On the zeros of polyanalytic polynomials

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,, 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.