Polynomial systems with few real zeroes

Details Polynomial systems with few real zeroes Polynomial systems with few real zeroes

2005-02-02

arxiv.org/abs/math/0502051v2

Mathematisches Zeitschrift, 253 (2006), no. 2, 361--385.

Mathematics

Algebraic Geometry

23 pages, 1 .eps figure. Revised Introduction

Scientific paper

We study some systems of polynomials whose support lies in the convex hull of a circuit, giving a sharp upper bound for their numbers of real solutions. This upper bound is non-trivial in that it is smaller than either the Kouchnirenko or the Khovanskii bounds for these systems. When the support is exactly a circuit whose affine span is ${\Z}^n$, this bound is $2n+1$, while the Khovanskii bound is exponential in $n^2$. The bound $2n+1$ can be attained only for non-degenerate circuits. Our methods involve a mixture of combinatorics, geometry, and arithmetic.

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