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∈{0,1,2,…,n2,∞} there exists a polyanalytic polynomial of degree...

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Published in:Journal of mathematical analysis and applications 2024-12, Vol.540 (1), p.128595, Article 128595
Main Authors: Sète, Olivier, Zur, Jan
Format: Article
Language:English
Subjects:
Fundamental theorem of algebra
Harmonic polynomial
Inclusion region
Polyanalytic polynomial
Wilmshurst's problem
Zeros
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description 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.
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subjects Fundamental theorem of algebra
Harmonic polynomial
Inclusion region
Polyanalytic polynomial
Wilmshurst's problem
Zeros
title On the zeros of polyanalytic polynomials
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