Geodesic
How many roots of a system of random Laurent polynomials are real?
Sbornik. Mathematics, Tome 213 (2022) no. 4, pp. 466-475 Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: trigonometric polynomial, proportion of real zeros, BKK theorem, mixed volume.
Mots-clés : Laurent polynomial 
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B. Ya. Kazarnovskii. How many roots of a system of random Laurent polynomials are real?. Sbornik. Mathematics, Tome 213 (2022) no. 4, pp. 466-475. http://geodesic.mathdoc.fr/item/SM_2022_213_4_a1/

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