Related papers: Continuous rational functions on real and $p$-adic…
A result of Graber, Harris, and Starr shows that a rationally connected variety defined over the function field of a curve over the complex numbers always has a rational point. Similarly, a separably rationally connected variety over a…
We single out some problems of Schubert calculus of subspaces of codimension 2 that have the property that all their solutions are real provided that the data are real. Our arguments explore the connection between subspaces of codimension 2…
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…
Differentially-algebraic (D-algebraic) functions are solutions of polynomial equations in the function, its derivatives, and the independent variables. We revisit closure properties of these functions by providing constructive proofs. We…
We study the computational complexity of fundamental problems over the $p$-adic numbers ${\mathbb Q}_p$ and the $p$-adic integers ${\mathbb Z}_p$. Gu\'epin, Haase, and Worrell proved that checking satisfiability of systems of linear…
We show that three problems involving linear difference equations with rational function coefficients are essentially equivalent. The first problem is the generalization of the classical Skolem-Mahler-Lech theorem to rational function…
Apart from an account of classical preliminaries, this volume contains a systematic introduction to Sobolev spaces and functions of bounded variation with selected applications. This is installment III of a four part discussion of certain…
We described a wide class of $p$-adic refinable equations generating $p$-adic multiresolution analysis. A method for the construction of $p$-adic orthogonal wavelet bases within the framework of the MRA theory is suggested. A realization of…
We prove that the subdifferential of any semi-algebraic extended-real-valued function on $\R^n$ has $n$-dimensional graph. We discuss consequences for generic semi-algebraic optimization problems.
The problem is considered as to whether a monotone function defined on a subset P of a Euclidean space can be strictly monotonically extended to the whole space. It is proved that this is the case if and only if the function is {\em…
The concepts of amenable and compatible functions have been introduced in a recent work, in order to state precise mathematical theorems that guarantee that a backward stable algorithm is also forward stable, and that the composition of two…
This paper investigates the p-adic valuation trees of degree-2 and degree-3 polynomials in two variables over any prime p, building upon prior research outlined in [14].
Using an adelic approach we simultaneously consider real and p-adic aspects of dynamical systems whose states are mapped by linear fractional transformations isomorphic to some subgroups of GL (2, Q), SL (2, Q) and SL (2, Z) groups. In…
The functional equation f(p(z))=g(q(z)) is studied, where p,q are polynomials and f,g are trancendental meromorphic functions in C. We find all the pairs p,q for which there exist nonconstant f,g satisfying our equation and there exist no…
We give two elementary proofs, at a level understandable by students with only pre-calculus knowledge of Algebra, of the well known fact that an irreducible irrational n-th root of a positive rational number cannot be solution of a…
In this paper we present some results related to the problem of finding periodic representations for algebraic numbers. In particular, we analyze the problem for cubic irrationalities. We show an interesting relationship between the…
Our main result is that any real cubic algebraic number has a continued fraction expansion with polynomial coefficients. Some generalizations are mentioned.
Classification theorems for linear differential equations in two real variables, possessing eigenfunctions in the form of the polynomials (the generalized Bochner problem) are given. The main result is based on the consideration of the…
We present algorithms revealing new families of polynomials allowing sub-exponential detection of p-adic rational roots, relative to the sparse encoding. For instance, we show that the case of honest n-variate (n+1)-nomials is doable in NP…
These informal notes are concerned with spaces of functions in various situations, including continuous functions on topological spaces, holomorphic functions of one or more complex variables, and so on.