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Counting Isolated Roots of Trinomial Systems in the Plane and Beyond

Tien-Yien Li·J. Maurice Rojas·Xiaoshen Wang
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Abstract

We prove that any pair of bivariate trinomials has at most 5 isolated roots in the positive quadrant. The best previous upper bounds independent of the polynomial degrees counted only non-degenerate roots and even then gave much larger bounds, e.g., 248832 via a famous general result of Khovanski. Our bound is sharp, allows real exponents, and extends to certain systems of n-variate fewnomials, giving improvements over earlier bounds by a factor exponential in the number of monomials. We also derive new bounds on the number of real connected components of fewnomial hypersurfaces.

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Tien-Yien Li, J. Maurice Rojas, Xiaoshen Wang. 2001-05-29. Counting Isolated Roots of Trinomial Systems in the Plane and Beyond. https://arxiv.org/abs/math/0008069

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