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Symbolic Summation of Multivariate Rational Functions.

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Title: Symbolic Summation of Multivariate Rational Functions.
Authors: Chen, Shaoshi1,2 (AUTHOR) schen@amss.ac.cn, Du, Lixin3 (AUTHOR) lx.du@hotmail.com, Fang, Hanqian1,2 (AUTHOR) hqfang_math@163.com
Source: Foundations of Computational Mathematics. Jun2026, Vol. 26 Issue 3, p1635-1697. 63p.
Subjects: Symbolic computation, Isotropy subgroups, Addition (Mathematics)
Abstract: Symbolic summation as an active research topic of symbolic computation provides efficient algorithmic tools for evaluating and simplifying different types of sums arising from mathematics, computer science, physics and other areas. Most of existing algorithms in symbolic summation are mainly applicable to the problem with univariate inputs. A long-term project in symbolic computation is to develop theories, algorithms and software for the symbolic summation of multivariate functions. This paper will give complete solutions to two challenging problems in symbolic summation of multivariate rational functions, namely the rational summability problem and the existence problem of telescopers for multivariate rational functions. Our approach is based on the structure of Sato's isotropy groups of polynomials, which enables us to reduce the problems to testing the shift equivalence of polynomials. Our results provide a complete solution to the discrete analogue of Picard's problem on differential forms and can be used to detect the applicability of the Wilf-Zeilberger method to multivariate rational functions. [ABSTRACT FROM AUTHOR]
Copyright of Foundations of Computational Mathematics is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: <searchLink fieldCode="AR" term="%22Chen%2C+Shaoshi%22">Chen, Shaoshi</searchLink><relatesTo>1,2</relatesTo> (AUTHOR)<i> schen@amss.ac.cn</i><br /><searchLink fieldCode="AR" term="%22Du%2C+Lixin%22">Du, Lixin</searchLink><relatesTo>3</relatesTo> (AUTHOR)<i> lx.du@hotmail.com</i><br /><searchLink fieldCode="AR" term="%22Fang%2C+Hanqian%22">Fang, Hanqian</searchLink><relatesTo>1,2</relatesTo> (AUTHOR)<i> hqfang_math@163.com</i>
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  Data: <searchLink fieldCode="JN" term="%22Foundations+of+Computational+Mathematics%22">Foundations of Computational Mathematics</searchLink>. Jun2026, Vol. 26 Issue 3, p1635-1697. 63p.
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  Data: Symbolic summation as an active research topic of symbolic computation provides efficient algorithmic tools for evaluating and simplifying different types of sums arising from mathematics, computer science, physics and other areas. Most of existing algorithms in symbolic summation are mainly applicable to the problem with univariate inputs. A long-term project in symbolic computation is to develop theories, algorithms and software for the symbolic summation of multivariate functions. This paper will give complete solutions to two challenging problems in symbolic summation of multivariate rational functions, namely the rational summability problem and the existence problem of telescopers for multivariate rational functions. Our approach is based on the structure of Sato's isotropy groups of polynomials, which enables us to reduce the problems to testing the shift equivalence of polynomials. Our results provide a complete solution to the discrete analogue of Picard's problem on differential forms and can be used to detect the applicability of the Wilf-Zeilberger method to multivariate rational functions. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Foundations of Computational Mathematics is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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              Text: Jun2026
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