Related papers: Universally and existentially definable subsets of…

For a positive proportion of primes $p$ and $q$, we prove that $\mathbb{Z}$ is Diophantine in the ring of integers of $\mathbb{Q}(\sqrt[3]{p},\sqrt{-q})$. This provides a new and explicit infinite family of number fields $K$ such that…

Let $G$ be a finite group and $K$ a normal subset consisting of odd-order elements. The rational closure of $K$, denoted $\mathbf D_K$, is the set of elements $x \in G$ with the property that $\langle x \rangle = \langle y \rangle$ for some…

We present a universal construction of Diophantine equations with bounded complexity in Isabelle/HOL. This is a formalization of our own work in number theory. Hilbert's Tenth Problem was answered negatively by Yuri Matiyasevich, who showed…

For a number field $K$, we extend the notion of the ring class field of an order in $K$ [C. Lv and Y. Deng, SciChina. Math., 2015] to that of an arbitrary number ring in $K$. We give both ideal-theoretic and idele-theoretic description of…

This paper is concerned with the extent to which the Skolem-Bang theorems in Diophantine approximations generalise from the standard setting of $<R,Z>$ to structures of the form $<F,I>$, where $F$ is an ordered field and $I$ is an integer…

Let K be a number field and let A be its ring of integers. Let G be a connected, noncommutative, absolutely almost simple algebraic K-group. If the K-rank of G equals 2, then G(A[t]) is not finitely presented.

We give model theoretic criteria for $\exists \forall$ and $\forall \exists$- formulas in the ring language to define uniformly the valuation rings $\mathcal{O}$ of models $(K, \mathcal{O})$ of an elementary theory $\Sigma$ of henselian…

We give first-order definitions of Campana and Darmon points in algebraic function fields in one variable over number fields. These sets are geometric generalizations of $n$-full integers (integers whose nonzero valuations are at least $n$)…

We prove the following result. Let k be an infinite perfect field of positive characteristic and assume that strong resolution of singularities holds over k. Let R be a localization of a commutative d-dimensional k-algebra of finite type…

We establish a "diagonal" ergodic theorem involving the additive and multiplicative groups of a countable field $K$ and, with the help of a new variant of Furstenberg's correspondence principle, prove that any "large" set in $K$ contains…

Let $k$ be a perfect field of characteristic $p>0$, $k(t)_{per}$ the perfect closure of $k(t)$ and $A$ a $k$-algebra. We characterize whether the ring $A\otimes_k k(t)_{per}$ is noetherian or not. As a consequence, we prove that the ring…

We provide a universal characterization of the construction taking a scheme $X$ to its stable $\infty$-category $\text{Mot}(X)$ of noncommutative motives, patterned after the universal characterization of algebraic K-theory due to…

We show that K2 of "sufficiently regular" localisations of local rings (e.g. inverting a sequence of regular parameters) can be described by the Steinberg presentation. The proof is inductive on the number of irreducible elements being…

Let $G$ be a finite 2-group and $K$ be a field satisfying that (i) $\fn{char}K\ne 2$, and (ii) $\sqrt{a}\in K$ for any $a\in K$. If $G$ acts on the rational function field $K(x,y,z)$ by monomial $K$-automorphisms, then the fixed field…

We study totally positive definite quadratic forms over the ring of integers $\mathcal{O}_K$ of a totally real biquadratic field $K=\mathbb{Q}(\sqrt{m}, \sqrt{s})$. We restrict our attention to classical forms (i.e., those with all…

Let $C$ be a smooth projective curve defined over the finite field $\mathbb{F}_q$ ($q$ is odd) and let $K=\mathbb{F}_q(C)$ be its function field. Removing one closed point $C^\text{af} = C-\{\infty\}$ results in an integral domain…

The classification of irreducible, spherical characters of the infinite-dimensional unitary/orthogonal/symplectic groups can be obtained by finding all possible limits of normalized, irreducible characters of the corresponding…

Let Rep(F;K) denote the category of functors from finite dimensional F-vector spaces to K-modules, where F is a field and K is a commutative ring. We prove that, if F is a finite field, and Char F is invertible in K, then the K-linear…

Let K be an arbitrary field. We will determine explicitly all the nontrivial finite groups of essential dimension one over K.

Let $\mathcal{G}$ be a smooth linear group scheme of finite type. For any positive integer $k$ and a finite field $\mathbb{F}$, let $W_k(\mathbb{F})$ be the ring of Witt vectors of length $k$ over $\mathbb{F}$. We show that the group…