Completing the structure theory of toric difference varieties: σ-torus classification, the orbit–face correspondence beyond face-saturated semimodules, and the Chow–resultant identity beyond transformally essential supports

Toric difference varieties, introduced by Gao–Huang–Wang–Yuan (2017), extend classical toric geometry to difference algebra: they are the Cohn closures of monomial parametrizations $y_i=t^{u_i}$ whose exponents $u_i$ lie in the Laurent polynomial lattice $\mathbb{Z}[x]^n$ (the indeterminate $x$ records the action of the distinguished endomorphism $\sigma$), and they are the irreducible building blocks of binomial difference ideals. The paper establishes a dictionary between affine $\mathbb{N}[x]$-semimodules and toric $\sigma$-varieties, but each of its central structure theorems rests on an auxiliary hypothesis: $\sigma$-tori are not classified (and, unlike algebraic tori, need not be isomorphic to a power of the invertibles $\mathbb{A}^*$); the orbit–face correspondence is proved only for face-saturated semimodules; and the identity between the $\sigma$-Chow form and the sparse difference resultant holds only for Laurent transformally essential supports. The open problem is to complete the dictionary: classify $\sigma$-tori up to $\sigma$-isomorphism and give a splitness criterion; extend the orbit–face correspondence to arbitrary affine $\mathbb{N}[x]$-semimodules together with a characterization of face-saturation; and extend the Chow–resultant identity and its Jacobi-style order bound to non-essential supports or precisely characterize their failure. No resolution of any of these components is known in the literature to date.

Background

A difference field is a field $k$ equipped with a distinguished endomorphism $\sigma$ (for example, the shift operator acting on a field of sequences); difference algebra studies systems of polynomial equations in indeterminates and their iterated transforms $\sigma^i(y)$. The difference polynomial ring $k\{y_1,\dots,y_m\}$ consists of polynomials in the indeterminates and all their transforms, and a $\sigma$-variety is the solution set of a system of such difference polynomials, with solutions taken in difference extension fields $K$ of $k$.

A Laurent difference monomial is a formal product $y_1^{u_1}\cdots y_m^{u_m}$ with exponents $u_i\in\mathbb{Z}[x]$, where $x$ acts on exponents as the operator $u(\sigma)=\sum_j a_j\sigma^j$; the exponent vectors thus live in the lattice $\mathbb{Z}[x]^m$. A $\mathbb{Z}[x]$-lattice $L\subseteq\mathbb{Z}[x]^m$ is the group of exponent relations of a system of Laurent binomial difference equations (equations of the form $y^{a}-y^{b}=0$); $L$ is called toric if it is $\mathbb{Z}[x]$-saturated, meaning $g\cdot u\in L$ with $0\neq g\in\mathbb{Z}[x]$ implies $u\in L$. The companion work of Gao–Huang–Yuan on binomial difference ideals shows that Laurent binomial $\sigma$-ideals are completely controlled by such support lattices, and that a binomial $\sigma$-ideal is toric exactly when its support lattice is toric.

The paper of Gao–Huang–Wang–Yuan builds the toric geometry of this setting. For a finite set $U=\{u_1,\dots,u_m\}\subset\mathbb{Z}[x]^n$, the toric $\sigma$-variety $X_U$ is the Cohn closure (the closure operation of difference algebra, named after R. M. Cohn) of the image of the monomial parametrization $y_i=t^{u_i}$ in $\mathbb{A}^m$. The quasi $\sigma$-torus $T_U^*$ is the corresponding solution set inside the invertibles $(\mathbb{A}^*)^m$, where $(\mathbb{A}^*)^m$ denotes the $\sigma$-variety of invertible $m$-tuples (the difference-algebraic analogue of the algebraic torus $(\mathbb{C}^*)^m$), realized via the equations $y_iz_i-1=0$. A $\sigma$-torus is defined as a $\sigma$-variety isomorphic to the Cohn $*$-closure of a quasi $\sigma$-torus. Theorem 5.5 of the paper shows that every $\sigma$-torus is of the form $\operatorname{Spec}_\sigma(k[L])$ for a toric lattice $L$, where $k[L]$ is the semigroup algebra spanned by the monomials with exponents in $L$, made into a $\sigma$-ring by $\sigma(T^u)=T^{x\cdot u}$. Unlike the algebraic situation, where every torus over an algebraically closed field is isomorphic to a power of $\mathbb{C}^*$, Example 5.6 of the paper exhibits a $\sigma$-torus that is not $\sigma$-isomorphic to $\mathbb{A}^*$: the one attached to the lattice generated by $2$ and $x$ in $\mathbb{Z}[x]$.

The paper then establishes a combinatorial dictionary. An affine $\mathbb{N}[x]$-semimodule $S$ is a finitely generated semimodule over the semiring $\mathbb{N}[x]$, i.e. a set of the form $\{\sum_i g_i u_i \mid g_i\in\mathbb{N}[x]\}$ inside some $\mathbb{Z}[x]^n$; the category of affine toric $\sigma$-varieties is antiequivalent to the category of affine $\mathbb{N}[x]$-semimodules. A face of $S$ is an $\mathbb{N}[x]$-subsemimodule $F\subseteq S$ such that $u_1+u_2\in F$ implies $u_1,u_2\in F$ and $x u\in F$ implies $u\in F$ (Definition 6.6 of the paper; the difference analogue of a face of an affine semigroup in classical toric geometry). The paper proves that irreducible $T$-invariant subvarieties of $X=\operatorname{Spec}_\sigma(k[S])$ correspond one-to-one with faces of $S$ unconditionally; but for the finer orbit–face statement — a one-to-one correspondence between the orbits of the $\sigma$-torus $T(K)$ acting on $X(K)$ and the faces of $S$ (Theorem 6.11) — the paper must assume that $S$ is face-saturated: every morphism $\varphi:F\to K^*$ from a face $F$ of $S$ (a multiplicative map compatible with $\sigma$) extends to a morphism $\tilde\varphi:S\to K^*$. A necessary condition for face-saturation is that the lattice $(F)_{\mathbb{Z}[x]}$ be $\mathbb{N}[x]$-saturated in $(S)_{\mathbb{Z}[x]}$, but this condition is not known to be sufficient, and Example 6.10 exhibits a semodule that is not face-saturated. In classical toric geometry the analogous orbit–face correspondence holds with no restrictive hypothesis, so the face-saturation assumption is a genuinely difference-theoretic obstruction.

Finally, the paper connects toric $\sigma$-varieties to difference elimination theory. The $\sigma$-Chow form of Li–Li is the difference analogue of the classical Chow form (the defining equation of the set of hyperplanes meeting a variety); the sparse difference resultant of Li–Yuan–Gao is the difference analogue of the sparse resultant of a sparse polynomial system, and it exists exactly for Laurent transformally essential systems — informally, systems whose symbolic support matrix has rank equal to the number $n$ of difference variables, so that $n$ of the Laurent $\sigma$-monomials are transformally independent. Under the Laurent transformally essential hypothesis $\operatorname{rk}(U)=n$, Theorem 7.1 of the paper proves that, up to a sign, the $\sigma$-Chow form $C_U$ of $X_U$ coincides with the sparse difference resultant $R_U$ of $n+1$ generic Laurent difference polynomials with support $U$, and derives from this a Jacobi-style bound on the transformal order of $X_U$, where the Jacobi number of a matrix is its maximal diagonal sum. Whether this identity and order bound survive without the essentiality hypothesis is not addressed.

Problem Statement

Complete the structure theory of toric difference varieties by removing the auxiliary hypotheses under which Gao–Huang–Wang–Yuan (2017) prove their main structure theorems and resolving the limitations those theorems leave open; the common objective is a lattice/semimodule-to-$\sigma$-variety dictionary valid without restrictions, and the three components jointly realize it.

1. Classification of $\sigma$-tori. Classify $\sigma$-tori up to $\sigma$-isomorphism, i.e. give a complete set of invariants separating non-isomorphic $\sigma$-tori. In particular, give necessary and sufficient conditions under which a $\sigma$-torus is $\sigma$-isomorphic to $(\mathbb{A}^*)^m$ for some $m$, and exhibit invariants that distinguish non-isomorphic $\sigma$-tori. (It is known that $\sigma$-tori defined by generators of the same lattice are isomorphic, and that the lattice generated by $2$ and $x$ in $\mathbb{Z}[x]$ yields a $\sigma$-torus not isomorphic to $\mathbb{A}^*$; nothing beyond this is known.)

2. Orbit–face correspondence and face-saturation. For an arbitrary affine $\mathbb{N}[x]$-semimodule $S$ that is not necessarily face-saturated, with $X=\operatorname{Spec}_\sigma(k[S])$ its toric $\sigma$-variety and $T$ the $\sigma$-torus of $X$, determine whether the $T(K)$-orbits of $X(K)$ are still in one-to-one correspondence with the faces of $S$ for each difference extension field $K$; if the correspondence fails in general, characterize precisely the semimodules for which it holds. As part of this, characterize the face-saturation property itself (every morphism $\varphi:F\to K^*$ from a face $F$ of $S$ extends to $S$) by necessary and sufficient conditions — the known condition, that $(F)_{\mathbb{Z}[x]}$ be $\mathbb{N}[x]$-saturated in $(S)_{\mathbb{Z}[x]}$, is only necessary.

3. Scope of the Chow–resultant identity. The identity $C_U=\pm R_U$ between the $\sigma$-Chow form of the toric $\sigma$-variety $X_U$ and the sparse difference resultant of $n+1$ generic Laurent difference polynomials with support $U$, and the Jacobi-style order bound for $X_U$ derived from it, are proved under the Laurent transformally essential hypothesis $\operatorname{rk}(U)=n$. Extend the identity and the order bound to supports $U\subset\mathbb{Z}[x]^n$ that are not Laurent transformally essential — which requires first giving a meaningful definition of the sparse difference resultant in the non-essential case — or precisely characterize the ways in which the identity and the bound fail for non-essential supports.

References

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