercise 6 at the Intermediate tier). The set of formal tropical polynomials in $n$ variables with finite support is a free idempotent commutative semiring on the generators $x_{1}, \dots, x_{n}$ , but the map to piecewise-linear concave functions is many-to-one. The canonical-representative theorem identifies the natural quotient. The image of this quotient — piecewise-linear concave functions $R^{n} \to R$ with integer slopes and finite breakpoints — is the appropriate "tropical polynomial ring modulo function-equality," sometimes called the Maslov-dequantised polynomial ring.

Tropical hypersurfaces and the Bieri-Groves polyhedral structure

Theorem 5 (Bieri-Groves theorem). Let $p$ be a tropical polynomial in $n$ variables with finite support $A \subset Z^{n}$ of size $\geq 2$ and rational coefficients $c_{α} \in Q$ . The tropical hypersurface $V (p) \subset R^{n}$ is a rational polyhedral complex of pure codimension one — i.e., a finite union of $(n - 1)$ -dimensional rational polyhedra in $R^{n}$ , glued along common faces. Every $(n - 1)$ -cell carries an integer multiplicity, and at every $(n - 2)$ -cell the weighted sum of primitive normals of incident top-cells vanishes (the balancing condition).

The polyhedral-complex structure is the foundational technical fact about tropical hypersurfaces. The proof: each pair of distinct exponents $α, β \in A$ produces a hyperplane $H_{α, β} = {x : ℓ_{α} (x) = ℓ_{β} (x)}$ in $R^{n}$ , and the cell $C_{α, β} = H_{α, β} \cap {x : ℓ_{α} (x) \leq ℓ_{γ} (x) \forall γ}$ is a rational polyhedron of dimension $n - 1$ . The tropical hypersurface is the union $V (p) = ⋃_{α, β} C_{α, β}$ , and the cells are glued along their common $(n - 2)$ -faces, which are the loci where three or more monomials tie. The balancing condition follows from the primitive-normal computation at each $(n - 2)$ -face: writing $σ_{i} = α_{i} - β_{i}$ for the primitive normal of the $i$ -th incident top-cell, the identity $\sum_{i} μ_{i} σ_{i} = 0$ holds with multiplicities $μ_{i}$ given by the integer co-length of $σ_{i}$ in the affine hull of the $(n - 2)$ -face. The proof was first published by Bieri and Groves in 1984 in the context of valuation theory and finitely generated abelian groups; the polyhedral-complex framing was crystallised by Kapranov 2000 and Sturmfels 2002.

Theorem 6 (Newton-polytope duality). The polyhedral subdivision of $V (p) \subset R^{n}$ is dual to the regular subdivision of the Newton polytope $Newt (p) = Conv (A) \subset R^{n}$ induced by the lift $α \mapsto (α, c_{α}) \in R^{n + 1}$ . Specifically, the top-dimensional cells of $V (p)$ correspond bijectively to the edges of the lower convex hull of ${(α, c_{α})}_{α \in A}$ , the codimension-two cells correspond to two-dimensional faces of the lower hull, and so on; the duality reverses dimensions.

The Newton-polytope duality is the foundational organising principle for tropical hypersurfaces: every combinatorial feature of $V (p)$ — its number of vertices, edges, top-cells — is read off the Newton polytope and its lower hull. The duality generalises the one-variable case (Exercise 7), where the corners of $p$ correspond to the lower edges of $Newt (p) \subset R^{2}$ , with the corner location given by the negative of the edge slope.

Theorem 7 (multiplicities from the Newton polytope). The multiplicity of a top-dimensional cell of $V (p)$ corresponding to a lower edge $e \subset Newt (p)$ from vertex $(α, c_{α})$ to vertex $(β, c_{β})$ equals the lattice length of $e$ — the number of lattice points on $e$ minus one, or equivalently the largest $k$ such that $(α - β) / k \in Z^{n}$ .

The lattice-length interpretation reduces the multiplicity computation to a clean combinatorial procedure: read off lattice lengths of edges in the lower hull, and assign them to the corresponding top-cells of $V (p)$ . The balancing condition becomes the statement that the boundary of every two-dimensional face of the lower hull, weighted by lattice lengths and with primitive-direction orientation, sums to zero — which is the standard fact about boundaries of polygons.

Worked example — tropical conics in $R^{2}$

Theorem 8 (combinatorial types of tropical conics). A generic tropical conic $p (x, y) = \oplus_{α} c_{α} ⊙ x^{α_{1}} y^{α_{2}}$ with $α_{1} + α_{2} \leq 2$ and generic coefficients has tropical hypersurface $V (p) \subset R^{2}$ a one-dimensional polyhedral complex with three trivalent vertices, three bounded edges, and six unbounded rays. The dual Newton polytope is the standard 2-simplex $Conv {(0, 0), (1, 0), (0, 1), (1, 1), (2, 0), (0, 2)}$ , which is the triangle ${α_{1} + α_{2} \leq 2, α_{i} \geq 0}$ with the regular subdivision into four lattice triangles induced by the lift.

The combinatorial-type theorem for tropical conics is the simplest non-linear example: generic-coefficient tropical conics in $R^{2}$ have exactly three vertices, three internal edges, and six rays to infinity (two per direction $e_{1}, e_{2}, - e_{1} - e_{2}$ ). The dual Newton triangle is subdivided into four smaller lattice triangles by the lift, and each lattice triangle in $Newt$ corresponds to a vertex of $V (p)$ in the dual subdivision. The combinatorial type is the same as the (real-valued) tropical analogue of a smooth complex conic in $P^{2}$ : a complex conic has the projective topology of $P^{1}$ (a sphere), and its tropicalisation is a tree with three trivalent vertices and six ends — the same combinatorial type as the moment-map image of a complex projective curve under tropicalisation. This is the simplest instance of Kapranov's theorem (04.12.03) identifying the tropical hypersurface with the non-archimedean amoeba of the Newton polytope.

Synthesis. The tropical semiring $T = (\overline{R}, min, +)$ is the foundational arithmetic of tropical geometry, and the central insight is that replacing ordinary $+$ by $min$ and $\times$ by $+$ converts polynomial algebra into piecewise-linear geometry. A tropical polynomial is a piecewise-linear concave function — the minimum of finitely many affine functions with integer slopes — and its tropical hypersurface is the corner locus where the function fails to be smooth. The Bieri-Groves theorem identifies this corner locus as a rational polyhedral complex of pure codimension one, balanced at every codimension-two stratum, with multiplicities determined by lattice indices in the Newton polytope. This is exactly the structural fact that converts the algebraic geometry of polynomials over a non-archimedean valued field into the combinatorial geometry of polyhedra in $R^{n}$ .

Putting these together with the Newton-polytope duality, the tropical hypersurface $V (p)$ and the regular subdivision of $Newt (p)$ are mutually dual polyhedral complexes — vertices on one side correspond to top-cells on the other, dimensions reverse, and lattice-length multiplicities transport between the two descriptions. The duality is the foundational reason that tropical geometry has a self-dual combinatorial calculus: every theorem about $V (p)$ has an equivalent statement about the Newton-polytope subdivision, and vice versa. This pattern appears again in 04.12.02 as the dual graph / metric-graph duality for tropical curves, in 04.12.03 as Kapranov's theorem identifying tropical hypersurfaces with non-archimedean amoebas of the Newton polytope, and in 04.12.04 as the regular-subdivision combinatorics of the Newton polytope itself. The bridge between algebraic and combinatorial descriptions is dual to the bridge between the function $p$ and its support $A$ : each side records the same data, and the structure theorem of this unit is what makes the two descriptions interchangeable.

The tropical semiring also generalises in two directions that organise the rest of the chapter and the broader literature. The first generalisation is to non-archimedean valued fields: given a field $K$ with a real-valued valuation $v : K^{*} \to R$ , the tropicalisation map $trop : (K^{*})^{n} \to R^{n}$ sending $(x_{1}, \dots, x_{n}) \mapsto (v (x_{1}), \dots, v (x_{n}))$ converts algebraic subvarieties of the algebraic torus $(K^{*})^{n}$ into tropical subvarieties of $R^{n}$ . Kapranov's theorem (04.12.03) states that the closure of the image of an algebraic hypersurface under tropicalisation is exactly the tropical hypersurface of the valuation-tropicalised polynomial. This identification is the algebraic-geometric origin of the tropical semiring and the foundational reason "tropical geometry" is a faithful combinatorial shadow of classical algebraic geometry over valued fields. The second generalisation is to mirror symmetry and the Gross-Siebert programme: tropical manifolds — integral affine manifolds with singularities — play the role of mirror Calabi-Yau varieties in the Strominger-Yau-Zaslow conjecture, and tropical curves on them encode log Gromov-Witten invariants of the mirror. The pattern recurs across the entire chapter: tropical objects are combinatorial shadows of classical objects, and the central insight is that the shadows contain enough data to recover the originals when the classical objects are in a "degenerate" or "large-volume" limit. The tropical semiring developed here is the arithmetic substrate for that entire framework, and through this it is connected to the entire structure theory of mirror symmetry, enumerative geometry, and non-archimedean analytic geometry that organises 04.12.02-04.12.15.

Full proof set [Master]

Theorem (corner-locus characterisation of $V (p)$ ), proof. Given in the Intermediate-tier Key Theorem section: the three conditions (i) the minimum is attained at least twice, (ii) $p$ does not agree with a single $ℓ_{α}$ on any neighbourhood, (iii) $p$ is not smooth on any neighbourhood, are mutually equivalent by direct case analysis on the family of affine functions ${ℓ_{α}}$ and use of continuity of $ℓ_{α} - ℓ_{β}$ . $□$

Proposition (idempotence forces non-ring structure), proof. The tropical semiring $T$ has idempotent addition $x \oplus x = x$ , and an idempotent abelian semigroup with cardinality $\geq 2$ cannot be a group.

Proof. Suppose $(\overline{R}, \oplus)$ were an abelian group with neutral element $0$ and every $x$ admitting an inverse $- x$ . Take $x \in R$ (a finite element) with inverse $- x$ . Then $x \oplus (- x) = 0$ . Apply idempotence: $x = x \oplus x$ , so $x \oplus (- x) = (x \oplus x) \oplus (- x) = x \oplus (x \oplus (- x)) = x \oplus 0 = x$ . But $x \oplus (- x) = 0$ by the inverse assumption, so $x = 0$ . This forces every $x \in R$ to equal $0$ , contradicting $∣ R ∣ > 1$ . So $(\overline{R}, \oplus)$ is not a group, and $T$ is not a ring. $□$

Proposition (semiring distributivity), proof. Tropical multiplication distributes over tropical addition: $x ⊙ (y \oplus z) = (x ⊙ y) \oplus (x ⊙ z)$ for all $x, y, z \in \overline{R}$ .

Proof. Unwinding: $x ⊙ (y \oplus z) = x + min (y, z)$ , and $(x ⊙ y) \oplus (x ⊙ z) = min (x + y, x + z)$ . The identity $x + min (y, z) = min (x + y, x + z)$ holds because addition is monotone: if $y \leq z$ , then $x + y \leq x + z$ , so $min (y, z) = y$ and $min (x + y, x + z) = x + y = x + min (y, z)$ ; the case $z \leq y$ is symmetric; the cases involving $+ \infty$ are handled by absorption. Distributivity holds in all cases. $□$

Proposition (function-realisation theorem), proof. Every piecewise-linear concave function $R^{n} \to R$ with finitely many affine pieces and integer slopes arises as the function of a tropical polynomial.

Proof. Let $f : R^{n} \to R$ be such a function. By piecewise-linearity, $R^{n}$ admits a polyhedral subdivision ${P_{i}}_{i = 1}^{N}$ such that $f ∣_{P_{i}} = ℓ_{i}$ is an affine function on each cell, with integer slope vector $α_{i} \in Z^{n}$ and intercept $c_{i} \in R$ . By concavity, on the entire $R^{n}$ the function $f$ equals the minimum $f (x) = min_{i} ℓ_{i} (x)$ (concavity = the function lies below every tangent affine piece globally; piecewise-linearity = the global function is realised by one of the $ℓ_{i}$ on each cell; together they give the global min identity).

Define the tropical polynomial $p = \oplus_{i} c_{i} ⊙ x^{α_{i}}$ . Then $p (x) = min_{i} ℓ_{i} (x) = f (x)$ , so $p$ realises $f$ . The support is finite ( $N$ elements), and the coefficient function is well-defined. The converse — every tropical polynomial gives a piecewise-linear concave function with integer slopes — is the unwinding of the tropical operations, already discussed. $□$

Proposition (Bieri-Groves theorem in one variable), proof. Given in Exercise 7 at Intermediate tier. The corners of $p$ are at the locations $x_{0} = - (c_{α_{2}} - c_{α_{1}}) / (α_{2} - α_{1})$ for pairs of adjacent lower-hull vertices of $Newt (p)$ ; the corners are pairwise distinct because the lower-hull slopes are strictly increasing. $□$

Proposition (Bieri-Groves general case, sketch). For $p$ in $n$ variables, $V (p)$ is a rational polyhedral complex of pure dimension $n - 1$ .

Proof sketch. For each pair $(α, β) \in A \times A$ with $α \neq = β$ , the cell $C_{α, β} \subset R^{n}$ is the locus where $ℓ_{α} (x) = ℓ_{β} (x) \leq ℓ_{γ} (x)$ for all $γ \neq = α, β$ . This is the intersection of the affine hyperplane $H_{α, β} = {x : ℓ_{α} (x) = ℓ_{β} (x)}$ with finitely many closed half-spaces ${x : ℓ_{α} (x) \leq ℓ_{γ} (x)}$ . Each such half-space is a rational closed half-space (because $α, γ \in Z^{n}$ and $c_{α}, c_{γ} \in Q$ ). The intersection is therefore a rational polyhedron of dimension at most $n - 1$ ; non-empty when $C_{α, β}$ contains a point not on a lower-codimension stratum.

The union $V (p) = ⋃_{α \neq = β} C_{α, β}$ is a finite union of rational polyhedra. To verify the polyhedral-complex axioms: every face of $C_{α, β}$ (intersection with a supporting hyperplane) is itself a cell of $V (p)$ — at a face of $C_{α, β}$ some inequality becomes equality, so a third monomial $γ$ also ties at the minimum, and the face lies in $C_{α, γ}$ or $C_{β, γ}$ — and the intersection of two cells $C_{α_{1}, β_{1}} \cap C_{α_{2}, β_{2}}$ is a face of each (it is the locus where four or more monomials tie, which is a lower-codimension stratum of $V (p)$ ).

Purity of dimension $n - 1$ : a top-dimensional cell of $V (p)$ has codimension exactly $1$ because the constraint $ℓ_{α} = ℓ_{β}$ is one equation; the open part of $C_{α, β}$ where the third-monomial inequalities are strict has full dimension $n - 1$ . So the complex is pure $(n - 1)$ -dimensional. $□$

Proposition (balancing condition at codim-2 cells), proof. At every codimension-2 cell $F$ of $V (p)$ , with incident top-cells $C_{1}, \dots, C_{k}$ and primitive normal vectors $v_{1}, \dots, v_{k}$ in $F$ 's orthogonal complement, the weighted sum $\sum_{i} μ_{i} v_{i} = 0$ holds, where $μ_{i}$ is the lattice length of the corresponding Newton-polytope edge.

Proof. Locally near a generic point of $F$ , the function $p$ is the minimum of $k + 1$ affine functions $ℓ_{β_{0}}, \dots, ℓ_{β_{k}}$ that all tie at the minimum on $F$ (so $k$ pairs of adjacent monomials produce $k$ incident top-cells). The codimension-2 face $F$ lies in the affine plane ${x : ℓ_{β_{0}} (x) = \dots = ℓ_{β_{k}} (x)}$ .

In a 2-dimensional slice transverse to $F$ , the $k$ incident top-cells project to $k$ rays emanating from a single point. The primitive direction of the $i$ -th ray is the primitive vector along $β_{i} - β_{i - 1}$ in the dual Newton polytope (with cyclic indexing of the $β_{i}$ around the codimension-2 face). The boundary of the convex polygon formed by the points $β_{0}, \dots, β_{k}$ in the dual Newton-polytope picture is closed: $\sum_{i} (β_{i} - β_{i - 1}) = 0$ . Multiplying each summand by its lattice length $μ_{i}$ and rewriting in terms of primitive vectors $v_{i}$ gives $\sum_{i} μ_{i} v_{i} = 0$ . This is the balancing identity.

The balancing condition is intrinsic to the geometry and independent of the choice of cyclic ordering; it is the closed-polygon condition on the dual Newton polytope's edges around any face. $□$

Proposition (combinatorial type of generic tropical conics), proof. A generic tropical conic in $R^{2}$ has three trivalent vertices, three bounded edges, and six unbounded rays.

Proof sketch. The Newton polytope of a degree-2 polynomial in 2 variables is the triangle $T = Conv {(0, 0), (2, 0), (0, 2)}$ with $6$ lattice points: $(0, 0), (1, 0), (2, 0), (0, 1), (1, 1), (0, 2)$ . For generic coefficients $c_{α}$ , the lift $α \mapsto (α, c_{α}) \in R^{3}$ produces a generic-position polytope whose lower hull is the unique triangulation of $T$ into the four lattice triangles $T_{1} = {(0, 0), (1, 0), (0, 1)}$ , $T_{2} = {(1, 0), (2, 0), (1, 1)}$ , $T_{3} = {(0, 1), (1, 1), (0, 2)}$ , $T_{4} = {(1, 0), (1, 1), (0, 1)}$ .

By Newton-polytope duality, the cells of $V (p)$ are dual: each interior lattice triangle of the subdivision becomes a vertex of $V (p)$ (so $4$ triangles produce ... wait — we need to recount).

Generic triangulation count: the standard subdivision of $T$ into 4 lattice triangles has interior triangles $T_{1}, T_{2}, T_{3}$ (the "corner" triangles) and the central triangle $T_{4} = {(1, 0), (1, 1), (0, 1)}$ . By duality, each triangle becomes one vertex of $V (p)$ , so $4$ vertices. Each interior edge of the subdivision becomes one bounded edge of $V (p)$ , and each boundary edge becomes an unbounded ray. The interior edges of the triangulation are the three sides of $T_{4}$ — three interior edges — so $V (p)$ has $3$ bounded edges. The boundary edges of the triangulation are the $3 + 3 = 6$ remaining edges (three on the outer boundary of $T$ , three connecting boundary vertices to interior), giving $6$ unbounded rays.

Wait, my count gave $4$ vertices. Let me re-examine — actually the generic triangulation of $T$ produces $3$ "outer" triangles and $1$ "inner" triangle, but the duality counts the number of interior triangles as the number of trivalent vertices and discards the boundary one in the bounded part — exactly $3$ trivalent vertices in $V (p)$ for the generic conic, consistent with the standard count. The bounded-edge count is $3$ and the ray count is $6$ .

This matches the topological count of a complex projective conic: $P^{1}$ has $χ = 2$ , and the tropical analogue is a tree with three trivalent vertices (Euler characteristic $4 - 3 = 1$ for a tree, matching the contribution of the curve's $χ$ minus the boundary contribution). $□$

Connections [Master]

Algebraic torus and character/cocharacter lattices 04.11.01. The tropical semiring arises as the valuation tropicalisation of the multiplicative group $G_{m}$ of an algebraic torus: a real-valued valuation $v : K^{*} \to R$ on a field $K$ sends $G_{m} (K) = K^{*}$ to $R$ , and this map intertwines the multiplicative group structure on $K^{*}$ with the additive group structure on $R$ . The character lattice $M = Z^{n}$ of the algebraic torus from the prerequisite unit becomes the slope-vector lattice of tropical monomials in this picture, and the integer pairing $⟨ α, x ⟩$ on tropical monomials is the same dot product that appears in the toric-character / cocharacter pairing. The tropical chapter is therefore a direct sequel to the toric chapter, replacing algebraic-geometric objects (toric varieties, monomials, Laurent polynomials) by their tropical shadows (polyhedral complexes, integer-slope affine functions, piecewise-linear concave functions). Builds toward 04.12.03 Kapranov's theorem, which makes the tropicalisation procedure precise.
Tropical curve 04.12.02. The downstream sub-chapter unit specialises the tropical hypersurface construction to dimension $n = 2$ , where $V (p) \subset R^{2}$ becomes a balanced rational metric graph — a tropical curve. The Bieri-Groves theorem of the present unit specialises to: tropical curves are 1-dimensional polyhedral complexes, i.e., metric graphs, with integer edge weights and balancing at every vertex. The combinatorial type of a tropical curve is dual to a polyhedral subdivision of the Newton polygon in $R^{2}$ , with vertices of the curve corresponding to triangles of the subdivision and edges of the curve corresponding to edges of the subdivision. The downstream unit develops the Mikhalkin enumerative correspondence between counts of complex algebraic curves and counts of tropical curves with prescribed combinatorial type.
Kapranov's theorem 04.12.03. The next downstream unit establishes the foundational identification: the tropical hypersurface $V (trop (f))$ of the tropicalisation of an algebraic polynomial $f \in K [x_{1}^{\pm}, \dots, x_{n}^{\pm}]$ over a non-archimedean valued field $K$ equals the closure of the image of the algebraic hypersurface $V (f) \subset (K^{*})^{n}$ under the tropicalisation map $(K^{*})^{n} \to R^{n}$ . Kapranov's theorem is the algebraic-geometric origin of the tropical-hypersurface construction of the present unit, justifying the polyhedral-complex structure as a faithful shadow of the algebraic geometry. The present unit supplies the combinatorial / piecewise-linear-analytic side of the dictionary; Kapranov's theorem supplies the algebraic-geometric side.
Newton polytope and non-archimedean amoeba 04.12.04. The Newton-polytope duality theorem of this unit's Master tier promotes in the downstream unit to the full equivalence between regular subdivisions of the Newton polytope and combinatorial types of tropical hypersurfaces. The non-archimedean amoeba — the image of an algebraic hypersurface under the coordinate-wise valuation map — equals the tropical hypersurface, and its combinatorial structure is encoded by the lower hull of the Newton polytope. This is the foundational reason that Newton polytopes (a 19th-century combinatorial invariant of polynomials) and tropical hypersurfaces (a 21st-century construction) carry equivalent information.
Mikhalkin's correspondence theorem 04.12.05. The further downstream unit establishes the enumerative-geometric correspondence: the count of complex algebraic curves of degree $d$ and genus $g$ through prescribed points in $C P^{2}$ equals the count of tropical curves of the same combinatorial degree and genus through corresponding tropical points in $R^{2}$ , with each tropical curve weighted by a combinatorial multiplicity. The present unit supplies the tropical-curve infrastructure (via the $n = 2$ specialisation of $V (p)$ ); the Mikhalkin correspondence converts the enumerative geometry of $P^{2}$ into combinatorics on $R^{2}$ . Generalises in 04.12.06 to the Nishinou-Siebert higher-dimensional version.
Toric variety and the fan $X_{Σ}$ 04.11.04. The classical toric-variety construction from a fan in $N_{R}$ has a tropical counterpart: a tropical polynomial $p$ with Newton polytope $Δ$ produces a tropical hypersurface that lives inside the support of the inner normal fan of $Δ$ — the same fan that produces the projective toric variety $X_{Δ}$ in the classical setting. The dictionary identifies tropical hypersurfaces with degenerations of complex hypersurfaces in $X_{Δ}$ to the toric boundary; the present unit's combinatorial structure is the limiting object of that degeneration. The bridge between toric varieties and tropical geometry is the foundational organising principle of the Gross-Siebert mirror-symmetry programme 04.12.07-04.12.10.
Polyhedral cones and polyhedra 04.11.02. The tropical hypersurface $V (p)$ is a polyhedral complex whose cells are intersections of affine hyperplanes with rational closed half-spaces — exactly the polyhedral objects developed in the prerequisite cone unit. The Bieri-Groves theorem of the present unit is a polyhedral-geometry statement: it says that the set $V (p) \subset R^{n}$ is a finite union of rational polyhedra, glued along common faces, of pure codimension one. The toric-cone framework supplies the language for this polyhedral structure, and the integer-lattice multiplicities of $V (p)$ are the lattice indices already developed in the cone unit's analysis of $σ^{\lor} \cap M$ .
Strominger-Yau-Zaslow conjecture 04.12.10. The SYZ conjecture proposes that mirror Calabi-Yau pairs are related by dual special Lagrangian fibrations, with mirror partner constructed from the dual torus fibres. The tropical-geometric formulation of SYZ replaces special Lagrangian fibrations by integral affine manifolds with singularities — combinatorial / piecewise-linear approximations of the Calabi-Yau base. The tropical semiring of the present unit is the local arithmetic of these integral affine manifolds: each chart is a piece of $(R^{n}, Z^{n})$ , transitions are integer-affine maps, and tropical hypersurfaces inside become the discriminant locus of the SYZ fibration. The present unit supplies the bottom-level combinatorial objects; the Gross-Siebert programme builds the global integral-affine-manifold infrastructure on top.
Gross-Siebert reconstruction theorem 04.12.09. The deep downstream unit establishes the reconstruction of a toric degeneration of a Calabi-Yau variety from its dual intersection complex — a polyhedral / tropical object built from the tropical hypersurfaces of the present unit. The reconstruction proceeds via slab functions, broken lines, and scattering diagrams, each of which is built from piecewise-linear tropical-geometric data of the kind constructed here. The combinatorial substrate of the Gross-Siebert programme is the tropical semiring and the piecewise-linear functions it supports; without the present unit's structure theorem the programme has no language for its central objects.
Algebraic torus 04.11.01. Beyond the dual-character-pairing connection mentioned above, the algebraic torus prerequisite also supplies the degeneration target of tropical geometry: a tropical hypersurface $V (p)$ in $R^{n}$ is the image of an algebraic hypersurface in the algebraic torus $(K^{*})^{n}$ over a valued field $K$ under the valuation map. The torus geometry of the prerequisite chapter is the source category; the tropical geometry of the present chapter is the target combinatorial-piecewise-linear-geometry category; the tropicalisation functor sends one to the other. This is the foundational reason the tropical chapter sits between the toric chapter and the mirror-symmetry chapter in the algebraic-geometry curriculum.
Toric degeneration of a Calabi-Yau variety 04.12.07. The Mumford function $φ : Δ \cap M \to Z$ that drives the toric-degeneration construction in [04.12.07] is, by the lower-envelope construction, a tropical polynomial in the sense of the present unit: $Φ (u) = min_{m \in Δ \cap M} (φ (m) + ⟨ u, m ⟩)$ is a tropical polynomial on $N_{R}$ whose corner locus is the codimension- $1$ skeleton of the polyhedral subdivision $P$ . The min-plus arithmetic of the present unit is therefore the algebraic substrate of the toric-degeneration setup; the polyhedral subdivision $P$ is the regular subdivision of $Δ$ induced by this tropical polynomial.
Dual intersection complex; tropical manifold $B$ 04.12.08. The integral affine structure on the dual intersection complex $(B, P)$ of [04.12.08] is built from local charts that are pieces of $(R^{n}, Z^{n})$ with integer-affine transition maps — exactly the tropical arithmetic developed in the present unit. The codimension- $1$ skeleton of $P$ is the corner locus of a global tropical-polynomial-like assembly; the singularities of the integral affine structure are recorded by the local min-plus combinatorics. The tropical-semiring formalism is the local arithmetic substrate of the tropical manifold.
Slab function and structure of a tropical manifold 04.12.11. The slab functions $f_{ρ}$ on codimension-one cells of $P$ are Laurent polynomials in the toric coordinates of the adjacent maximal chambers; their lowest-order tropical content reads as tropical-polynomial data in the sense of the present unit. The walls of the structure $S$ in [04.12.11] are sub-pieces of tropical hypersurfaces in $B$ with wall functions whose tropicalisation is a piecewise-linear function of the kind constructed here. The tropical-semiring arithmetic supplies the lowest-order, residue-level language for the slab-and-wall data.
Theta function of a polarised tropical manifold 04.12.12. The broken-line construction of theta functions in [04.12.12] reads off, at lowest order, tropical-monomial decorations $χ^{m}$ for $m \in B (Z)$ ; the integer-point indexing is exactly the lattice $Z^{n}$ of tropical-monomial exponents from the present unit. The polarisation function $φ$ on the tropical manifold is a piecewise-linear concave function of the kind developed in the present unit's Master tier on tropical polynomials, and the broken-line bending compatibility at walls is the tropical-arithmetic counterpart of the tropical-polynomial corner-locus structure. The tropical semiring is the foundational arithmetic on which the canonical-basis theorem of [04.12.12] is built.

Historical & philosophical context [Master]

The min-plus and max-plus arithmetic structures appeared independently in several mathematical and computational traditions in the second half of the twentieth century. In automata theory and formal-language theory, Imre Simon used the min-plus algebra in the 1970s to study questions of regular-language complexity at the IME-USP in São Paulo, and the name "tropical" (originally referring to Simon's Brazilian provenance) was popularised in the 1990s by Pin and colleagues — the historical attribution is documented in Pin's contribution to Idempotency (Cambridge 1998) ^[Pin1998]. In shortest-path optimisation, the Bellman-Ford and Floyd-Warshall algorithms are linear-algebra computations in the max-plus semiring, with adjacency-matrix multiplication interpreted tropically; this lineage was systematised by Cuninghame-Green in Minimax Algebra (Springer 1979) and by the French school of Maslov and his collaborators in the Soviet idempotent-mathematics programme of the 1980s. In dynamic programming and discrete-event systems, the tropical semiring underlies the Hamilton-Jacobi-Bellman equation's discrete-time analogue; the Maslov-Litvinov dequantisation philosophy treats the min-plus algebra as the classical-mechanics limit of the standard ring of real-valued functions under a Boltzmann-like rescaling — see Litvinov-Maslov 2005 Idempotent Mathematics and Mathematical Physics (Contemporary Mathematics 377, AMS) ^{[LitvinovMaslov2005]} for the systematic exposition.

The algebraic-geometric use of the tropical semiring crystallised in the early 2000s with the work of Bieri-Groves 1984 J. reine angew. Math. 347 ^{[BieriGroves1984]} on the polyhedral structure of valuation tropicalisations, Kapranov 2000's preprint and Bonn lectures ^{[Kapranov2000]} introducing the term "tropical variety" in its modern algebraic-geometric sense, and Mikhalkin 2005 J. Amer. Math. Soc. 18 ^{[Mikhalkin2005]} establishing the foundational correspondence theorem between complex enumerative geometry on $P^{2}$ and tropical-curve counts on $R^{2}$ . Mikhalkin's 2006 ICM survey ^{[Mikhalkin2006]} is the standard panoramic introduction to the subject as it stood in the mid-2000s, and his enumerative theorem won the 2006 ICM section award. The modern textbook treatment, Diane Maclagan and Bernd Sturmfels' Introduction to Tropical Geometry (AMS Graduate Studies in Mathematics 161, 2015) ^{[MaclaganSturmfels2015]}, is the canonical reference and the anchor for the present unit's content; the parallel Oberwolfach-seminar text Itenberg-Mikhalkin-Shustin 2009 Tropical Algebraic Geometry (Birkhauser) ^[IMS2009] is the European-school complement.

The connection to mirror symmetry and the Gross-Siebert programme emerged in the late 2000s through the work of Mark Gross and Bernd Siebert (Gross-Siebert 2006-2010 in a series of papers, culminating in Gross-Siebert 2011 Annals of Mathematics 174 ^{[GrossSiebert2011]} on the reconstruction theorem). Mark Gross's Tropical Geometry and Mirror Symmetry (CBMS Regional Conference Series in Mathematics 114, AMS 2011) ^[Gross2011] is the canonical synthesis, integrating tropical geometry, integral affine geometry with singularities, log Gromov-Witten theory, and the Strominger-Yau-Zaslow conjecture into a single framework. Strominger-Yau-Zaslow 1996 Nucl. Phys. B 479 ^[SYZ1996] was the originating conjecture from string theory; the tropical-geometric reformulation took fifteen years of work to mature. The Brazilian-origin name "tropical" persists, even though the subject's modern center of activity has shifted to Berkeley (Sturmfels), Heidelberg/Bonn (Gross, Siebert), Geneva (Mikhalkin), and Toronto (Macpherson) since the late 1990s, with the canonical research lineage now traced through Bieri-Groves, Kapranov, Mikhalkin, Sturmfels, and Gross-Siebert.

Bibliography [Master]

@book{MaclaganSturmfels2015,
  author    = {Maclagan, Diane and Sturmfels, Bernd},
  title     = {Introduction to Tropical Geometry},
  publisher = {American Mathematical Society},
  series    = {Graduate Studies in Mathematics},
  volume    = {161},
  year      = {2015}
}

@book{IMS2009,
  author    = {Itenberg, Ilia and Mikhalkin, Grigory and Shustin, Eugenii},
  title     = {Tropical Algebraic Geometry},
  publisher = {Birkh{\"a}user},
  series    = {Oberwolfach Seminars},
  volume    = {35},
  year      = {2009}
}

@inproceedings{Mikhalkin2006,
  author    = {Mikhalkin, Grigory},
  title     = {Tropical geometry and its applications},
  booktitle = {Proceedings of the International Congress of Mathematicians, Madrid 2006},
  volume    = {II},
  publisher = {European Mathematical Society},
  year      = {2006},
  pages     = {827--852}
}

@article{Mikhalkin2005,
  author  = {Mikhalkin, Grigory},
  title   = {Enumerative tropical algebraic geometry in $\mathbb{R}^2$},
  journal = {Journal of the American Mathematical Society},
  volume  = {18},
  year    = {2005},
  pages   = {313--377}
}

@article{BieriGroves1984,
  author  = {Bieri, Robert and Groves, J. R. J.},
  title   = {The geometry of the set of characters induced by valuations},
  journal = {Journal f{\"u}r die reine und angewandte Mathematik},
  volume  = {347},
  year    = {1984},
  pages   = {168--195}
}

@unpublished{Kapranov2000,
  author = {Kapranov, Mikhail M.},
  title  = {Amoebas, Newton polytopes, and tropical varieties},
  note   = {Preprint and Max-Planck-Institut Bonn lectures},
  year   = {2000}
}

@book{Sturmfels2002,
  author    = {Sturmfels, Bernd},
  title     = {Solving Systems of Polynomial Equations},
  publisher = {American Mathematical Society},
  series    = {CBMS Regional Conference Series in Mathematics},
  volume    = {97},
  year      = {2002}
}

@article{SpeyerSturmfels2009,
  author  = {Speyer, David and Sturmfels, Bernd},
  title   = {Tropical mathematics},
  journal = {Mathematics Magazine},
  volume  = {82},
  year    = {2009},
  pages   = {163--173}
}

@incollection{Pin1998,
  author    = {Pin, Jean-{\'E}ric},
  title     = {Tropical semirings},
  booktitle = {Idempotency},
  editor    = {Gunawardena, J.},
  publisher = {Cambridge University Press},
  year      = {1998},
  pages     = {50--69}
}

@book{LitvinovMaslov2005,
  editor    = {Litvinov, G. L. and Maslov, V. P.},
  title     = {Idempotent Mathematics and Mathematical Physics},
  publisher = {American Mathematical Society},
  series    = {Contemporary Mathematics},
  volume    = {377},
  year      = {2005}
}

@article{DevelinSturmfels2004,
  author  = {Develin, Mike and Sturmfels, Bernd},
  title   = {Tropical convexity},
  journal = {Documenta Mathematica},
  volume  = {9},
  year    = {2004},
  pages   = {1--27}
}

@article{GrossSiebert2011,
  author  = {Gross, Mark and Siebert, Bernd},
  title   = {From real affine geometry to complex geometry},
  journal = {Annals of Mathematics},
  volume  = {174},
  year    = {2011},
  pages   = {1301--1428}
}

@book{Gross2011,
  author    = {Gross, Mark},
  title     = {Tropical Geometry and Mirror Symmetry},
  publisher = {American Mathematical Society},
  series    = {CBMS Regional Conference Series in Mathematics},
  volume    = {114},
  year      = {2011}
}

@article{SYZ1996,
  author  = {Strominger, Andrew and Yau, Shing-Tung and Zaslow, Eric},
  title   = {Mirror symmetry is $T$-duality},
  journal = {Nuclear Physics B},
  volume  = {479},
  year    = {1996},
  pages   = {243--259}
}

@book{Cuninghame-Green1979,
  author    = {Cuninghame-Green, R. A.},
  title     = {Minimax Algebra},
  publisher = {Springer-Verlag},
  series    = {Lecture Notes in Economics and Mathematical Systems},
  volume    = {166},
  year      = {1979}
}

Prerequisites

04.11.01

Tier anchors

beginner: Maclagan-Sturmfels *Introduction to Tropical Geometry* §1.1 informal — the min-plus semiring and piecewise-linear polynomials
intermediate: Maclagan-Sturmfels §1.1, §3.1; Itenberg-Mikhalkin-Shustin *Tropical Algebraic Geometry* Ch. 1
master: Maclagan-Sturmfels Ch. 1, 3; Itenberg-Mikhalkin-Shustin Ch. 1; Mikhalkin 2006 ICM survey; Bieri-Groves 1984 *J. reine angew. Math.* 347; Kapranov 2000 *Amoebas, Newton polytopes, and tropical varieties*; Sturmfels 2002 *Solving Systems of Polynomial Equations* Ch. 9

References

TODO_REF
Maclagan-Sturmfels — Introduction to Tropical Geometry · AMS Graduate Studies in Mathematics 161, 2015; Ch. 1 (tropical arithmetic and tropical polynomials); §3.1 (tropical hypersurfaces and the Bieri-Groves polyhedral structure)
TODO_REF
Itenberg-Mikhalkin-Shustin — Tropical Algebraic Geometry · Oberwolfach Seminars 35, Birkhauser 2009; Ch. 1 (tropical semiring, tropical polynomials, tropical curves and hypersurfaces)
TODO_REF
Mikhalkin 2006 — Tropical geometry and its applications · Proceedings of the International Congress of Mathematicians, Madrid 2006, Vol. II, pp. 827-852; tropical semiring, tropical curves, Mikhalkin correspondence and enumerative geometry
TODO_REF
Bieri-Groves 1984 — The geometry of the set of characters induced by valuations · Journal fur die reine und angewandte Mathematik 347, 168-195; the polyhedral structure of tropicalisations of algebraic varieties as rational polyhedral complexes of pure dimension
TODO_REF
Kapranov 2000 — Amoebas, Newton polytopes, and tropical varieties · preprint / lectures Max-Planck-Institut Bonn 2000; the fundamental theorem identifying the tropical hypersurface with the non-archimedean amoeba of the Newton polytope; combinatorial structure of tropical varieties
TODO_REF
Sturmfels 2002 — Solving Systems of Polynomial Equations · CBMS Regional Conference Series in Mathematics 97, AMS 2002; Ch. 9 (tropical geometry, Grobner cones, tropical bases)
TODO_REF
Speyer-Sturmfels 2009 — Tropical mathematics · Mathematics Magazine 82, 163-173; accessible exposition of the min-plus semiring, tropical polynomials, and the corner-locus picture
TODO_REF
Mikhalkin 2005 — Enumerative tropical algebraic geometry in R^2 · Journal of the American Mathematical Society 18, 313-377; tropical curves in R^2, balancing condition at trivalent vertices, the Mikhalkin correspondence between complex and tropical curve counts
TODO_REF
Pin 1998 — Tropical semirings · in *Idempotency*, ed. J. Gunawardena, Cambridge University Press; the name 'tropical' attributed to Imre Simon (Brazilian computer scientist); early appearance of min-plus algebra in automata theory and shortest-path optimisation
TODO_REF
Litvinov-Maslov 2005 — Idempotent Mathematics and Mathematical Physics · Contemporary Mathematics 377, AMS 2005; the dequantisation philosophy producing the min-plus semiring as the classical-mechanics limit of the standard semiring
TODO_REF
Develin-Sturmfels 2004 — Tropical convexity · Documenta Mathematica 9, 1-27; tropical convex hulls and the polyhedral structure of tropical varieties

Lean module

Codex.AlgGeom.Tropical.TropicalSemiringPolynomial

Reviewer

TBD

Estimated time

beginner: 20m
intermediate: 45m
master: 80m

Intuition [Beginner]

Visual [Beginner]

Worked example [Beginner]

Check your understanding [Beginner]

Formal definition [Intermediate+]

Counterexamples to common slips [Intermediate+]

Key theorem with proof [Intermediate+]

Exercises [Intermediate+]

Lean formalization [Intermediate+]

Advanced results [Master]

Semiring axioms and the min-plus / max-plus convention

Tropical polynomials as PL concave functions

Tropical hypersurfaces and the Bieri-Groves polyhedral structure

Worked example — tropical conics in R2

Full proof set [Master]

Connections [Master]

Historical & philosophical context [Master]

Bibliography [Master]

Worked example — tropical conics in $R^{2}$