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摘要翻译：
Kushnirenko和Bernstein的一个定理表明，环面上多项式系的孤立根个数以给定多项式的牛顿多边形的混合体积为界，并且这个上界是一般精确的。通过引入多项式的精化组合不变量和凸体混合体积的推广&凹函数的混合积分，改进了这一结果。这一证明是基于新的技术和相对多曲面几何的结果。
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英文标题：
《A refinement of the Kushnirenko-Bernstein estimate》
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作者：
Patrice Philippon and Martin Sombra
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最新提交年份：
2007
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分类信息：
一级分类：Mathematics 数学
二级分类：Algebraic Geometry 代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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一级分类：Mathematics 数学
二级分类：Combinatorics 组合学
分类描述：Discrete mathematics, graph theory, enumeration, combinatorial optimization, Ramsey theory, combinatorial game theory
离散数学，图论，计数，组合优化，拉姆齐理论，组合对策论
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英文摘要：
A theorem of Kushnirenko and Bernstein shows that the number of isolated roots of a system of polynomials in a torus is bounded above by the mixed volume of the Newton polytopes of the given polynomials, and this upper bound is generically exact. We improve on this result by introducing refined combinatorial invariants of polynomials and a generalization of the mixed volume of convex bodies: the mixed integral of concave functions. The proof is based on new techniques and results from relative toric geometry.
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PDF链接：
https://arxiv.org/pdf/0709.3306
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