An algorithm for deciding the summability of bivariate rational functions
| العنوان: | An algorithm for deciding the summability of bivariate rational functions |
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| المؤلفون: | Rong-Hua Wang, Qing-Hu Hou |
| المصدر: | Advances in Applied Mathematics. 64:31-49 |
| بيانات النشر: | Elsevier BV, 2015. |
| سنة النشر: | 2015 |
| مصطلحات موضوعية: | Computer Science - Symbolic Computation, FOS: Computer and information sciences, Discrete mathematics, Gosper's algorithm, 33F10, 39A04, 68W30, Differential equation, Applied Mathematics, Univariate, Bivariate analysis, Rational function, Symbolic Computation (cs.SC), Combinatorics, Bivariate polynomials, Factorization, Mathematics - Classical Analysis and ODEs, Classical Analysis and ODEs (math.CA), FOS: Mathematics, Algebraically closed field, Algorithm, Mathematics |
| الوصف: | Let $\Delta_x f(x,y)=f(x+1,y)-f(x,y)$ and $\Delta_y f(x,y)=f(x,y+1)-f(x,y)$ be the difference operators with respect to $x$ and $y$. A rational function $f(x,y)$ is called summable if there exist rational functions $g(x,y)$ and $h(x,y)$ such that $f(x,y)=\Delta_x g(x,y) + \Delta_y h(x,y)$. Recently, Chen and Singer presented a method for deciding whether a rational function is summable. To implement their method in the sense of algorithms, we need to solve two problems. The first is to determine the shift equivalence of two bivariate polynomials. We solve this problem by presenting an algorithm for computing the dispersion sets of any two bivariate polynomials. The second is to solve a univariate difference equation in an algebraically closed field. By considering the irreducible factorization of the denominator of $f(x,y)$ in a general field, we present a new criterion which requires only finding a rational solution of a bivariate difference equation. This goal can be achieved by deriving a universal denominator of the rational solutions and a degree bound on the numerator. Combining these two algorithms, we can decide the summability of a bivariate rational function. Comment: 18 pages |
| تدمد: | 0196-8858 |
| DOI: | 10.1016/j.aam.2014.11.002 |
| URL الوصول: | https://explore.openaire.eu/search/publication?articleId=doi_dedup___::916e8a3e9d5245c54b07dbc25e259c9d https://doi.org/10.1016/j.aam.2014.11.002 |
| Rights: | OPEN |
| رقم الانضمام: | edsair.doi.dedup.....916e8a3e9d5245c54b07dbc25e259c9d |
| قاعدة البيانات: | OpenAIRE |
| تدمد: | 01968858 |
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| DOI: | 10.1016/j.aam.2014.11.002 |