@article{SHKARAVSKA202122,
    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−τ1),{\ldots},P(x−τs))+G0(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 G0 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 fl(u0) such that if there exists a (minimal) index l0 with fl0(u0) being a non-zero polynomial, then the degree d is one of its roots or d≤l0, or d<deg⁡(G0). Moreover, the existence of such l0 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.},
    Author = {Shkaravska, Olha and {van Eekelen}, Marko},
    File = {Polynomial solutions of algebraic difference equations and homogeneous symmetric polynomials - 1-s2.0-S074771711930135X-main.pdf},
    ISSN = {0747-7171},
    Journal = {Journal of Symbolic Computation},
    Keywords = {Algebraic difference equation, Power-sum symmetric polynomial, Partition, Homogeneous symmetric polynomial},
    Pages = {22-45},
    Title = {Polynomial solutions of algebraic difference equations and homogeneous symmetric polynomials},
    URL = {https://www.sciencedirect.com/science/article/pii/S074771711930135X},
    Volume = {103},
    Year = {2021},
    bdsk-url-1 = {https://www.sciencedirect.com/science/article/pii/S074771711930135X},
    bdsk-url-2 = {https://doi.org/10.1016/j.jsc.2019.10.022},
    date-added = {2023-07-19 07:35:15 +0200},
    date-modified = {2023-07-19 07:35:15 +0200},
    doi = {10.1016/j.jsc.2019.10.022}
}

@article{SHKARAVSKA202122, 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−τ1),{\ldots},P(x−τs))+G0(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 G0 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 fl(u0) such that if there exists a (minimal) index l0 with fl0(u0) being a non-zero polynomial, then the degree d is one of its roots or d≤l0, or d<deg⁡(G0). Moreover, the existence of such l0 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.}, Author = {Shkaravska, Olha and {van Eekelen}, Marko}, File = {Polynomial solutions of algebraic difference equations and homogeneous symmetric polynomials - 1-s2.0-S074771711930135X-main.pdf}, ISSN = {0747-7171}, Journal = {Journal of Symbolic Computation}, Keywords = {Algebraic difference equation, Power-sum symmetric polynomial, Partition, Homogeneous symmetric polynomial}, Pages = {22-45}, Title = {Polynomial solutions of algebraic difference equations and homogeneous symmetric polynomials}, URL = {https://www.sciencedirect.com/science/article/pii/S074771711930135X}, Volume = {103}, Year = {2021}, bdsk-url-1 = {https://www.sciencedirect.com/science/article/pii/S074771711930135X}, bdsk-url-2 = {https://doi.org/10.1016/j.jsc.2019.10.022}, date-added = {2023-07-19 07:35:15 +0200}, date-modified = {2023-07-19 07:35:15 +0200}, doi = {10.1016/j.jsc.2019.10.022} }