相关论文: Some aspects of analysis related to p-adic numbers

We begin by defining general hypergeometric functions over finite fields and obtaining a finite field analogue of a classical symmetry in their complex counterparts. We give a geometric proof for the symmetry by constructing isomorphisms…

In this paper, we will show that the $p$-adic valuation (where $p$ is a given prime number) of some type of rational numbers is unusually large. This generalizes the very recent results by the author and by A. Dubickas, which are both…

We give a proposal for future development of the model theory of valued fields. We also summarize some recent results on p-adic numbers.

We design algorithms for computing values of many p-adic elementary and special functions, including logarithms, exponentials, polylogarithms, and hypergeometric functions. All our algorithms feature a quasi-linear complexity with respect…

We investigate the computability-theoretic properties of valued fields, and in particular algebraically closed valued fields and $p$-adically closed valued fields. We give an effectiveness condition, related to Hensel's lemma, on a valued…

We present a version of Smale's $\alpha$-theory for ultrametric fields, such as the $p$-adics and their extensions, which gives us a multivariate version of Hensel's lemma.

Let $p$ be a prime. We discuss $p$-adic properties of various arithmetical functions related to the coefficients of modular form and generating functions. Modular forms are considered as a tool of solving arithmetical problems. Examples of…

In this article, we introduce pseudo-absolute values, which generalise usual absolute values. Roughly speaking, a pseudo-absolute value on a field $K$ is a map $|\cdot| : K \to [0,+\infty]$ satisfying axioms similar to those of usual…

We describe all metric spaces that have sufficently many affine functions. As an application we obtain a metric characterization of linear-convex subsets of Banach spaces.

We study the set of algebraic numbers of bounded height and bounded degree where an analytic transcendental function takes algebraic values.

The study of prime divisibility plays a crucial role in number theory. The $p$-adic valuation of a number is the highest power of a prime, $p$, that divides that number. Using this valuation, we construct $p$-adic valuation trees to…

The $p$-adic completion $\mathbb{Q}_p$ of the rational numbers induces a different absolute value $|\cdot|_p$ than the typical $| \cdot |$ we have on the real numbers. In this paper we compare and contrast functions $f \colon \mathbb{R}^{+}…

We characterize those valued fields for which the image of the valuation ring under every polynomial in several variables contains an element of maximal value, or zero.

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…

In this brief note, we consider p-adic unit roots or poles of L-functions of exponential sums defined over finite fields. In particular, we look at the number of unit roots or poles, and a congruence relation on the units. This raises a…

Existing machine learning frameworks operate over the field of real numbers ($\mathbb{R}$) and learn representations in real (Euclidean or Hilbert) vector spaces (e.g., $\mathbb{R}^d$). Their underlying geometric properties align well with…

Functions whose composition with every metric is a metric are said to be metric-preserving. In this article, we investigate a variation of the concept of metric-preserving functions where metrics are replaced by ultrametrics.

We show that Hermite's approximations to values of the exponential function at given algebraic numbers are nearly optimal when considered from an adelic perspective. We achieve this by taking into account the ratio of these values whenever…

We generalize the concept of a field by allowing addition to be a partial operation. We show that elements of such a "partially additive field" share many similarities with physical quantities. In particular, they form subsets of mutually…

The article is devoted to approximate, global and along curves differentiability of functions over non-archimedean infinite fields with non-trivial valuations. Fields with zero and non-zero characteristics are considered. Spaces of…