A nullstellensatz for linear partial differential equations with polynomial coefficients

TL;DR

This paper establishes a general formula for the Weyl closure of systems of linear partial differential equations with polynomial coefficients, extending previous results to cases without finite rank assumptions and including real coefficients.

Contribution

It proves that the Weyl closure formula holds in full generality for systems of linear PDEs with polynomial coefficients, regardless of module rank or coefficient field.

Findings

The Weyl closure formula is valid in general for polynomial coefficient PDE systems.

The approach applies to both real and complex coefficients.

The proof is constructive, based on Riquier-Janet theory.

Abstract

In this paper an equation means a homogeneous linear partial differential equation in $n$ unknown functions of $m$ variables which has real or complex polynomial coefficients. The solution set consists of all $n$-tuples of real or complex analytic functions that satisfy the equation. For a given system of equations we would like to characterize its Weyl closure, i.e. the set of all equations that vanish on the solution set of the given system. It is well-known that in many special cases the Weyl closure is equal to $B_m(F)N \cap A_m(F)^n$ where $F$ is either the field of real or complex numbers, $A_m(F)$ (respectively $B_m(F)$) consists of all linear partial differential operators with coefficients in $F[x_1,\ldots,x_m]$ (respectively $F(x_1,\ldots,x_m)$) and $N$ is the submodule of $A_m(F)^n$ generated by the given system. Our main result is that this formula holds in general. In particular, we do not assume that the module $A_m(F)^n/N$ has finite rank which used to be a standard assumption. Our approach works also for the real case which was not possible with previous methods. Moreover, our proof is constructive as it depends only on the Riquier-Janet theory.