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Integro-differential equations and functional analysis
Necessary and sufficient conditions for the existence of rational solutions to homogeneous difference equations with constant coefficients
Pavel V. Trishin Siberian Federal University, Krasnoyarsk, Russian Federation
Abstract:
A necessary and a sufficient condition for solvability of homogeneous difference equations with constant coefficients in the class of rational functions are obtained. The necessary condition is a restriction on the Newton polyhedron of the characteristic polynomial. In the two-dimensional case, this condition is the existence of parallel sides on the polygon. The sufficient condition is the equality to zero of certain sums of the coefficients of the equation. If the sufficient condition is satisfied, the solution is the class of rational functions whose denominators form a subring in the ring of polynomials. This subring can be associated with an edge of the Newton polyhedron of the characteristic polynomial of the equation.
Keywords:
difference equations, rational functions, Newton's polyhedron.
Received: 02.10.2023 Revised: 14.12.2023 Accepted: 18.12.2023
Citation:
Pavel V. Trishin, “Necessary and sufficient conditions for the existence of rational solutions to homogeneous difference equations with constant coefficients”, Bulletin of Irkutsk State University. Series Mathematics, 47 (2024), 47–62
Linking options:
https://www.mathnet.ru/eng/iigum554 https://www.mathnet.ru/eng/iigum/v47/p47
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Abstract page: | 26 | Full-text PDF : | 13 | References: | 4 |
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