Polynomial systems with few real zeroes |
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Authors: | Benoît Bertrand Frédéric Bihan Frank Sottile |
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Affiliation: | (1) Section de mathematiques, Université de Genève, 2-4, rue du Lièvre, Case postale 64, 1211 Genève 4, Suisse;(2) Laboratoire de Mathématiques, Université de Savoie, Le Bourget-du-Lac Cedex, 73376, France;(3) Department of Mathematics, Texas A&M University, College Station, TX 77843, USA |
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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 ℤn, this bound is 2n+1, while the Khovanskii bound is exponential in n2. The bound 2n+1 can be attained only for non-degenerate circuits. Our methods involve a mixture of combinatorics, geometry, and arithmetic. Part of work done at MSRI was supported by NSF grant DMS-9810361. Work of Sottile is supported by the Clay Mathematical Institute. Sottile and Bihan were supported in part by NSF CAREER grant DMS-0134860. Bertrand is supported by the European research network IHP-RAAG contract HPRN-CT-2001-00271. |
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