Multidimensional Systems and Signal Processing

, Volume 30, Issue 1, pp 363–372 | Cite as

A nullstellensatz for linear partial differential equations with polynomial coefficients

  • J. CimpričEmail author


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 \(B_m(\mathbb {F})N \cap A_m(\mathbb {F})^n\) where \(\mathbb {F}\in \{\mathbb {R},\mathbb {C}\}\), the algebra \(A_m(\mathbb {F})\) (respectively \(B_m(\mathbb {F})\)) consists of all linear partial differential operators with coefficients in \(\mathbb {F}[x_1,\ldots ,x_m]\) (respectively \(\mathbb {F}(x_1,\ldots ,x_m)\)) and N is the submodule of \(A_m(\mathbb {F})^n\) generated by the given system. Our main result is that this formula holds in general. In particular, we do not assume that the module \(A_m(\mathbb {F})^n/N\) 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.


Algebraic theory of systems Symbolic methods for systems Weyl algebra Nullstellensatz Riquier bases 


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Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Mathematics and PhysicsUniversity of LjubljanaLjubljanaSlovenia

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