Related papers: Power Series with Coefficients from a Finite Set

We obtain a fully explicit quantitative version of the Eisenstein theorem on algebraic power series which is more suitable for certain applications than the existing version due to Dwork, Robba, Schmidt and van der Poorten. We also treat…

We give an easy proof of Sch\"utzenberger's Theorem stating that non-commutative formal power series are rational if and only if they are recognisable. A byproduct of this proof is a natural metric on a subgroup of invertible rational…

We consider D-finite power series $f(z)=\sum a_n z^n$ with coefficients in a number field $K$. We show that there is a dichotomy governing the behaviour of $h(a_n)$ as a function of $n$, where $h$ is the absolute logarithmic Weil height. As…

Suppose that we are given a formal power series of many variables with coefficients in $\mathbb{R}$ (or $\mathbb{C}$) and we want to compute its $n$-th (multiplicative) root. As can be expected coefficients of the root have to satisfy a…

We consider Knizhnik-Zamolodchikov system of linear differential equations. The coefficients of this system are rational functions. We prove that under some conditions the solution of KZ system is rational too. This assertion confirms…

In the work we have considered p-adic functional series with binomial coefficients and discussed its p-adic convergence. Then we have derived a recurrence relation following with a summation formula which is invariant for rational argument.…

The problem to decide whether a given multivariate (quasi-)rational function has only positive coefficients in its power series expansion has a long history. It dates back to Szego in 1933 who showed certain quasi-rational function to be…

A sequence of coefficients that appeared in the evaluation of a rational integral has been shown to be unimodal. An alternative proof is presented.

A (noncommutative) P\'olya series over a field $K$ is a formal power series whose nonzero coefficients are contained in a finitely generated subgroup of $K^\times$. We show that rational P\'olya series are unambiguous rational series,…

In [A. Berele, Computing super matrix invariants, {\it Advances in Applied Math. \bf48} (2012), 273--289.] we defined integrals that approximated the Poincar\'e series of the invariants and concomitants of the general linear Lie supergroup…

We study properties of coefficients of a linear form, originating from a multiple integral. As a corollary, we prove Vasilyev's conjecture, connected with the problem of irrationality of the Riemann zeta function at odd integers.

In this paper we study the set of rational solutions of equations defined by power sums symmetric polynomials with coefficients in a finite field. We do this by means of applying a methodology which relies on the study of the geometry of…

Polynomial interpretations are a useful technique for proving termination of term rewrite systems. They come in various flavors: polynomial interpretations with real, rational and integer coefficients. As to their relationship with respect…

We say a power series $\sum_{n=0}^\infty a_n q^n$ is multiplicative if the function $n\mapsto a_n/a_1$ ($n\ge 1$) is so. In this paper, we consider multiplicative power series $f$ such that $f^2$ is also multiplicative. We find various…

A system of multivariate formal power series $\varphi$ with a homogeneous decomposition $\varphi=\sum_{k=0}^\infty\varphi_k$ is invertible under composition if $\varphi_0=0$ and $\mathrm{det}(\varphi_1)\ne 0.$ All invertible series over a…

Let k be a number field. It is well known that the set of sequences composed by Taylor coefficients of rational functions over k is closed under component-wise operations, and so it can be equipped with a ring structure. A conjecture due to…

In this short note, a general result concerning the positivity, under some conditions, of the coefficients of a power series is proved. This allows us to answer positively a question raised by Guo (2010) about the sign of the coefficients…

In 1923 Schur considered the following problem. Let f(X) be a polynomial with integer coefficients that induces a bijection on the residue fields Z/pZ for infinitely many primes p. His conjecture, that such polynomials are compositions of…

Let $X \subset \mathbb{P}(w_0, w_1, w_2, w_3)$ be a quasismooth well-formed weighted projective hypersurface and let $L = lcm(w_0,w_1,w_2,w_3)$. We characterize when $X$ is rational under the assumption that $L$ divides $deg(X)$ by…

Let $(W,S)$ be a Coxeter system of finite rank and let $J,K\subset S$. We study the rationality of the Poincar\'e series of the set of representatives of minimal length of $(W_J,W_K)$-double cosets of $W$: we conclude that it depends mostly…