Counting Real Connected Components of Trinomial Curve Intersections and m-nomial Hypersurfaces

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dc.creatorLi, Tien-Yien
dc.creatorRojas, J. Maurice
dc.creatorWang, Xiaoshen
dc.date2002-12-12
dc.date.accessioned2026-07-07T04:53:45Z
dc.date.available2026-07-07T04:53:45Z
dc.descriptionWe 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 were much larger, e.g., 248832 (for just the non-degenerate roots) via a famous general result of Khovanski. Our bound is sharp, allows real exponents, allows degeneracies, 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 analogous sharpened bounds on the number of connected components of the real zero set of a single n-variate m-nomial.
dc.description27 pages, 2 figures. Extensive revision of math.CO/0008069. To appear in Discrete and Computational Geometry. Technique from main theorem (Theorem 1) now pushed as far as it will go. In particular, Theorem 1 now covers certain fewnomial systems of type (n+1,...,n+1,m) and certain non-sparse fewnomial systems. Also, a new result on counting non-compact connected components of fewnomial hypersurfaces (Theorem 3) has been added
dc.identifierhttps://arxiv.org/abs/math/0212178
dc.identifierhttp://arxiv.org/abs/math/0212178
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/65976
dc.subjectAlgebraic Geometry
dc.subjectCombinatorics
dc.titleCounting Real Connected Components of Trinomial Curve Intersections and m-nomial Hypersurfaces
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