Related papers: Counting mod n in pseudofinite fields

While modal extensions of decidable fragments of first-order logic are usually undecidable, their monodic counterparts, in which formulas in the scope of modal operators have at most one free variable, are typically decidable. This only…

An old question of Erdos asks if there exists, for each number N, a finite set S of integers greater than N and residue classes r(n) mod n for n in S whose union is all the integers. We prove that if $\sum_{n\in S} 1/n$ is bounded for such…

A vector bundle E on a projective variety X is called finite if it satisfies a nontrivial polynomial equation with integral coefficients. A theorem of Nori implies that E is finite if and only if the pullback of E to some finite etale…

We generalise the constructions of Brady and Lodha to give infinite families of hyperbolic groups, each having a finitely presented subgroup that is not of type $F_3$. By calculating the Euler characteristic of the hyperbolic groups…

We prove, assuming resolution of singularities in positive characteristic, an analogue of Siegel's theorem on sum of squares in positive characteristic. The method of proof combines techniques from central simple algebras with model theory…

For every natural number k we introduce the notion of k-th order convolution of functions on abelian groups. We study the group of convolution preserving automorphisms of function algebras in the limit. It turns out that such groups have…

A family of subsets $\mathcal{F} \subseteq \mathcal{P}(\{1, 2, \ldots, n\})$ has the disparate union property if any two disjoint subfamilies $\mathcal{F}_1, \mathcal{F}_2 \subseteq \mathcal{F}$ have distinct unions $\bigcup \mathcal{F}_1…

The polynomial-time computability of the permanent over fields of characteristic 3 for k-semi-unitary matrices (i.e. square matrices such that the differences of their Gram matrices and the corresponding identity matrices are of rank k) in…

The Kochen-Specker theorem states that a 3-dimensional complex Euclidean space admits a finite configuration of projective lines such that the corresponding quantum observables (the orthogonal projectors) cannot be assigned with 0 and 1…

We study the question of Eulerianity (factorizability) for Fourier coefficients of automorphic forms, and we prove a general transfer theorem that allows one to deduce the Eulerianity of certain coefficients from that of another…

Let n be a positive integer, and let R be a finitely presented (but not necessarily finite dimensional) associative algebra over a computable field. We examine algorithmic tests for deciding (1) if every n-dimensional representation of R is…

We prove an identity relating the product of two opposite Schubert varieties in the (equivariant) quantum K-theory ring of a cominuscule flag variety to the minimal degree of a rational curve connecting the Schubert varieties. We deduce…

We describe supertraces on ``queerifications'' (see arxiv:2203.06917) of the algebras of matrices of ``complex size'', algebras of observables of Calogero-Moser model, Vasiliev higher spin algebras, and (super)algebras of…

We show that for any $k\in\omega$, the structure $(H_k,\in)$ of sets that are hereditarily of size at most $k$ is decidable. We provide a transparent complete axiomatization of its theory, a quantifier elimination result, and tight bounds…

To an arbitrary variety over a field of characteristic zero, we associate a complex of Chow motives, which is, up to homotopy, unique and bounded. We deduce that any variety has a natural Euler characteristic in the Grothendieck group of…

We give a simple example of a set that is weakly Dedekind infinite (= can be mapped onto omega) but dually Dedekind finite (=cannot be mapped noninjectively onto itself), namely, the power set of a superamorphous set. (A infinite set is…

Federer's characterization of sets of finite perimeter states (in Euclidean spaces) that a set is of finite perimeter if and only if the measure-theoretic boundary of the set has finite Hausdorff measure of codimension one. In complete…

We study cofinal systems of finite subsets of $\omega_1$. We show that while such systems can be NIP, they cannot be defined in an NIP structure. We deduce a positive answer to a question of Chernikov and Simon from 2013: in an NIP theory,…

Given a number field $k$, a finitely generated subgroup $\mathcal{A}\subseteq k^\times$, and an integer $n\geq 3$, we study the distribution of $S_n$-extensions of $k$ such that the elements of $\mathcal{A}$ are norms. For $n\leq 5$, and…

We establish exponential laws for certain spaces of differentiable functions over a valued field K. For example, we show that the topological vector spaces C^{r,s}(U x V,E) and C^r(U,C^s(V,E)) are isomorphic if U and V are open subsets of…