Related papers: Kakeya-Type Sets in Local Fields with Finite Resid…

Local fields, and fields complete with respect to a discrete valuation, are essential objects in commutative algebra, with applications to number theory and algebraic geometry. We formalize in Lean the basic theory of discretely valued…

Using the polynomial method of Dvir \cite{dvir}, we establish optimal estimates for Kakeya sets and Kakeya maximal functions associated to algebraic varieties $W$ over finite fields $F$. For instance, given an $n-1$-dimensional projective…

We study Kakeya sets over local non-archimedean fields with a probabilistic point of view: we define a probability measure on the set of Kakeya sets as above and prove that, according to this measure, almost all non-archimedean Kakeya sets…

In the finite field setting, we show that the restriction conjecture associated to any one of a large family of $d=2n+1$ dimensional quadratic surfaces implies the $n+1$ dimensional Kakeya conjecture (Dvir's theorem). This includes the case…

We say that a planar set $A$ has the Kakeya property if there exist two different positions of $A$ such that $A$ can be continuously moved from the first position to the second within a set of arbitrarily small area. We prove that if $A$ is…

We classify all possible extensions of a valuation from a ground field $K$ to a rational function field in one or several variables over $K$. We determine which value groups and residue fields can appear, and we show how to construct…

The singularity category of a ring makes only the modules of finite projective dimension vanish among the modules, so the singularity category is expected to characterize a homological property of modules of infinite projective dimension.…

A Kakeya set is a compact subset of $\mathbb{R}^n$ that contains a unit line segment pointing in every direction. The Kakeya conjecture asserts that such sets must have Hausdorff and Minkowski dimension $n$. There is a special class of…

The article is devoted to the investigation of particular classes of quasi-invariant descending at infinity measures on linear spaces over non-Archimedean fields such that measures are with values in non-Archimedean fields also. Their…

The restriction and Kakeya problems in Euclidean space have received much attention in the last few decades, and are related to many problems in harmonic analysis, PDE, and number theory. In this paper we initiate the study of these…

We introduce a non real-valued measure on the definable sets contained in the finite part of a cartesian power of an o-minimal field $R$. The measure takes values in an ordered semiring, the Dedekind completion of a quotient of $R$. We show…

In this paper, we present the Nullstellensatz in case of the coordinate rings of a nonempty subset of Kn where K is a finite field Fq. Some applications of the Nullstellensatz are also discussed.

In recent decades, the defect of finite extensions of valued fields has emerged as the main obstacle in several fundamental problems in algebraic geometry such as the local uniformization problem. Hence, it is important to identify…

The authors establish a connection between the Quillen K-theory of certain local fields and the de Rham-Witt complex of their rings of integers with logarithmic poles at the maximal ideal. They consider fields K that are complete discrete…

In the present paper, we propose a new construction of quantum Markov fields on arbitrary connected, infinite, locally finite graphs. The construction is based on a specific tessellation on the considered graph, that allows us to express…

Discrete integrable equations over finite fields are investigated. The indeterminacy of the equation is resolved by treating it over a field of rational functions instead of the finite field itself. The main discussion concerns a…

In this article, we study the structure of finitely ramified mixed characteristic valued fields. For any two complete discrete valued fields $K_1$ and $K_2$ of mixed characteristic with perfect residue fields, we show that if the $n$-th…

We study the structure of the zero set of a finite point charge electrical field $F = (X,Y,Z)$ in $\mathbb R^3$. Indeed, mostly we focus on a finite point charge electrical field $F =(X,Y)$ in $\mathbb R^2$. The well-known conjecture is…

A problem of constructing of local definitions for formations of finite groups is discussed in the article. The author analyzes relations between local definitions of various types. A new proof of existence of an $\omega$-composition…

We obtain several results concerning the concept of isotypic structures. Namely we prove that any field of finite transcendence degree over a prime subfield is defined by types; then we construct isotypic but not isomorphic structures with…