Tropical Geometry: Foundations¶

Job: the tropical semiring, tropical varieties as balanced polyhedral complexes, Kapranov’s theorem linking amoebas to tropical hypersurfaces, correspondence theorems for enumerative geometry, and core applications.

Core¶

Tropical semiring¶

The tropical semiring (min convention) is $$ \mathbb{T}=(\mathbb{R}\cup{\infty},\oplus,\odot),\qquad a\oplus b=\min(a,b),\qquad a\odot b=a+b, $$ with $\infty$ additive identity and $0$ multiplicative identity. Idempotent: $a\oplus a=a$. (Max convention: $\max$ and dualize.)

Tropical polynomial: $f=\bigoplus_{u\in A} c_u\odot x^{\odot u}$ with $A\subset\mathbb{Z}^n$ finite, $c_u\in\mathbb{R}$, i.e. piecewise-linear convex $$ f(x)=\min_{u\in A}\bigl(c_u+\langle u,x\rangle\bigr). $$ Tropical hypersurface $V(f)$: locus where the minimum is achieved at least twice (nondifferentiability locus of $f$).

Tropical varieties¶

For an ideal $I\subset K[x_1^{\pm 1},\ldots,x_n^{\pm 1}]$ over a valued field $(K,\operatorname{val})$ (e.g. Puiseux series $\mathbb{C}\{\{t\}\}$ with $\operatorname{val}(t)=1$), the tropicalization is $$ \operatorname{trop}(V(I))=\overline{\operatorname{val}(V(I)\cap (K^\times)^n)}\subset\mathbb{R}^n $$ (closure in the Euclidean topology). Structure theorem: $\operatorname{trop}(V)$ is a pure-dimensional balanced weighted polyhedral complex of dimension $\dim V$, dual to a Gröbner complex / initial degeneration fan data.

Balancing. For each codim-1 cell $\tau$, if $\sigma_i$ are adjacent facets with primitive lattice vectors $u_i$ outward in the quotient lattice $N/N_\tau$ and weights $m_{\sigma_i}$, $$ \sum_i m_{\sigma_i} u_i = 0. $$

Object	Notes
Linear spaces	tropical linear spaces $\leftrightarrow$ valuated matroids
Curves	balanced metric graphs (abstract) or 1-complexes in $\mathbb{R}^n$
Grassmannian	$\operatorname{trop}(\operatorname{Gr}(k,n))$ Speyer–Sturmfels

Kapranov’s theorem¶

Let $f=\sum_{u\in A} a_u x^u\in K[x_1^{\pm},\ldots,x_n^{\pm}]$, $a_u\neq 0$. Define tropical polynomial $\operatorname{trop}(f)=\min_u\bigl(\operatorname{val}(a_u)+\langle u,x\rangle\bigr)$.

Theorem (Kapranov). For a hypersurface, $$ \operatorname{trop}(V(f))=V(\operatorname{trop}(f)). $$ Thus non-Archimedean amoebas equal tropical hypersurfaces of the valued coefficient polynomial.

Complex amoebas. For $V\subset(\mathbb{C}^\times)^n$, $\mathcal{A}(V)=\{\bigl(\log|z_1|,\ldots,\log|z_n|\bigr):z\in V\}$. As one rescales toward the spine (or takes $t\to 0$ limits of families), amoebas collapse onto tropical complexes — Gelfand–Kapranov–Zelevinsky / Mikhalkin pictures.

Correspondence theorems¶

Mikhalkin correspondence (curve counts in toric surfaces). For suitable genus $g$ and degree class in a toric surface, the number of algebraic curves of given genus/degree through point conditions equals a weighted count of tropical curves in $\mathbb{R}^2$ through corresponding tropical points — weights from vertex multiplicities $$ m_v=|\det(u_1,u_2)| $$ (complex multiplicity) and lattice lengths of edges. Real enumerative variants use Welschinger signs / real multiplicities.

Philosophy. Degenerate the complex structure / coefficients to a non-Archimedean limit; curves $\to$ tropical curves; intersection theory $\to$ balancing and stable intersections; recover classical numbers by multiplicity.

Stable intersection. Tropical varieties can be perturbed to intersect properly; the balanced sum of intersection cells (with multiplicities) is independent of small perturbation — tropical replacement for Bézout / Chow.

Applications¶

Area	Use
Enumerative geometry	GW/Severi numbers via tropical counts; floor diagrams
Mirror symmetry	SYZ bases, Gross–Siebert program, scattering diagrams
Toric degenerations	initial ideals; special fibers as unions of toric strata
Optimization / scheduling	tropical eigenvalues; discrete event systems
Phylogenetics	tree spaces as tropical linear spaces (e.g. $\operatorname{Gr}(2,n)$)
Computational AG	Gröbner / tropical bases; feasibility certificates

Connection to mirror symmetry. The dual intersection complex of a toric degeneration of a CY is a tropical manifold (affine structure with singularities) — the combinatorial backbone of SYZ and Gross–Siebert reconstruction of mirrors from tropical data.

See it¶

Bézout in the plane. Two tropical curves of degrees $d,e$ in $\mathbb{R}^2$ (compactified in tropical $\mathbb{P}^2$) intersect in $de$ points counted with multiplicity (stable intersection) — tropical Bézout. Example: two general tropical lines meet in one weight-1 point; a tropical line and a tropical quadric meet in two points counting mult.

Valued field computation. Over Puiseux series, solve $t x + y + 1=0$ in $(K^\times)^2$: $\operatorname{val}$ images fill the tropical line $\min(1+x,y,0)$ nondifferentiable locus — Kapranov in coordinates.

Check¶

Verify balancing for the standard tropical line $V(\min(x,y,0))$ in $\mathbb{R}^2$.

Three rays with primitive directions $v_1=(-1,0)$, $v_2=(0,-1)$, $v_3=(1,1)$, each weight $1$. Sum: $(-1,0)+(0,-1)+(1,1)=(0,0)$. Balanced.

Why is the tropical semiring not a ring, and why does that matter for ‘ideals’?

No additive inverses ($a\oplus b=\min$ has no subtraction). Tropical ‘ideals’ are defined via algebraic circuits / bend relations or as tropicalizations of classical ideals — not as kernels of semiring homs in the naive sense. Foundations use either the valuation route or Maclagan–Sturmfels axiomatic tropical ideals.

Pitfalls¶

Convention clash: min-plus (optimization literature) vs max-plus (some algebraic texts).
Embedded image of an abstract tropical curve may have edge overlaps; multiplicities and parameterizations matter.
Not every balanced complex is realizable as $\operatorname{trop}(V)$ over a given field — realizability is subtle (especially in higher codimension).
Floor diagrams / lattice paths are computational models, not the definition of tropical curves.

Refs¶

Maclagan–Sturmfels, Introduction to Tropical Geometry
Mikhalkin, “Enumerative tropical algebraic geometry in $\mathbb{R}^2$”
Speyer–Sturmfels, “The tropical Grassmannian”
Gross–Siebert, logarithmic / tropical mirror symmetry series

Next¶

TG index · Berkovich & Skeletons · Matroids & Linear Spaces · Moduli & Enumerative · Scattering & Mirror · Algebraic geometry · Mirror foundations · Complex geometry · Symplectic geometry · Discrete math