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ALGEBRAIC STRUCTURE OF THE RANGE OF A TRIGONOMETRIC POLYNOMIAL

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

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.

References

Bleher, P. M., Homma, Y., Ji, L. L. and Roeder, R. K. W., ‘Counting zeros of harmonic rational functions and its application to gravitational lensing’, Int. Math. Res. Not. IMRN 2014(8) (2014), 2245–2264.CrossRefGoogle Scholar
Khavinson, D. and Neumann, G., ‘On the number of zeros of certain rational harmonic functions’, Proc. Amer. Math. Soc. 134(4) (2006), 1077–1085.CrossRefGoogle Scholar
Kovalev, L. V. and Kalmykov, S., ‘Self-intersections of Laurent polynomials and the density of Jordan curves’, Proc. Amer. Math. Soc. (2019), to appear, arXiv:1902.02468.Google Scholar
Orzech, G. and Orzech, M., Plane Algebraic Curves: An Introduction Via Valuations, Monographs and Textbooks in Pure and Applied Mathematics, 61 (Marcel Dekker, Inc., New York, 1981).Google Scholar
Quine, J. R., ‘On the self-intersections of the image of the unit circle under a polynomial mapping’, Proc. Amer. Math. Soc. 39 (1973), 135–140.CrossRefGoogle Scholar
Quine, J. R., ‘Some consequences of the algebraic nature of p (e i𝜃)’, Trans. Amer. Math. Soc. 224(2) (1976), 437–442.Google Scholar
van den Essen, A., Polynomial Automorphisms and the Jacobian Conjecture, Progress in Mathematics, 190 (Birkhäuser, Basel, 2000).CrossRefGoogle Scholar
Walker, R. J., Algebraic Curves (Springer, New York, 1978). Reprint of the 1950 edition.CrossRefGoogle Scholar
Whitney, H., ‘Elementary structure of real algebraic varieties’, Ann. of Math. (2) 66 (1957), 545–556.CrossRefGoogle Scholar
Wilmshurst, A. S., ‘The valence of harmonic polynomials’, Proc. Amer. Math. Soc. 126(7) (1998), 2077–2081.CrossRefGoogle Scholar