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The number of components of complements to level surfaces of partially harmonic polynomials. (English. Russian original) Zbl 1033.14035
Math. Notes 62, No. 6, 697-700 (1997); translation from Mat. Zametki 62, No. 6, 831-835 (1997).
The author proves the upper bound $$2m^{n-1}+O(m^{n-2})$$ for the number of connected components of the complement in $$\mathbb{R}^n$$ to the zero set of a real $$k$$-harmonic polynomial of degree $$m$$ in $$n$$ variables, where $$1\leq k\leq n$$ and $$n\geq 2$$ (compare with the fact that for arbitrary real polynomials of degree $$m$$ in $$n$$ variables the upper bound is $$m^n+O(m^{n-1})$$, and the exponent cannot be diminished). A real polynomial $$F$$ in $$n$$ variables is $$k$$-harmonic if it satisfies the Laplace equation with respect to $$k$$ variables: $(\partial^2/ \partial x^2_1+\cdots +\partial^2/ \partial x^2_k)F=0.$ The result follows from two observations:
(1) The complement in $$\mathbb{R}^n$$ to the zero set of a real $$k$$-harmonic polynomial in $$n$$, variables does not have bounded components (according to the maximum principle).
(2) The number of connected components of the complement in the sphere $$S^{n-1}$$ to the zero set of a real polynomial of degree $$m$$ in $$n$$ variables does not exceed $$2m^{n-1}+ O(m^{n-2})$$.
The author also proves sharper estimates under the assumptions that the singular set of the zero locus is compact or that the leading homogeneous part of the $$k$$-harmonic polynomial is nondegenerate.

##### MSC:
 14P25 Topology of real algebraic varieties 14J99 Surfaces and higher-dimensional varieties 14F45 Topological properties in algebraic geometry 57R45 Singularities of differentiable mappings in differential topology 58C25 Differentiable maps on manifolds
zero set
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##### References:
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