Bounds on the number of real solutions to polynomial equations

Daniel J. Bates; Frédéric Bihan; Frank Sottile

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arXiv · 0706.4134

Bounds on the number of real solutions to polynomial equations

Daniel J. Bates·Frédéric Bihan·Frank Sottile

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Abstract

We use Gale duality for polynomial complete intersections and adapt the proof of the fewnomial bound for positive solutions to obtain the bound (e^4+3) 2^(k choose 2) n^k/4 for the number of non-zero real solutions to a system of n polynomials in n variables having n+k+1 monomials whose exponent vectors generate a subgroup of Z^n of odd index. This bound exceeds the bound for positive solutions only by the constant factor (e^4+3)/(e^2+3) and it is asymptotically sharp for k fixed and n large.

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BibTeX RIS

Daniel J. Bates, Frédéric Bihan, Frank Sottile. 2007-10-03. Bounds on the number of real solutions to polynomial equations. https://arxiv.org/abs/0706.4134

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