The existence of linear differential resolvents for z for any root z of an ordinary polynomial with coefficients in a given ordinary differential field has been established, where α is an indeterminate constant with respect to the derivation of the given field. In this paper we consider several alphas. We will call a finite sum of indeterminate powers of a variable v a pseudopolynomial in v . W...

Let $alpha$ be an endomorphism and $delta$ an $alpha$-derivationof a ring $R$.  In this paper we  study the relationship between an$R$-module $M_R$ and the  general polynomial module  $M[x]$ over theskew polynomial ring $R[x;alpha,delta]$. We introduce the notionsof skew-Armendariz modules and skew quasi-Armendariz modules whichare generalizations of $alpha$-Armendariz modules and extend thecla...

Given a finite set of polynomial, multivariate, and vector-valued functions, we show that their span can be written as the solution set of a linear system of partial differential equations (PDE) with polynomial coefficients. We present two different but equivalent ways to construct a PDE system whose solution set is precisely the span of the given trajectories. One is based on commutative algeb...

Polynomial and rational solutions for linear ordinary differential equations can be obtained by algorithmic methods. For instance, the maple package DEtools provides efficient functions polysols and ratsols to find polynomial and rational solutions for a given linear ordinary differential equation with rational function coefficients. A natural analogue of the notion of linear ordinary different...

An analogue of the Eisenstein irreducibility criterion is developed for linear differential operators, or, more generally, noncommutative polynomials, and is applied to a few simple examples. Introduction. The question of the irreducibility of an ordinary linear differential operator is of interest in the Picard-Vessiot theory (see Kolchin [1, §22]). Indeed, the operator is irreducible if and o...

Let $T$ be a polynomially bounded o-minimal theory extending the of real closed ordered fields. $K$ model equipped with $T$-convex valuation ring and $T$-derivation. If this derivation is continuous respect to topology, then we call $T$-differential field. We show that every field has an immediate strict extension which spherically complete. In some important cases, assumption polynomial bounde...