Partial Differential Equations in Module of Copolynomials of Several Variables over a Commutative Ring
S. L. Gefter, A. L. Piven'
TL;DR
The paper develops an algebraic analogue of the Malgrange-Ehrenpreis framework for copolynomials over a commutative ring, formulating linear differential operators of infinite order on the module of copolynomials and establishing existence, uniqueness, and convolution representations of solutions. It constructs the fundamental and Cauchy-solutions via convolution with a fundamental copolynomial and the Laplace transform, and proves that solvability and solution representations depend on whether the base ring contains the rational numbers. The approach yields explicit formulas for fundamental solutions in various settings (heat, Helmholtz, transport) and clarifies the interplay between operator theory and time-evolution problems in the copolynomial context. These results provide a purely algebraic, coefficient-wise framework for infinite-order PDEs with polynomial data, enriching the theory with exact, ring-theoretic solvability criteria and constructive solution formulas.
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.