On the zeros of polyanalytic polynomials

Abstract

We give sufficient conditions under which a polyanalytic polynomial of degree n has (i) at least one zero, and (ii) finitely many zeros. In the latter case, we prove that the number of zeros is bounded by n2. We then show that for all k∈{0,1,2,…,n2,∞} there exists a polyanalytic polynomial of degree n with exactly k distinct zeros. Moreover, we generalize the Lagrange and Cauchy bounds from analytic to polyanalytic polynomials and obtain inclusion disks for the zeros. Finally, we construct a harmonic and thus polyanalytic polynomial of degree n with n nonzero coefficients and the maximum number of n2 zeros.