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An algebraic theorem of E. Fischer, and the holomorphic Goursat problem. (English) Zbl 0706.35034
The basic problem discussed in this paper is whether the differential operator $$f\mapsto P(D)(Qf)$$ is injective, surjective or both, as a map from $$E_ n$$ to itself. Here $$E_ n$$ denotes the space of entire functions of n complex variables. P and Q are polynomials on n letters with complex coefficients and $$D=(D_ 1,...,D_ n)$$, $$D_ k=\partial /\partial z_ k$$. Theorem 1 shows, using a classical algebraic theorem of E. Fischer, that this map is bijective when P and Q are homogeneous and $$P=Q^*$$ where $$Q^*$$ denotes the polynomial derived from Q by conjugating its coefficients. Theorem 3 shows that bijectivity persists when Q is homogeneous and $$P=Q^*-1.$$
In a corollary to this theorem a proof of bijectivity is sketched in the case P homogeneous and $$Q=P^*-1$$. As remarked on p. 536 the hypothesis of homogeneity can be weakened to generalized homogeneity based on positive weights; this idea also is found in the work of E. Fischer. Relations of the indicated results to real initial value problems are indicated, and to eigenfunction expansions in $$E_ n$$. Many open questions are stated.
Reviewer: H.S.Shapiro

##### MSC:
 35E15 Initial value problems for PDEs and systems of PDEs with constant coefficients 35A10 Cauchy-Kovalevskaya theorems 35A20 Analyticity in context of PDEs
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