TL;DR
This paper extends Bezout-type theorems to semidegrees and subdegrees, providing explicit formulas and inequalities, and proves a Bernstein-type theorem for solutions of polynomial pairs on surfaces using convex geometry.
Contribution
It introduces new Bezout-type theorems for semidegrees and subdegrees, including explicit formulas and inequalities, and establishes a Bernstein-type theorem for polynomial solutions on surfaces.
Findings
Explicit formula for iterated semidegrees.
Inequality for subdegrees.
Bernstein-type theorem relating solutions to mixed volumes.
Abstract
In this sequel to arxiv:arXiv:1012.0835 we develop Bezout type theorems for semidegrees (including an explicit formula for {\em iterated semidegrees}) and an inequality for subdegrees. In addition we prove (in case of surfaces) a Bernstein type theorem for the number of solutions of two polynomials in terms of the mixed volume of planar convex polygons associated to them (via the theory of Kaveh-Khovanskii and Lazarsfeld-Mustata.
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Taxonomy
TopicsPolynomial and algebraic computation · Advanced Numerical Analysis Techniques · Commutative Algebra and Its Applications