On the zeros of polyanalytic polynomials
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866910484390739968 |
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| author | Sète, Olivier Zur, Jan |
| author_facet | Sète, Olivier Zur, Jan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_04591 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the zeros of polyanalytic polynomials Sète, Olivier Zur, Jan Complex Variables 30C55, 31A05, 30C10, 30C15 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. |
| title | On the zeros of polyanalytic polynomials |
| topic | Complex Variables 30C55, 31A05, 30C10, 30C15 |
| url | https://arxiv.org/abs/2403.04591 |