Home | Quick Search | Advanced Search | Bibliography submission | Bibliography submission using bibtex | Bibliography submission using bibtex file | Links | Help | Internal

Details:

   
TitleUnivariate polynomial solutions of algebraic difference equations
Author(s) M. van Eekelen, O. Shkaravska
TypeArticle in Journal
AbstractAbstract Contrary to linear difference equations, there is no general theory of difference equations of the form G ( P ( x − τ_1 ) , … , P ( x − τ_s ) ) + G_0 ( x ) = 0 , with τ_i ∈ K , G_( x 1 , … , x s ) ∈ K [ x 1 , … , x s ] of total degree D ⩾ 2 and G 0 ( x ) ∈ K [ x ] , where K is a field of characteristic zero. This article concerns the following problem: given τ_i , G and G_0 , find an upper bound on the degree d of a polynomial solution P ( x ) , if it exists. In the presented approach the problem is reduced to constructing a univariate polynomial for which d is a root. The authors formulate a sufficient condition under which such a polynomial exists. Using this condition, they give an effective bound on d, for instance, for all difference equations of the form G ( P ( x − a ) , P ( x − a − 1 ) , P ( x − a − 2 ) ) + G_0 ( x ) = 0 with quadratic G, and all difference equations of the form G ( P ( x ) , P ( x − τ ) ) + G_0 ( x ) = 0 with G having an arbitrary degree.
KeywordsDifference equation, Elementary symmetric polynomials, Power-sum symmetric polynomials, Newton–Girard formulæ, System of linear equations
ISSN0747-7171
URL http://www.sciencedirect.com/science/article/pii/S0747717113001296
LanguageEnglish
JournalJournal of Symbolic Computation
Volume60
Number0
Pages15 - 28
Year2014
Edition0
Translation No
Refereed No
Webmaster