Random polynomials with prescribed Newton polytope
Random polynomials with prescribed Newton polytope
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具有指定牛顿多面体的随机多项式
DOI：
10.1090/s0894-0347-03-00437-5
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发表时间：
2002
影响因子：
3.9
通讯作者：
S. Zelditch
中科院分区：
文献类型：
--
作者：
B. Shiffman;S. Zelditch
众所周知，多项式f的牛顿多面体Pf对它的零点有很大的影响，如关于给定牛顿多面体的m个多项式的同时零点的个数的Kouchnirenko-Bernstein定理。在这篇文章中，我们表明，Pf也有一个或几个多项式的零点分布的强烈影响。本文用通常的SU（m + 1）不变Gauss测度来刻画m复变N次（全纯）多项式空间，然后考虑其条件测度|NP在其牛顿多面体PfNP的多项式的子空间上导出。当P = λ，单位单形，那么很明显，众所周知，期望的零点分布Zf 1，.，fk（被认为是当前的）k个多项式f1，...，次数N的fk相对于Fubini-Study形式是一致的。我们的主要结果涉及条件期望E| NP（Zf1，...，fk）的零点的k多项式与牛顿多面体NPNp（其中p = deg P）;这些结果是渐近的比例因子N！1.我们表明，E| NP（Zf1，...，fk）通过矩映射μ在开标度多面体1 P的逆像AP上是渐近一致的：CP m！� 关于Projective Space然而，零点在AP之外具有奇异分布，并且当k = m时（Kouchnirenko-Bernstein定理的情况），AP之外的零点的百分比接近0，因为N！1.
The Newton polytope Pf of a polynomial f is well known to have a strong impact on its zeros, as in the Kouchnirenko-Bernstein theorem on the number of simultaneous zeros of m polynomials with given Newton polytopes. In this article, we show that Pf also has a strong impact on the distribution of zeros of one or several polynomials. We equip the space of (holomorphic) polynomials of degreeN in m complex variables with its usual SU(m + 1)-invariant Gaussian measure and then consider the conditional measures |NP induced on the subspace of polynomials whose Newton polytope PfNP. When P = �, the unit simplex, then it is obvious and well-known that the expected distribution of zeros Zf1,...,fk (regarded as a current) of k polynomials f1,...,fk of degree N is uniform relative to the Fubini-Study form. Our main results concern the conditional expectation E|NP(Zf1,...,fk ) of zeros of k polynomials with Newton polytope NPNp� (where p = deg P); these results are asymptotic as the scaling factor N ! 1. We show that E|NP(Zf1,...,fk ) is asymptotically uniform on the inverse image AP of the open scaled polytope 1 P ◦ via the moment map µ : CP m ! � for projective space. However, the zeros have an exotic distribution outside of AP and when k = m (the case of the Kouchnirenko-Bernstein theorem) the percentage of zeros outside AP approaches 0 as N ! 1.
环面流形拓扑
DOI：
--
发表时间：
2010
期刊：
影响因子：
--
作者：
Rappleye;J.;S.Choi
通讯作者：
S.Choi
可积本征函数的分布定律
DOI：
--
发表时间：
2005
期刊：
Annales de l'Institut Fourier 54 no5(予定)
影响因子：
--
作者：
B.Shiffman;T.Tate;S.Zelditch
通讯作者：
S.Zelditch