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On projectivization of zeros of parabolic polynomials. (English. Russian original) Zbl 0964.35028
Dokl. Math. 55, No. 1, 61-62 (1997); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 352, No. 3, 307-308 (1997).
Summary: It is shown that the projective closure and the difference between the projective closure and the affine part of a level hypersurface of a parabolic polynomial in \(\mathbb{R}^n\) are connected. (A polynomial is called parabolic if it satisfies the heat equation.) This result is obtained under the assumption that the number of critical points of the parabolic polynomial \(F\) in \(\mathbb{R}^n\) is finite. The proof is based on the maximum principle.
MSC:
35E20 General theory of PDEs and systems of PDEs with constant coefficients
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35B50 Maximum principles in context of PDEs
35K05 Heat equation