Real Algebraic and Analytic Geometry Preprint Server

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Submission: 2005, July 18.

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.

Mathematics Subject Classification (2000): 12D10, 14M25.

Keywords and Phrases: polynomial systems, real solutions, toric geometry, Fewnomials.

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