The expected number of real zeros of an algebraic polynomial ao a1x a2x · · · anx with random coefficient aj , j 0, 1, 2, . . . , n is known. The distribution of the coefficients is often assumed to be identical albeit allowed to have different classes of distributions. For the nonidentical case, there has been much interest where the variance of the jth coefficient is var aj ( n j ) . It is sh...

Our interest lies in describing the zero behaviour of Gauss hypergeometric polynomials F (−n, b; c; z) where b and c are arbitrary parameters. In general, this problem has not been solved and even when b and c are both real, the only cases that have been fully analyzed impose additional restrictions on b and c. We review recent results that have been proved for the zeros of several classes of h...

We show a Condition Number Theorem for the condition number of zero counting for real polynomial systems. That is, we show that this condition number equals the inverse of the normalized distance to the set of ill-posed systems (i.e., those having multiple real zeros). As a consequence, a smoothed analysis of this condition number follows.

We present a number of conjectures involving rook polynomials having only real zeros. Many of these generalize a previous conjecture of the author, K. Ono, and D. G. Wagner, namely that if A is a real n n matrix which is weakly increasing down columns, then the permanent of zA + Jn has only real zeros. In some cases these conjectures are motivated by factorization theorems for Ferrers boards. S...

|We present an eecient and precise numerical method in computational complexity and reliability to compute the number of zeros of a real polynomial in the unit disk.