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Multiplicity operators

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Abstract

For functions of a single complex variable, zeros of multiplicity greater than k are characterized by the vanishing of the first k derivatives. There are various quantitative generalizations of this statement, showing that for functions that are in some sense close to having a zero of multiplicity greater than k, the first k derivatives must be small.

In this paper we aim to generalize this situation to the multi-dimensional setting. We define a class of differential operators, the multiplicity operators, which act on maps from ℂn to ℂn and satisfy properties analogous to those described above. We demonstrate the usefulness of the construction by applying it to some problems in the theory of Noetherian functions.

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Toronto, Toronto, Canada

    Gal Binyamini

  2. Department of Mathematics, Weizmann Institute of Science, Rehovot, 76100, Israel

    Dmitry Novikov

Authors
  1. Gal Binyamini
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  2. Dmitry Novikov
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Corresponding author

Correspondence to Gal Binyamini.

Additional information

The first author was supported by the Banting Postdoctoral Fellowship and the Rothschild Fellowship.

Supported by the Minerva foundation with funding from the Federal German Ministry for Education and Research.

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Binyamini, G., Novikov, D. Multiplicity operators. Isr. J. Math. 210, 101–124 (2015). https://doi.org/10.1007/s11856-015-1247-8

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  • DOI: https://doi.org/10.1007/s11856-015-1247-8

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