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arXiv · 2403.04591

On the zeros of polyanalytic polynomials

Olivier Sète·Jan Zur
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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 $n^2$. We then show that for all $k \in \{0,\dots, n^2, \infty\}$ 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 $n^2$ zeros.

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Olivier Sète, Jan Zur. 2024-06-13. On the zeros of polyanalytic polynomials. https://doi.org/10.1016/j.jmaa.2024.128595

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