具有多项式增长系数的随机多项式的实根：比较原理及应用
Real roots of random polynomials with coefficients of polynomial growth: a comparison principle and applications
摘要：
本文进一步研究了非中心系数随机多项式的实根的分布。我们专注于多项式的系数的典型值有权力的增长和计数的实零点的平均数。以往的结果几乎都要求系数的均值为零，将这些结果推广到一般情况是很有意义的。我们的方法是基于一种新的比较原则，减少了一般情况下的均值为零的设置。作为应用，我们得到了Kac多项式、双曲随机多项式及其导数的新结果，以及这些多项式的推广。证明功能新的对数可积性估计随机多项式（本地和全球）和相当尖锐的估计本地的实零点的数量。
英文摘要：
This paper seeks to further explore the distribution of the real roots of random polynomials with non-centered coefficients. We focus on polynomials where the typical values of the coefficients have power growth and count the average number of real zeros. Almost all previous results require coefficients with zero mean, and it is non-trivial to extend these results to the general case. Our approach is based on a novel comparison principle that reduces the general situation to the mean-zero setting. As applications, we obtain new results for the Kac polynomials, hyperbolic random polynomials, their derivatives, and generalizations of these polynomials. The proof features new logarithmic integrability estimates for random polynomials (both local and global) and fairly sharp estimates for the local number of real zeros.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
