Difference Chow varieties: existence of a parameter-space theory and the missing structural theory of the difference Chow form

In algebraic geometry, the Chow form of an irreducible variety encodes it by the coefficients of generic hyperplanes meeting it, and Chow varieties parametrize varieties of fixed dimension and degree. This picture was completed for differential varieties: the differential Chow form has a full structural theory, including a Poisson-type factorization formula, and differential Chow varieties exist as genuine parameter spaces. For difference (discrete) algebra — the framework for systems of difference equations and recurrence relations — Li and Li (J. Algebra, 2015) proved a generic intersection theorem and defined the difference Chow form of an irreducible difference variety with its basic properties, but stated explicitly that the structural theory is incomplete: no Poisson-type factorization is known, the correspondence between difference varieties and their Chow forms is not one-to-one, and whether a theory of difference Chow varieties can be developed is open. Does a difference-algebraic parameter space for irreducible difference varieties of fixed dimension and order exist, together with the missing structure theory of the difference Chow form, or is there a structural obstruction?

Background

Classical elimination theory represents an irreducible algebraic variety $V$ of dimension $d$ in affine or projective $n$-space by its Chow form (also called the Cayley–Chow form): the irreducible polynomial $F(u_0,\dots,u_d)$ in the coefficients of $d+1$ generic hyperplanes that vanishes exactly when those hyperplanes meet $V$. Chow forms are basic tools in elimination theory, transcendence theory, and algebraic complexity, and Chow varieties — parameter spaces whose points encode algebraic cycles of fixed dimension and degree — are a classical construction going back to Chow and van der Waerden.

Difference algebra provides the analogous framework for systems of equations in unknowns and their iterates under a shift. A difference field $(F,\sigma)$ is a field with a distinguished endomorphism $\sigma$ (for example $\mathbb{Q}(x)$ with $\sigma(f(x))=f(x+1)$); the difference polynomial ring $F\{Y\}$ in indeterminates $Y=\{y_1,\dots,y_n\}$ is the polynomial ring in all transforms $\sigma^k y_i$ with $k\geq 0$. Solution sets of systems of difference polynomial equations are difference varieties, and the irreducible ones correspond to reflexive prime difference ideals (prime ideals closed under $\sigma$ such that $\sigma(f)\in I$ forces $f\in I$). Two numerical invariants govern an irreducible difference variety $V$ with generic point $\eta$: its difference dimension $d$ and its order $h$, both read off from the difference dimension polynomial $\varphi_I(t)=\operatorname{tr.deg} F(\eta^{[t]})/F=d(t+1)+h$ for large $t$, where $\eta^{[t]}$ collects the transforms of $\eta$ up to level $t$ (Kondratieva–Levin–Mikhalev–Pankratiev). Computations with difference ideals are organized by characteristic sets: a ranking is a total order on the transforms $\sigma^k y_i$ compatible with $\sigma$, and every reflexive prime difference ideal has a characteristic set — a canonical triangular system — under any ranking. Theorem 3.5 of the source paper (Li–Li, 2015) shows that the dimension and the relative order of the ideal (the transcendence degree of the generic point over the transformal field generated by a maximal transformally independent set of variables) can be recovered from a characteristic set under an arbitrary ranking, generalizing Cohn's elimination-ranking result and the differential theorem of Cluzeau–Hubert.

The differential counterpart of the Chow-form picture is complete. Gao–Li–Yuan (Trans. AMS, 2013) proved a generic intersection theorem for differential varieties, defined the differential Chow form, established its main properties — including a Poisson-type factorization formula, the analogue of the classical Poisson formula for resultants describing the multiplicative/factorization behavior of the form — and constructed a differential Chow quasi-variety parametrizing differential varieties of fixed dimension and order. Freitag–Li–Scanlon–Johnson (2017) then proved that differential Chow varieties exist as genuine parameter spaces; algorithms for computing differential Chow forms (Li–Li, 2016) and a partial-differential extension (Li, 2020) followed.

For the difference case, Li–Li (J. Algebra, 2015) proved the generic intersection theorem — intersecting an irreducible difference variety of dimension $d>0$ and order $h$ with a generic difference hypersurface of order $s$ yields an irreducible difference variety of dimension $d-1$ and order $h+s$ — and defined the difference Chow form of an irreducible difference variety $V$ as the unique irreducible difference polynomial $F(u_0,\dots,u_d)$ in the coefficients of $d+1$ generic difference hyperplanes that generates the codimension-one elimination ideal of the incidence system. They proved that this form is transformally homogeneous (invariant, up to a factor that is a difference monomial, under scaling each hyperplane's coefficient vector by a new difference indeterminate) and that its order equals the order of $V$. But the theory stops there. No Poisson-type factorization is known. Moreover, the correspondence between irreducible difference varieties and their Chow forms fails to be one-to-one: Example 6.4 of that paper exhibits two distinct reflexive prime difference ideals over $\mathbb{Q}(x)$ with $\sigma(x)=x+1$, namely $I=[y_1^2+1,\,y_1-y_1^{(1)}]$ and $J=[y_1^2+1,\,y_1+y_1^{(1)}]$ (writing $y_1^{(1)}$ for the first transform $\sigma(y_1)$), which share the same difference Chow form, although the correspondence at the level of difference Chow ideals (the full elimination ideals) remains bijective. The authors state that whether a theory of difference Chow varieties — a difference-algebraic analogue of Chow coordinates — can be developed is still open, and attribute the difficulty to the fact that extensions of difference fields are much more complicated than in the differential case.

Problem Statement

The problem posed at the end of Li–Li (2015), in its structural core: establish whether a theory of difference Chow varieties can be developed, completing the structural theory of the difference Chow form to the level of its differential counterpart. Concretely, for fixed difference dimension $d$ and order $h$: (1) determine whether the collection of difference Chow ideals of irreducible difference varieties of dimension $d$ and order $h$ — the full elimination ideals, which remain in bijection with those varieties even though the correspondence through difference Chow forms alone is provably not one-to-one (Example 6.4) — constitutes a difference variety, that is, a difference-algebraic set in an explicit coefficient space: resolve this positively by giving its difference-polynomial defining equations and proving that its points are exactly the encodings of irreducible difference varieties of dimension $d$ and order $h$, with a precise account of which objects the parametrization distinguishes and how the non-injectivity phenomenon is classified, or negatively by proving that no such difference variety exists, a result admissible as the structural obstruction sought in (3); (2) establish the missing structural properties of the difference Chow form, in particular a Poisson-type factorization formula analogous to the differential case — items (1) and (2) together reflect the source's single open problem of enriching the properties of the difference Chow form and, especially, establishing a theory of difference Chow varieties; or (3) prove rigorously that no such difference-algebraic parameter space and factorization theory can exist, identifying the obstruction arising from the structure of difference field extensions. The differential theory of Gao–Li–Yuan (2013) and Freitag–Li–Scanlon–Johnson (2017) is the parity benchmark for the difference-side theory. Two further problems posed in the same conclusion — an efficient algorithm for computing difference Chow forms with complexity polynomial in input size and geometric degree, and the conjecture that the relative effective order of a reflexive prime difference ideal is readable from characteristic sets under arbitrary rankings — are related open problems but have separate completion boundaries and fall outside the scope of this problem.

References

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