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The number of components in the complement to a level surface of a partially parabolic polynomial. (English. Russian original) Zbl 1073.14561
Math. Notes 69, No. 5, 730-732 (2001); translation from Mat. Zametki 69, No. 5, 798-800 (2001).
Summary: It is known that the number of components in the complement to the set of zeros of a degree \(m\) polynomial in \(\mathbb{R}^n\) is not greater than \(m^n+ O(m^{n-1})\) [see D. A. Gudkov, Russ. Math. Surveys 29, 3–79 (1974; Zbl 0316.14018)]. In the present paper, we study \(l\)-parabolic polynomials \(F\) in \(\mathbb{R}^n\), i.e., polynomials satisfying the following equation \[ {\partial F\over\partial x_1}= {\partial^2 F\over\partial x^2_2}+\cdots+ {\partial^2 F\over\partial x^2_l},\quad n\geq 3,\quad 2\leq l\leq n, \] which is the heat conduction equation in a part of the variables. We show that the number of components in the complement to a level surface of \(F\) does not exceed \(2m^{n-1}+ O(m^{n-2})\). We also deduce an upper bound for the number of connected components of a level hypersurface of \(F\). The results of the present paper generalize and sharpen those of our previous article [Russ. Acad. Sci., Dokl., Math. 50, 90–91 (1995; Zbl 0849.12001)].
MSC:
14P25 Topology of real algebraic varieties
57R45 Singularities of differentiable mappings in differential topology
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