## Abstract

This article addresses the problem of computing an upper bound of the degree d of a polynomial solution P(x) of an algebraic difference equation of the form G(x)(P(x−τ_{1}),…,P(x−τ_{s}))+G_{0}(x)=0 when such P(x) with the coefficients in a field K of characteristic zero exists and where G is a non-linear s-variable polynomial with coefficients in K[x] and G_{0} is a polynomial with coefficients in K. It will be shown that if G is a quadratic polynomial with constant coefficients then one can construct a countable family of polynomials f_{l}(u_{0}) such that if there exists a (minimal) index l_{0} with f_{l0}(u_{0}) being a non-zero polynomial, then the degree d is one of its roots or d≤l_{0}, or d<deg(G_{0}). Moreover, the existence of such l_{0} will be proven for K being the field of real numbers. These results are based on the properties of the modules generated by special families of homogeneous symmetric polynomials. A sufficient condition for the existence of a similar bound of the degree of a polynomial solution for an algebraic difference equation with G of arbitrary total degree and with variable coefficients will be proven as well.

Original language | English |
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Pages (from-to) | 22-45 |

Number of pages | 24 |

Journal | Journal of Symbolic Computation |

Volume | 103 |

DOIs | |

Publication status | Published - Mar 2021 |

## Keywords

- Algebraic difference equation
- Homogeneous symmetric polynomial
- Partition
- Power-sum symmetric polynomial