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...

Full description

Saved in:
Bibliographic Details
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
Citations: Items that this one cites
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2024.128595