Email this article How many roots of a system of random Laurent polynomials are real?
- Authors: Kazarnovskii B.Y.1
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Affiliations:
- Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute)
- Issue: Vol 213, No 4 (2022)
- Pages: 27-37
- Section: Articles
- URL: https://journals.rcsi.science/0368-8666/article/view/133435
- DOI: https://doi.org/10.4213/sm9559
- ID: 133435
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Abstract
We say that a zero of a Laurent polynomial that lies on the unit circle with centre $0\in\mathbb C$ is real. We also say that a Laurent polynomial that is real on this circle is real. In contrast with ordinary polynomials, it is known that for random real Laurent polynomials of increasing degree the average proportion of real roots tends to $1/\sqrt 3$ rather than to $0$. We show that this phenomenon of the asymptotically nonvanishing proportion of real roots also holds for systems of Laurent polynomials of several variables. The corresponding asymptotic formula is obtained in terms of the mixed volumes of certain convex compact sets determining the growth of the system of polynomials. Bibliography: 11 titles.
About the authors
Boris Yakovlevich Kazarnovskii
Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute)
Email: kazbori@gmail.com
Candidate of physico-mathematical sciences, no status
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