Real zeros of random algebraic polynomialswithbinomial elements

[摘要] This paper provides an asymptotic estimate for the expected numberof real zeros of a random algebraic polynomiala0+a1x+a2x2+…+an−1xn−1. The coefficientsaj(j=0,1,2,…,n−1)are assumed to be independent normal randomvariables with mean zero. For integersmandk=O(log⁡n)2thevariances of the coefficients are assumed to have nonidenticalvaluevar⁡(aj)=(k−1j−ik), wheren=k⋅mandi=0,1,2,…,m−1. Previous results are mainly for identicallydistributed coefficients or whenvar⁡(aj)=(nj). Weshow that the latter is a special case of our general theorem.

[学科分类] 应用数学