Integral Affine Stratification

• Integral affine stratification is a decomposition method that partitions arithmetic, semigroup, and tropical objects into tractable affine pieces with well-defined lattice structures.
• It plays a key role in organizing additive character sums, fiber decompositions in monoid morphisms, and tropical contractions to integral affine manifolds with singularities.
• By structuring singular or degenerate behaviors into affine strata, this framework supports both quantitative bounds and robust geometric or combinatorial analysis.

Integral affine stratification is a term used in several mathematically distinct but structurally related senses. In arithmetic geometry, it denotes a decomposition of the dual affine frequency space into homogeneous subvarieties that control additive character sums in a polynomial sieve for counting integral points in affine thin sets of type II (Bonolis et al., 16 May 2025). In combinatorial commutative algebra, it denotes a finite disjoint union of lattice-translated normal affine semigroups describing subsets of $\mathbb{Z}^d$, especially fibers of monoid morphisms and winning-position sets in lattice games (Miller, 2010). In tropical geometry and the Gross–Siebert program, it denotes the canonical decomposition of a tropical variety into strata carrying integral affine structures and mapping by a contraction to an integral affine manifold with singularities (Yamamoto, 2021). The term therefore does not refer to a single universal definition; rather, it indexes a family of stratified integral or lattice-theoretic decompositions whose affine pieces encode arithmetic cancellation, monoidal structure, or tropical monodromy.

1. Terminological range and common formal pattern

The three principal usages differ in ambient category, but each organizes an integral or lattice-valued object into affine pieces whose geometry is rigid enough to support structural or quantitative theorems.

Setting	Object stratified	Basic strata
Thin sets of type II	$\mathbb{A}^n$ in frequency variables $\mathbf{u}$	Homogeneous subvarieties $V_i$
Affine semigroups	Subsets of $\mathbb{Z}^d$ or fibers in $Q$	Translates $a_i+S_i$
Tropical degenerations	$\mathrm{Trop}(X)$	Cell-indexed pieces $X_\tau^\circ$

In the arithmetic setting, the stratification is not imposed on the original parameter space of integral points but on the dual frequency space arising after Poisson summation; the depth of a stratum measures the size of the corresponding character sums (Bonolis et al., 16 May 2025). In the affine-semigroup setting, the strata are explicitly integral polyhedral objects: translates of normal affine semigroups, equivalently of cone–lattice intersections $C \cap L$ inside $\mathbb{A}^n$0 (Miller, 2010). In the tropical setting, the strata are compatible with integral affine charts and singular affine monodromy, and they are tied to a global contraction from the tropicalization to the dual intersection complex of a toric degeneration (Yamamoto, 2021).

A plausible unifying description is that integral affine stratification packages nontrivial global objects into finitely many affine pieces whose combinatorics and lattice structure remain tractable. The data supporting that description, however, are context-dependent.

2. Arithmetic stratification in affine thin sets of type II

For an absolutely irreducible polynomial

$\mathbb{A}^n$1

that is monic in $\mathbb{A}^n$2, with $\mathbb{A}^n$3, and such that $\mathbb{A}^n$4 is a polynomial in $\mathbb{A}^n$5 for some $\mathbb{A}^n$6, the counting function is

$\mathbb{A}^n$7

This is Serre’s integral counting problem for affine thin sets of type II. The structured form kept throughout is

$\mathbb{A}^n$8

with $\mathbb{A}^n$9, $\mathbf{u}$0, each $\mathbf{u}$1, and $\mathbf{u}$2 (Bonolis et al., 16 May 2025).

The principal nondegeneracy conditions are formulated in terms of genuine function-field extensions. A finite nontrivial extension $\mathbf{u}$3 is called $\mathbf{u}$4-genuine if, for every presentation

$\mathbf{u}$5

with $\mathbf{u}$6 absolutely irreducible, one has $\mathbf{u}$7 and $\mathbf{u}$8 for each $\mathbf{u}$9. An absolutely irreducible $V_i$0 monic in $V_i$1 is $V_i$2-allowable if for every $V_i$3 the extension

$V_i$4

is $V_i$5-genuine. It is strongly $V_i$6-allowable if the corresponding extension is strongly $V_i$7-genuine for every $V_i$8, meaning that every strict subextension remains $V_i$9-genuine. Equivalently, for every $\mathbb{Z}^d$0,

$\mathbb{Z}^d$1

so $\mathbb{Z}^d$2 is algebraically closed in $\mathbb{Z}^d$3 (Bonolis et al., 16 May 2025).

Under these hypotheses, the main bounds improve substantially on Serre’s general estimate. For $\mathbb{Z}^d$4 and strong $\mathbb{Z}^d$5-allowability,

$\mathbb{Z}^d$6

and under GRH the same bound holds for $\mathbb{Z}^d$7. If $\mathbb{Z}^d$8 is $\mathbb{Z}^d$9-allowable but not strongly $Q$0-allowable, then

$Q$1

These are contrasted with Serre’s general bound $Q$2 for some $Q$3 (Bonolis et al., 16 May 2025).

The stratification itself arises from additive character sums

$Q$4

attached to the morphism

$Q$5

The Katz–Laumon stratification provides homogeneous subvarieties

$Q$6

with $Q$7, such that for primes outside a finite set and $Q$8,

$Q$9

A companion uniform formulation yields constants $a_i+S_i$0 and $a_i+S_i$1 depending only on $a_i+S_i$2 and the total degree bound $a_i+S_i$3, together with bounds on $a_i+S_i$4, the number of irreducible components, and homogeneous defining equations $a_i+S_i$5 satisfying

$a_i+S_i$6

(Bonolis et al., 16 May 2025).

In this arithmetic usage, “integral affine stratification” means grouping the integral frequencies $a_i+S_i$7 according to their membership in $a_i+S_i$8, $a_i+S_i$9, and related integral loci. The sieve then splits contributions into global/global, local/local, and global/local terms. Degenerate components lying in proper linear subspaces are handled by second-moment bounds over hyperplanes,

$\mathrm{Trop}(X)$0

while nondegenerate components are controlled via Pila’s determinant method. The method requires no nonsingularity property of $\mathrm{Trop}(X)$1, and the stratification is explicitly described as simultaneously capturing arithmetic and geometric singular behaviors in the projection $\mathrm{Trop}(X)$2 (Bonolis et al., 16 May 2025).

3. Affine-semigroup stratifications and fibers of monoid morphisms

In the affine-semigroup literature, the relevant ambient object is a finitely generated submonoid $\mathrm{Trop}(X)$3, called an affine semigroup. Its universal group is $\mathrm{Trop}(X)$4. A semigroup $\mathrm{Trop}(X)$5 is normal if it is integrally closed in its group, equivalently if

$\mathrm{Trop}(X)$6

For a subset $\mathrm{Trop}(X)$7, an affine stratification is a finite partition

$\mathrm{Trop}(X)$8

where each $\mathrm{Trop}(X)$9 is a normal affine semigroup and each $X_\tau^\circ$0 (Miller, 2010).

This notion is used to analyze fibers of monoid morphisms

$X_\tau^\circ$1

into an arbitrary commutative monoid. For $X_\tau^\circ$2, the fiber is $X_\tau^\circ$3. The central result states that every fiber admits an affine stratification: there exist $X_\tau^\circ$4 and normal affine semigroups $X_\tau^\circ$5 such that

$X_\tau^\circ$6

Moreover, the strata can be chosen with $X_\tau^\circ$7, where each $X_\tau^\circ$8 is an intersection of associated lattices arising from the mesoprimary decomposition of the induced congruence on $X_\tau^\circ$9 (Miller, 2010).

A second major result gives equivalent characterizations of when a subset $C \cap L$0 possesses such a stratification. The equivalent conditions include being a finite union of translates of affine semigroups, a finite union of translates of normal affine semigroups, and a finite disjoint union of translates of not necessarily normal affine semigroups. The proof proceeds through toric filtrations, polyhedral refinements, and module structures over affine semigroups (Miller, 2010).

Mesoprimary decomposition is the decisive structural input. The congruence induced by $C \cap L$1 decomposes into mesoprimary components with associated faces and associated lattices; congruence classes in localized settings become finite unions of sets of the form $C \cap L$2, and intersections across components produce cosets of intersection lattices. Nil fibers appear as ideals in $C \cap L$3 and decompose into translates of faces of $C \cap L$4 [(Miller, 2010); (Kahle et al., 2011)].

The integral character of the theory is explicit. Normal affine semigroups are precisely lattice-point sets in rational cones after restriction to a lattice, so the strata are integral polyhedral pieces. Illustrative examples include residue fibers in $C \cap L$5,

$C \cap L$6

which become a single translate of the normal affine semigroup

$C \cap L$7

and diagonal fibers

$C \cap L$8

(Miller, 2010).

These stratifications feed directly into lattice-game theory. When the misère quotient is finite, the set of $C \cap L$9-positions in a lattice game admits an affine stratification. In the cone case this is Corollary 4.5; in the polyhedral case it is Theorem 4.6. The finite-misère-quotient hypothesis is decisive in the application (Miller, 2010).

4. Tropical contractions to integral affine manifolds with singularities

In tropical geometry, the ambient object is not a semigroup subset or a frequency space but the tropicalization of a toric degeneration. An integral affine manifold with singularities consists of a topological manifold $\mathbb{A}^n$00 together with an integral affine structure on

$\mathbb{A}^n$01

where the discriminant locus $\mathbb{A}^n$02 is a locally finite union of submanifolds of codimension at least $\mathbb{A}^n$03. On $\mathbb{A}^n$04 one has the locally constant sheaves $\mathbb{A}^n$05 of integral tangent vectors and $\mathbb{A}^n$06 of integral cotangent vectors, together with the affine exponential exact sequence

$\mathbb{A}^n$07

whose extension class equals $\mathbb{A}^n$08, where $\mathbb{A}^n$09 is the radiance obstruction (Yamamoto, 2021).

For toric degenerations of Batyrev–Borisov Calabi–Yau complete intersections in the Gross–Siebert program, one obtains two spaces: a tropical variety $\mathbb{A}^n$10 and the dual intersection complex

$\mathbb{A}^n$11

which is an integral affine manifold with singularities. The compact tropical part satisfies

$\mathbb{A}^n$12

and there exists a global contraction

$\mathbb{A}^n$13

that is locally modeled on explicit local contractions (Yamamoto, 2021).

The strata are indexed by cells $\mathbb{A}^n$14 in the polyhedral decomposition $\mathbb{A}^n$15 of $\mathbb{A}^n$16. Over each stratum, one has

$\mathbb{A}^n$17

and the local maps glue to give the global contraction. This is the tropical meaning of integral affine stratification: $\mathbb{A}^n$18 is decomposed into pieces subordinate to the cells of the integral affine base, and the contraction respects integral affine charts. For $\mathbb{A}^n$19 in a local model, the fiber is described by

$\mathbb{A}^n$20

so the fibers are tropical torus cones in divisorial directions (Yamamoto, 2021).

The contraction is also cohomologically rigid. It induces isomorphisms between tropical cotangent sheaves and affine cotangent sheaves on the base, yielding

$\mathbb{A}^n$21

and analogous Borel–Moore statements. On wave groups, the eigenwave $\mathbb{A}^n$22 maps to the radiance obstruction $\mathbb{A}^n$23 (Yamamoto, 2021).

This stratified picture is compatible with singular affine monodromy. In local models, the monodromy transformation around a codimension-$\mathbb{A}^n$24 cell has the form

$\mathbb{A}^n$25

and positivity is established under the stated hypotheses (Yamamoto, 2021).

5. Singularities, degeneracy, and exceptional loci

A notable feature across these three literatures is that stratification is used to organize singular or non-generic behavior rather than to remove it.

In the arithmetic setting, no nonsingularity assumption is imposed on $\mathbb{A}^n$26. The stratification $\mathbb{A}^n$27 measures the “singularity complexity” of the projection $\mathbb{A}^n$28 in the frequency parameter $\mathbb{A}^n$29, and deeper strata permit larger character sums. Degenerate components lying in hyperplanes are handled by second-moment estimates averaged over the hyperplane, with exceptional primes controlled by a bound of size

$\mathbb{A}^n$30

(Bonolis et al., 16 May 2025).

In the affine-semigroup setting, singular behavior is replaced by congruence-theoretic complications. Nil fibers become ideals in $\mathbb{A}^n$31 and decompose into translates of faces; non-nil fibers are described via associated faces and lattices extracted from mesoprimary decomposition. The result is not a smoothness statement but a structural decomposition that survives nilpotent and non-free behavior of the quotient monoid (Miller, 2010).

In the tropical setting, the singular object is the integral affine manifold with singularities itself. The discriminant locus $\mathbb{A}^n$32 has codimension at least $\mathbb{A}^n$33, and monodromy is encoded in the radiance obstruction. The contraction $\mathbb{A}^n$34 does not remove these singularities; it organizes $\mathbb{A}^n$35 by strata over the singular affine base, with fibers thickening along directions determined by the degeneration (Yamamoto, 2021).

This suggests that “integral affine stratification” often functions as a method for retaining quantitative or categorical control in the presence of singularities, degeneracies, or nontrivial monodromy.

6. Scope, applications, and interpretive cautions

The arithmetic application is quantitative. For strongly $\mathbb{A}^n$36-allowable polynomials, the stratified sieve yields the exponent

$\mathbb{A}^n$37

unconditionally for $\mathbb{A}^n$38 and under GRH for $\mathbb{A}^n$39, while for allowable but not strongly allowable polynomials one obtains $\mathbb{A}^n$40 up to logarithmic factors. Genericity is also established: strongly $\mathbb{A}^n$41-allowable polynomials form a Zariski-open dense subset in the relevant moduli spaces, including structured subfamilies such as polynomials in $\mathbb{A}^n$42 and even within the locus of singular polynomials (Bonolis et al., 16 May 2025).

The affine-semigroup application is structural and algorithmic. Fibers of arbitrary monoid morphisms from affine semigroups admit affine stratifications, linear images preserve stratifications, and finite unions of affinely stratified sets remain affinely stratified. In lattice games, finite misère quotient implies affine stratifiability of the set of $\mathbb{A}^n$43-positions (Miller, 2010).

The tropical application is geometric and cohomological. The contraction from $\mathbb{A}^n$44 to the Gross–Siebert dual intersection complex preserves tropical cohomology, identifies tropical cotangent data with affine cotangent data, and sends the eigenwave to the radiance obstruction. This ties the polyhedral geometry of tropicalization to the affine differential-topological data of the degeneration base (Yamamoto, 2021).

Several misconceptions are therefore avoided by the literature itself. First, integral affine stratification is not a single standard construction across fields. Second, in the arithmetic usage the stratified object is the dual frequency space, not the original set of integral parameters. Third, in the affine-semigroup usage the strata are not arbitrary polyhedra but translates of normal affine semigroups. Fourth, in the tropical usage the stratification is inseparable from the contraction map and from the sheaf-theoretic structures $\mathbb{A}^n$45, $\mathbb{A}^n$46, and the radiance obstruction.

Taken together, these works show that integral affine stratification is best understood as a family of highly structured decompositions of integral or tropical objects into affine pieces, each adapted to a specific mathematical problem: polynomial sieves for thin sets, mesoprimary analysis of monoid fibers, and tropical contraction to integral affine manifolds with singularities [(Bonolis et al., 16 May 2025); (Miller, 2010); (Yamamoto, 2021)].