We consider multivariable polynomials over a fixed number field, linear in some of the variables. For system such satisfying certain technical conditions we prove existence search bounds for simultaneous zeros with respect to height. single polynomial, height lying outside prescribed algebraic set. also obtain case homogeneous multilinear polynomials, which are related so-called “sparse” versio...

We show that the zeros of random sequences of Gaussian systems of polynomials of increasing degree almost surely converge to the expected limit distribution under very general hypotheses. In particular, the normalized distribution of zeros of systems of m polynomials of degree N , orthonormalized on a regular compact set K ⊂ Cm, almost surely converge to the equilibrium measure on K as N → ∞.

It is shown that the zeros of the Faber polynomials generated by a regular m-star are located on the m-star. This proves a recent conjecture of J. Bartolomeo and M. He. The proof uses the connection between zeros of Faber polynomials and Chebyshev quadrature formulas.

In this paper we obtain sharp bounds for the zeros of classical orthogonal polynomials of a discrete variable, considered as functions of a parameter, by using a theorem of A. Markov and the so-called HellmannFeynman theorem. Comparisons with previous results for zeros of Hahn, Meixner, Kravchuk and Charlier polynomials are also presented.