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«THE BULLETIN OF IRKUTSK STATE UNIVERSITY». SERIES «MATHEMATICS»
«IZVESTIYA IRKUTSKOGO GOSUDARSTVENNOGO UNIVERSITETA». SERIYA «MATEMATIKA»
ISSN 1997-7670 (Print)
ISSN 2541-8785 (Online)

List of issues > Series «Mathematics». 2024. Vol 47

Necessary and Sufficient Conditions for the Existence of Rational Solutions to Homogeneous Difference Equations with Constant Coefficients

Author(s)
Pavel V. Trishin1

1Siberian 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.
About the Authors
Pavel V. Trishin, Siberian Federal University, Krasnoyarsk, 660041, Russian Federation, me@trishin.xyz
For citation
Trishin P. V. Necessary and Sufficient Conditions for the Existence of Rational Solutions to Homogeneous Difference Equations with Constant Coefficients. The Bulletin of Irkutsk State University. Series Mathematics, 2024, vol. 47, pp. 47–62. https://doi.org/10.26516/1997-7670.2024.47.47
Keywords
difference equations, rational functions, Newton’s polyhedron
UDC
517.55+517.96
MSC
39A06, 32A10, 39A14
DOI
https://doi.org/10.26516/1997-7670.2024.47.47
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