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

We prove that any pair of bivariate trinomials has at most five 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 shar...

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Bibliographic Details
Published in:Discrete & computational geometry 2003-09, Vol.30 (3), p.379-414
Main Authors: Li, Tien-Yien, Rojas, J. Maurice, Wang, Xiaoshen
Format: Article
Language:English
Subjects:
Geometry
Mathematical models
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Summary:We prove that any pair of bivariate trinomials has at most five 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. [PUBLICATION ABSTRACT]
ISSN:0179-5376
1432-0444
DOI:10.1007/s00454-003-2834-8