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arXiv · 1008.4518

The sharp bound for the number of real solutions to polynomial equation systems

Sheng-Ming Ma
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Abstract

This paper solves the open problem on the sharp bound for the number of isolated solutions in $\mathbf{R}_*^n$ to the real system of $n$ polynomial equations in $n$ variables, i.e., the real $n$ by $n$ fewnomial system. For an unmixed system of $n$ polynomial equations in $n$ variables, this paper shows that the number of its positive solutions in $\mathbf{R}_*^n$ is sharply bounded by that of the simplex configurations in the triangulation of its support generically. The proof is based on a homotopic argument and an inductive triangulation of the support of the system via a hierarchy of pyramid configurations of different orders. For the mixed system of $n$ polynomial equations in $n$ variables, this paper shows that the maximal number of positive solutions in $\mathbf{R}_*^n$ to the systems with the same support is a symmetric multilinear function of the support generically and hence can be computed via the polarization identity.

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Sheng-Ming Ma. 2010-10-05. The sharp bound for the number of real solutions to polynomial equation systems. https://arxiv.org/abs/1008.4518

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