Real Algebraic and Analytic Geometry

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210. Frédéric Bihan, Frank Sottile:
New Fewnomial Upper Bounds from Gale Dual Polynomial Systems.

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Submission: 2007, January 29.

We show that there are fewer than (e^2+3)/4 \cdot 2^{k\atop 2}n^k positive solutions to a fewnomial system consisting of n polynomials in n variables having a total of n+k+1 distinct monomials. This is significantly smaller than Khovanskii’s fewnomial bound 2^{n+k\atop 2}(n+1)^{n+k}. We reduce the original system to a system of k equations in variables which depends upon the vector configuration Gale dual to the exponents monomials in the original system. We then bound the number of solutions to this system. We adapt these methods to show that a hypersurface in the positive orthant R^n defined by a polynomial with n+k+1 monomials has at most C(k)n^{k-1} compact connected components. Our results hold for polynomials with real exponents.

Full text, 20p.: pdf 213k.

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