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ALGEBRAIC STRUCTURE OF THE RANGE OF A TRIGONOMETRIC POLYNOMIAL

Part of: Harmonic analysis in one variable Polynomials, rational functions Two-dimensional theory

Published online by Cambridge University Press:  08 January 2020

LEONID V. KOVALEV Open the ORCID record for LEONID V. KOVALEV [Opens in a new window]  and
XUERUI YANG Open the ORCID record for XUERUI YANG [Opens in a new window]
LEONID V. KOVALEV*
Affiliation:
215 Carnegie, Mathematics Department, Syracuse University, Syracuse, NY13244, USA email [email protected]
XUERUI YANG
Affiliation:
215 Carnegie, Mathematics Department, Syracuse University, Syracuse, NY13244, USA email [email protected]
*
✉[email protected]
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Abstract

The range of a trigonometric polynomial with complex coefficients can be interpreted as the image of the unit circle under a Laurent polynomial. We show that this range is contained in a real algebraic subset of the complex plane. Although the containment may be proper, the difference between the two sets is finite, except for polynomials with a certain symmetry.

Keywords

Laurent polynomialtrigonometric polynomialalgebraic set

MSC classification

Primary: 26C05: Polynomials
Secondary: 26C15: Rational functions 31A05: Harmonic, subharmonic, superharmonic functions 42A05: Trigonometric polynomials, inequalities, extremal problems
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 102 , Issue 2 , October 2020 , pp. 251 - 260
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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Footnotes

The first author was supported by the National Science Foundation grant DMS-1764266; the second author was supported by a Young Research Fellow award from Syracuse University.

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