Polynomial systems with few real zeroes

Benoit Bertrand; Frederic Bihan; Frank Sottile

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Polynomial systems with few real zeroes

Benoit Bertrand·Frederic Bihan·Frank Sottile

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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 $2n+1$, 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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BibTeX RIS

Benoit Bertrand, Frederic Bihan, Frank Sottile. 2005-04-05. Polynomial systems with few real zeroes. https://arxiv.org/abs/math/0502051

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