Partial Differential Equations in Module of Copolynomials of Several Variables over a Commutative Ring

TL;DR

This paper extends the theory of linear differential equations to copolynomials over a commutative ring, establishing existence, uniqueness, and fundamental solutions for infinite order differential operators.

Contribution

It introduces an algebraic framework for differential equations in copolymer modules, generalizing classical theorems to infinite order operators over rings.

Findings

Proved existence and uniqueness of solutions for infinite order differential equations.

Derived fundamental solutions and convolution representations.

Established Cauchy problem solutions in formal power series modules.

Abstract

We study the copolynomials of $n$ variables, i.e. $K$-linear mappings from the ring of polynomials $K[x_1,...,x_n]$ into the commutative ring $K$. We prove an existence and uniqueness theorem for a linear differential equation of infinite order which can be considered as an algebraic version of the classical Malgrange-Ehrenpreis theorem for the existence of the fundamental solution of a linear differential operator with constant coefficients. We find the fundamental solutions of linear differential operators of infinite order and show that the unique solution of the corresponding inhomogeneous equation can be represented as a convolution of the fundamental solution of this operator and the right-hand side. We also prove the existence and uniqueness theorem of the Cauchy problem for some linear differential equations in the module of formal power series with copolynomial coefficients.