# Polynomial systems with few real zeroes

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## Abstract

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 2*n*+1, while the Khovanskii bound is exponential in *n* ^{2}. The bound 2*n*+1 can be attained only for non-degenerate circuits. Our methods involve a mixture of combinatorics, geometry, and arithmetic.

## Keywords

Convex Hull Real Solution Polynomial System Real Zero Affine Span
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