Related papers: Ordinary differential equations with only entire s…

The aim of this paper is a quantitative analysis of the solution set of a system of polynomial nonlinear differential equations, both in the ordinary and partial case. Therefore, we introduce the differential counting polynomial, a common…

We propose a sufficient condition of the convergence of a generalized power series formally satisfying an algebraic (polynomial) ordinary differential equation. The proof is based on the majorant method.

Polynomials related to rational solutions of Painleve' equations satisfy certain difference equations. Conditions are given to acertain that all solutions really are polynomials.

Some properties and relations satisfied by the polynomial solutions of a bispectral problem are studied. Given a finite order differential operator, under certain restrictions, its polynomial eigenfunctions are explicitly obtained, as well…

Let $\mathcal{A}(p)$ be the class consisting of functions $f$ that are holomorphic in $\ID\setminus \{p\}$, $p\in (0,1)$ possessing a simple pole at the point $z=p$ with nonzero residue and normalized by the condition $f(0)=0=f'(0)-1$. In…

We give an optimal necessary and sufficient condition for the quotient polynomial and remainder in the division algorithm to have positive coefficients.

Assume that $n$ is a positive integer, $p_{j}$ ($j=1,2, \cdots, 6)$ are polynomials, $p$ is an irreducible polynomial, and $f$ is an entire function on $\mathbb{C}^{n}.$ Let $ L(f)=\sum_{j=1}^s q_{t_j}f_{z_{t_j}}$ and…

We investigate various properties of p-adic differential equations which have as a solution an analytic function of the form $F_k (x) = \sum_{n\geq 0} n! P_k (n) x^n$, where $P_k (n) = n^k + C_{k-1} n^{k-1} + ...+ C_0$ is a polynomial in n…

In present paper we propose an approach based on examination of the structure of the general solution of equations of the type dy/dx=P(x,y)/Q(x,y), with P and Q polynomials only in y. Under the term structure we mean the dependency…

In order to prove that the P of problems is different to the NP class, we consider the satisfability problem of propositional calculus formulae, which is an NP-complete problem. It is shown that, for every search algorithm A, there is a set…

In this paper we show that there is no algorithm to decide whether an arbitrarily given polynomial equation $P(z_1,\ldots,z_{52})=0$ (with integer coefficients) over the Gaussian ring $\mathbb Z[i]$ is solvable.

We consider systems of ordinary differential equations with quadratic homogeneous right hand side. We give a new simple proof of a result already obtained in [8,10] which gives the necessary conditions for the existence of polynomial first…

We construct polynomial solutions of the KZ differential equations over a finite field $F_p$ as analogs of hypergeometric solutions.

We prove that the binary complexity of solving ordinary polynomial differential equations in terms of Puiseux series is single exponential in the number of terms in the series. Such a bound was given by Grigoriev [10] for Riccatti…

Properties of partial integrals such as real and complex-valued polynomial, multiple polynomial, exponential, and conditional for ordinary differential systems are studied. The possibilities of constructing first integrals and last…

Let $\mathbb{Z}_p[x]$ be the set of all functions whose coefficients are in the field of $p$-adic integers $\mathbb{Z}_p$. This work considers a problem of finding a root of a polynomial equation $P(x)=0$ where $P(x)\in\mathbb{Z}_p[x]$. The…

Differential-elimination algorithms apply a finite number of differentiations and eliminations to systems of partial differential equations. For systems that are polynomially nonlinear with rational number coefficients, they guarantee the…

We show that for every non-negative integer d, there exist differential equations w''+Pw=0, where P is a polynomial of degree d, such that some non-trivial solution w has all zeros real.

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 establish the existence of a fully nontrivial solution with nonnegative components for a weakly coupled competitive system for the $p$-Laplacian in $\mathbb{R}^N$ whose nonlinear terms are purely critical. We also show that the purely…