We demonstrate counterexamples toWilmshurst's conjecture on the valence of harmonic polynomials in the plane, and we conjecture a bound that is linear in the analytic degree for each fixed anti-analytic degree. Then, we initiate a discussion of Wilmshurt's theorem in more than two dimensions, showing that if the zero set of a polynomial harmonic field is bounded, then it must have codimension at least 2. Examples are provided to show that this conclusion cannot be improved.

Remarks on Wilmshurst’s Theorem / Lee, S. Y.; Lerario, Antonio; Lundberg, E. E.. - In: INDIANA UNIVERSITY MATHEMATICS JOURNAL. - ISSN 0022-2518. - 64:4(2015), pp. 1153-1167. [10.1512/iumj.2015.64.5526]

Remarks on Wilmshurst’s Theorem

Lerario, Antonio;
2015

Abstract

We demonstrate counterexamples toWilmshurst's conjecture on the valence of harmonic polynomials in the plane, and we conjecture a bound that is linear in the analytic degree for each fixed anti-analytic degree. Then, we initiate a discussion of Wilmshurt's theorem in more than two dimensions, showing that if the zero set of a polynomial harmonic field is bounded, then it must have codimension at least 2. Examples are provided to show that this conclusion cannot be improved.
64
4
1153
1167
https://arxiv.org/abs/1308.6474
Lee, S. Y.; Lerario, Antonio; Lundberg, E. E.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/20.500.11767/32548
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