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;
2015-01-01
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.| File | Dimensione | Formato | |
|---|---|---|---|
|
LLLsubmit3.pdf
accesso aperto
Tipologia:
Documento in Pre-print
Licenza:
Non specificato
Dimensione
584.92 kB
Formato
Adobe PDF
|
584.92 kB | Adobe PDF | Visualizza/Apri |
I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
