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Dragan Stankov

Approximation of the number of roots that do not lie on the unit circle of a self-reciprocal polynomial

THE BOOK OF ABSTRACTS XIV SYMPOSIUM "MATHEMATICS AND APPLICATIONS” December, 6–7, 2024 Belgrade, Serbia

Miljan Knežević, Aleksandra Delić

Univerzitet u Beogradu, Matematički fakultet

2024

We introduce the ratio of the number of roots not equal to 1 in modulus of a reciprocal polynomial Rd(x) to its degree d. For some sequences of reciprocal polynomials we show that the ratio has a limit L when d tends to infinity. Each of these sequences is defined using a two variable polynomial P(x,y) so that Rd(x) = P(x,xn). For P(x,y) we present the theorem for the limit ratio which is analogous to the Boyd-Lawton limit formula for Mahler measure. We present a double integral formula for approximation the limit ratio. In a previous paper we have calculated the exact value of the limit ratio of polynomials correlated to many bivariate polynomials having small Mahler measure introduced by Boyd and Mossinghoff. We demonstrate here that the double integral formula gives the value very close to the exact value (the error is < 10−5. We show that the limit ratio of the sequence P(x,xn) is not always equal to the limit ratio of the sequence P(yn,y) unlike Mahler measure.

44

44

978-86-7589-197-0

Reciprocal polynomial, Envelope, Unimodular roots.

COBISS.SR-ID 158252041

https://simpozijum.matf.bg.ac.rs/KNJIGA_APSTRAKATA_2024.pdf

М60

М64

Partially supported by Serbian Ministry of Education and Science, Project 174032

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