Related papers: Solutions of equations involving the modular $j$ f…
We analyze the polynomial solutions of a nonlinear integral equation, generalizing the work of C. Bender and E. Ben-Naim. We show that, in some cases, an orthogonal solution exists and we give its general form in terms of kernel…
In this paper, we define finitely additive, probability and modular functions over semiring-like structures. We investigate finitely additive functions with the help of complemented elements of a semiring. We also generalize some classical…
A well-known generalisation of positional numeration systems is the case where the base is the residue class of $x$ modulo a given polynomial $f(x)$ with coefficients in (for example) the integers, and where we try to construct finite…
Effective computation of resultants is a central problem in elimination theory and polynomial system solving. Commonly, we compute the resultant as a quotient of determinants of matrices and we say that there exists a determinantal formula…
One of the aims of this article is to provide a class of polynomial mappings for which the Jacobian conjecture is true. Also, we state and prove several global univalence theorems and present a couple of applications of them.
We classify general systems of polynomial equations with a single solution, or, equivalently, collections of lattice polytopes of minimal positive mixed volume. As a byproduct, this classification provides an algorithm to evaluate the…
We prove an extension of the Bourgain-Sarnak-Ziegler theorem and then apply it to bound certain polynomial exponential sums with modular coefficients.
We consider the theory of algebraically closed fields of characteristic zero with multivalued operations $x\mapsto x^r$ (raising to powers). It is in fact the theory of equations in exponential sums. In an earlier paper we have described…
In this paper we define a class of polynomial functors suited for constructing coalgebras representing processes in which uncertainty plays an important role. In these polynomial functors we include upper and lower probability measures,…
The aim of the work is to construct new polynomial systems, which are solutions to certain functional equations which generalize the second-order differential equations satisfied by the so called classical orthogonal polynomial families of…
A general structure theorem on higher order invariants is proven. For an arithmetic group, the structure of the corresponding Hecke module is determined. It is shown that the module does not contain any irreducible submodule. This explains…
In this note, we study the arithmetic nature of values of modular functions, meromorphic modular forms and meromorphic quasi-modular forms with respect to arbitrary congruence subgroups, that have algebraic Fourier coefficients. This…
In this paper we prove the generalized Kaplansky conjecture for the Jordan algebras of the type $J_n$ in particular for self adjoint $2\times 2$ matrices over $\R$, over $\C$, $\HH$ and $\Oct$. In fact, we prove that the image of…
The known facts about solvability of equations over groups are considered from a more general point of view. A generalized version of the theorem about solvability of unimodular equations over torsion-free groups is proved. In a special…
The theory of partition congruences has been a fascinating and difficult subject for over a century now. In attempting to prove a given congruence family, multiple possible complications include the genus of the underlying modular curve,…
In this paper, we first prove the coefficient conjecture of Clunie and Sheil-Small for a class of univalent harmonic functions which includes functions convex in some direction. Next, we prove growth and covering theorems and some related…
We develop new polynomial methods for studying systems of word equations. We use them to improve some earlier results and to analyze how sizes of systems of word equations satisfying certain independence properties depend on the lengths of…
We investigate whether the group algebra of a finite group over a localisation of the integers is semiperfect. The main result is a necessary and sufficient arithmetic criterion in the ordinary case. In the modular case, we propose a…
Every abelian (and even every nilpotent) group contains a solution of any finite unimodular system of equations over itself. However, this is not true for infinite systems. We deduced a criterion for a periodic abelian group to contain a…
A general theory of vector-valued modular functions, holomorphic in the upper half-plane, is presented for finite dimensional representations of the modular group. This also provides a description of vector-valued modular forms of arbitrary…