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Polynomial systems with few real zeroes

Bihan, Frédéric
Sottile, Frank
Published in Mathematische Zeitschrift. 2006, vol. 253, no. 2, p. 361 - 385
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.
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BERTRAND, Benoît, BIHAN, Frédéric, SOTTILE, Frank. Polynomial systems with few real zeroes. In: Mathematische Zeitschrift, 2006, vol. 253, n° 2, p. 361 - 385. doi: 10.1007/s00209-005-0912-8 https://archive-ouverte.unige.ch/unige:12116

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Deposited on : 2010-10-15

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