Related papers: The Geometry of Locally Bounded Rational Functions

We characterize functions which are growth types of Riemannian manifolds of bounded geometry.

This paper is devoted to proving the general {\L}ojasiewicz inequality, in both the definable and subanalytic cases, under the most relaxed assumptions. It means that we drop the usual continuity and compactness assumptions. In the second…

In our previous work, we have constructed explicit smooth real algebraic functions which may have both compact and non-compact preimages on smooth real algebraic manifolds. This paper presents its variant. Our result is new in obtaining…

Starting from a description of various generalized function algebras based on sequence spaces, we develop the general framework for considering linear problems with singular coefficients or non linear problems. Therefore, we prove…

In this paper, we determine explicit bases for Riemann--Roch spaces of linearized function fields, and we give a lower bound for the minimum distance of generalized algebraic geometry codes. As a consequence, we construct generalized…

We prove that a $k$-regulous function defined on a two-dimensional non-singular affine variety can be extended to an ambient variety. Additionally we derive some results concerning sums of squares of $k$-regulous functions; in particular we…

We consider A-hypergeometric functions associated to normal sets in the plane. We give a classification of all point configurations for which there exists a parameter vector such that the associated hypergeometric function is algebraic. In…

Variational analysis presents a unified theory encompassing in particular both smoothness and convexity. In a Euclidean space, convex sets and smooth manifolds both have straightforward local geometry. However, in the most basic hybrid case…

In this paper, an algebraic theory for local rings of finite embedding dimension is developed. Several extensions of (Krull) dimension are proposed, which are then used to generalize singularity notions from commutative algebra. Finally,…

One distinguishing feature of rational curves is that they have algebraic parameterizations. Arc spaces are a way of describing approximations to parameterizations of all curves in some fixed space. Playing on these descriptions, this paper…

We study the mean-median map as a dynamical system on the space of finite sets of piecewise-affine continuous functions with rational coefficients. We determine the structure of the limit function in the neighbourhood of a distinctive…

In this review article we present regularity properties of generalized functions which are useful in the analysis of non-linear problems. It is shown that Schwartz distributions embedded into our new spaces of generalized functions, with…

The \emph{Noetherian class} is a wide class of functions defined in terms of polynomial partial differential equations. It includes functions appearing naturally in various branches of mathematics (exponential, elliptic, modular, etc.). A…

We survey elementary results in Minkowski spaces (i.e. finite dimensional Banach spaces) that deserve to be collected together, and give simple proofs for some of them. We place special emphasis on planar results. Many of these results have…

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…

Given an Euclidean space, this paper elucidates the topological link between the partial derivatives of the Minkowski functional associated to a set (assumed to be compact, convex, with a differentiable boundary and a non-empty interior)…

We consider the functions that bound the dimensions of finite-dimensional associative or Lie algebras in terms of the dimensions of their commutative subalgebras. It is proved that these functions have quadratic growth. As a result, we also…

We obtain (essentially sharp) boundedness results for certain generalized local maximal operators between fractional weighted Sobolev spaces and their modifications. Concrete boundedness results between well known fractional Sobolev spaces…

computable functions are defined by abstract finite deterministic algorithms on many-sorted algebras. We show that there exist finite universal algebraic specifications that specify uniquely (up to isomorphism) (i) all abstract computable…

This article is an interdisciplinary review and an on-going progress report over the last few years made by myself and collaborators in certain fundamental subjects on two major theoretic branches in mathematics and theoretical physics:…