Non-expansion in polynomial automorphisms of $\mathbb{C}^2$
We treat the higher-dimensional Elekes-Szabó problem in the case of the action of Aut(C^2) on C^2.
math.CO↗
arXiv subjects
Tingxiang Zou
Publications and source records attributed to Tingxiang Zou.
Elekes-Szabó for collinearity on cubic surfaces
We study the orchard problem on cubic surfaces. We classify possibly reducible cubic surfaces $X\subseteq \mathbb{P}^3(\C)$ with smooth components on which there exist families of finite sets (of unbounded size) with quadratically many 3-rich lines which do not concentrate (in a natural sense) on any projective plane. Namely, we prove that such a family exists precisely when $X$ is a union of three planes sharing a common line. Along the way, we obtain a general result about nilpotency of groups admitting an algebraic action satisfying an Elekes-Szabó condition, and we prove the following purely algebrogeometric statement: if the composition of four Geiser involutions through sufficiently generic points $a,b,c,d$ on a smooth irreducible cubic surface has infinitely many fixed points, then a single plane contains $a,b,c,d$ and all but finitely many of the fixed points.
math.LO↗
A group-action Szemerédi-Trotter theorem and applications to orchard problems in all characteristics
We establish a group-action version of the Szemerédi-Trotter theorem over any field, extending Bourgain's result for the group $\mathrm{SL}_2(k)$. As an Elekes-Szabó-type application, we obtain quantitative bounds on the number of collinear triples on reducible cubic surfaces in $\mathbb{P}^3(k)$, where $k = \mathbb{F}_{q}$ and $k = \mathbb{C}$, thereby improving a recent result by Bays, Dobrowolski, and the second author.
math.CO↗
An asymmetric version of Elekes-Szab\'o via group actions
We consider when finite families $F \subseteq \mathbb{C}[t]$ of bounded degree polynomials, or more generally of bounded complexity finite-to-finite correspondences on $\mathbb{C}$, can exhibit non-expansion of the form $|F(A)| = O(|A|^{1+\eta})$ in their actions on finite sets $A \subseteq \mathbb{C}$ with $|F| \gg |A|^\eps \gg 1$, for a fixed $\eps>0$ and arbitrarily small $\eta>0$. Our conclusions generalise the Elekes-R\'onyai and Elekes-Szab\'o theorems, which correspond to the case that $F$ is parametrised by a single complex variable and $|F|=|A|$. Our result also applies to families of correspondences between varieties of arbitrary dimension if we impose a general position assumption on $A$. In all cases, the conclusion is that a commutative algebraic group structure is responsible. As a special case, we obtain asymmetric versions of Elekes-R\'onyai and Elekes-Szab\'o, with explicit bounds on exponents. Our methods originate in model theory.
math.CO↗
Lang-Weil Type Estimates in Finite Difference Fields
We prove a uniform estimate of the number of points for difference algebraic varieties in finite difference fields in the spirit of Lang-Weil. More precisely, we give uniform lower and upper bounds for the number of rational points of a difference variety in terms of its transformal dimension. As a main technical ingredient, we prove an equidimensionality result for Frobenius reductions of difference varieties.
math.NT↗
Elekes-Szabó for groups, and approximate subgroups in weak general position
We show that with a suitable weak notion of general position, the Elekes-Szabó condition on the group operation of a connected complex algebraic group characterises nilpotence of the group. Along the way, we prove a Mordell-Lang result for generic finitely generated subgroups of commutative complex algebraic groups.
math.CO↗
The model theory of geometric random graphs
We study the logical properties of infinite geometric random graphs, introduced by Bonato and Janssen. These are graphs whose vertex set is a dense ``generic'' subset of a metric space, where two vertices are adjacent with probability $p>0$ provided the distance between them is bounded by some constant number. We prove that for a large class of metric spaces, including circles, spheres and the complete Urysohn space, almost all geometric random graphs on a given space are elementary equivalent. Moreover, their first-order theory can reveal geometric properties of the underlying metric space.
math.LO↗
Dimension and measure in pseudofinite H-structures
We study H-structures associated to SU-rank 1 measurable structures. We prove that the SU-rank of the expansion is continuous and that it is uniformly definable in terms of the parameters of the formulas. We also introduce notions of dimension and measure for definable sets in the expansion and prove they are uniformly definable in terms of the parameters of the formulas.
math.LO↗
Pseudofinite difference fields and counting dimensions
We study a family of ultraproducts of finite fields with the Frobenius automorphism in this paper. Their theories have the strict order property and TP2. But the coarse pseudofinite dimension of the definable sets is definable and integer-valued. Moreover, we also discuss the possible connection between coarse dimension and transformal transcendence degree in these difference fields.
math.LO↗
Schlichting's Theorem for Approximate Subgroups
We prove Schlichting's theorem for approximate subgroups: if $\mathcal{X}$ is a uniform family of commensurable approximate subgroups in some ambient group, then there exists an invariant approximate subgroup commensurable with $\mathcal{X}$.
math.GR↗
Pseudofinite primitive permutation groups acting on one-dimensional sets
Working in a theory with an integer-valued dimension on interpretable sets, we classify pseudofinite definably primitive permutation groups acting on one-dimensional sets which satisfy a version of chain condition on centralizers and on point-wise stabilizers. This generalises the classification of pseudofinite definably primitive permutation groups in supersimple theories of finite rank to supersimple theories of infinite rank.
math.LO↗
Pseudofinite H-structures and groups definable in supersimple H-structures
In this paper we explore some properties of H-structures. We describe a construction of H-structures based on one-dimensional asymptotic classes which preserves pseudo-finiteness. That is, the H-structures we construct are ultraproducts of finite structures. We also prove that under the assumption that the base theory is supersimple of SU-rank one, there are no new definable groups in H-structures.
math.LO↗
Classical and Relative Realizability
We show that every abstract Krivine structure in the sense of Streicher can be obtained, up to equivalence of the resulting tripos, from a filtered opca (A,A') and a subobject of 1 in the relative realizability topos RT(A',A); the topos is always a Booleanization of a closed subtopos of RT(A',A). We exhibit a range of non-localic Boolean subtoposes of the Kleene-Vesley topos.
math.CT↗