Partial Differential Polynomials Research Articles

Some Results Regarding Partial Differential Polynomials and the Uniqueness of Meromorphic Functions in Several Variables

Some Results Regarding Partial Differential Polynomials and the Uniqueness of Meromorphic Functions in Several Variables

Ritt-Wu Characteristic Set Method for Laurent Partial Differential Polynomial Systems

In this paper, a Ritt-Wu characteristic set method for Laurent partial differential polynomial systems is presented. The concept of Laurent regular differential chain is defined and its basic properties are proved. The authors give a partial method to decide whether a Laurent differential chain A is Laurent regular. The decision for whether A is Laurent regular is reduced to the decision of whether a univariate differential chain A1 is Laurent regular. For a univariate differential chain A1, the authors first give a criterion for whether A1 is Laurent regular in terms of its generic zeros and then give partial results on deciding whether A1 is Laurent regular.

Effective uniform bounding in partial differential fields

Motivated by the effective bounds found in [12] for ordinary differential equations, we prove an effective version of uniform bounding for fields with several commuting derivations. More precisely, we provide an upper bound for the size of finite solution sets of partial differential polynomial equations in terms of data explicitly given in the equations and independent of parameters. Our methods also produce an upper bound for the degree of the Zariski closure of solution sets, whether they are finite or not.

Jacobi’s bound for systems of algebraic differential equations

This review paper is devoted to the Jacobi bound for systems of partial differential polynomials. We prove the conjecture for the system of n partial differential equations in n differential variables which are independent over a prime differential ideal \(\mathfrak{p}\). On the one hand, this generalizes our result about the Jacobi bound for ordinary differential polynomials independent over a prime differential ideal \(\mathfrak{p}\) and, on the other hand, the result by Tomasovic, who proved the Jacobi bound for linear partial differential polynomials.

Jacobi’s bound for independent systems of algebraic partial differential equations

In this paper we prove Jacobi’s bound for systems of n partial differential polynomials in n differential variables, which are independent over a prime differential ideal . This generalizes on the one hand our result (Kondratieva et al. in On Jacobi’s bound for systems of differential polynomials, algebra. Moscow University Press, Moscow, pp 79–85, 1982) about Jacobi’s bound for ordinary differential polynomials independent over a prime differential ideal and, on the other hand, the result by Tomasovic (A generalized Jacobi conjecture for arbitrary systems of algebraic differential equations. PhD thesis, Columbia University, 1976), who proposed a version of Jacobi’s bound for partial differential polynomials and proved it in the linear case. In Kondratieva et al. (Differential and difference dimension polynomials, Kluwer, Dordrecht, 1999) we gave another proof of this result by Tomasovic. The exposition in the present paper follows our proof in Kondratieva et al. (Differential and difference dimension polynomials, Kluwer, Dordrecht, pp 273–280, 1999).

The uniqueness problem and meromorphic solutions of partial differential equations

A general uniqueness theorem is proved for meromorphic functions in Cn which share three distinct small functions with their linear partial differential polynomials. As a consequence, a necessary and sufficient condition in terms of shared values for a meromorphic function to be a solution of a linear partial differential equation of constant coefficients is obtained.

On the shared values of entire functions and their partial differential polynomials in

On the shared values of entire functions and their partial differential polynomials in

A resolution of the Euler operator II

AbstractAn exact sequence resolving the Euler operator of the calculus of variations for partial differential polynomials in several dependent and independent variables is described. This resolution provides a solution to the ‘Inverse problem of the calculus of variations’ for systems of polynomial partial equations.That problem consists of characterizing those systems of partial differential equations which arise as the Euler-Lagrange equations of some variational principle. It can be embedded in the more general problem of finding a resolution of the Euler operator. In (3), hereafter referred to as I, a solution of this problem was given for the case of one independent and one dependent variable. Here we generalize this resolution to several independent and dependent variables simultaneously. The methods employed are similar in spirit to the algebraic techniques associated with the Gelfand-Dikii transform in I, although are considerably complicated by the appearance of several variables. In particular, a simple algebraic proof of the local exactness of a complex considered by Takens(5), Vinogradov(6), Anderson and Duchamp(1), and others appears as part of the resolution considered here.

The Low Power Theorem for Partial Differential Polynomials

In a recent paper of J. F. Ritt' it is shown that every essential irreducible manifold in the manifold of a partial differential polynomial (p.d.p.) is the gene, ral solution of some p.d.p.; or, equivalently, that the prime ideal corresponding to each component has a basic set consisting of a single p.d.p. However, the general solution of a p.d.p. might be an irreducible manifold in the manifold of a given p.d.p. and yet not be essential. This paper presents a method for determining whether or not such a manifold is essential, thereby extending to p.d.p.'s results previously obtained for ordinary d.p.'s by Ritt2 and the author3. Let 9f be a partial differential field of characteristic zero with mn operations of differentiation. Let (f{jyi, ... **, I = WFAy} denote the ring of p.d.p.'s in the unknowns y', *.. , yn with coefficients in St. It is assumed that the yi and their derivatives are completely ordered by means of marks. Unless the contrary is stated, any p.d.p. mentioned below is supposed to belong to Ifly). Let F and A be two algebraically irreducible p.d.p.'s, such that the general solution of A is contained in the manifold of F. Let S be the separant of A. By slightly modifying a result of M.D.P. we shall show that there is an integer t such that StF has a unique representation as a polynomial in A and certain of its partial derivatives Ai

On the Manifolds of Partial Differential Polynomials

In a paper which will follow shortly in these Annals, we shall treat the analytical theory of the singular solutions of partial differential equations of the first order. It appears desirable to make clear first the nature of the irreducible components of the manifold of a partial differential polynomial (p.d.p. or even d.p.). We first define and discuss the general solution of an algebraically irreducible p.d.p. It is then proved that every irreducible component of the manifold of a p.d.p. is the general solution of some p.d.p.; that is, the prime ideal corresponding to the component has a basic set composed of a single d.p. This result generalizes a theorem previously established for ordinary differential polynomials.2