Algebraic dependency of roots of multivariate polynomials and its applications to linear functional equations
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In this paper we prove that a multivariate polynomial has algebraically dependent roots if and only if the coefficients are algebraic numbers up to a common proportional term; for the problem see section 4.4 in Varga-Vincze (On the characteristic polynomials of linear functional equations, Period Math Hungar 71(2):250–260, 2015). The case of univariate polynomials belongs to basic algebra. As far as we know the case of multivariate polynomials is not discussed in the literature. As an application we formulate a sufficient and necessary condition for the existence of non-trivial solutions of special types of linear functional equations. The criteria is based only on the algebraic properties of the parameters in the functional equation.
KeywordsAlgebraic dependent systems Polynomials Linear functional equations Spectral analysis Characteristic polynomials of linear functional equations
Mathematics Subject Classification12D10 39B22
The work is supported by the University of Debrecen’s internal research project RH/885/2013. The author wishes to thank Professor Miklós Laczkovich and Professor Róbert Szőke for their contribution to simplify the proof of Theorem 2.1. Complex analytic tools make the argument as short as possible. The author is especially grateful to Professor M. Laczkovich for the discussion of alternative arguments to prove Theorem 2.1.
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