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Denominator Bounds for Systems of Recurrence Equations Using \(\varPi \varSigma \)-Extensions

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Advances in Computer Algebra (WWCA 2016)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 226))

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Abstract

We consider linear systems of recurrence equations whose coefficients are given in terms of indefinite nested sums and products covering, e.g., the harmonic numbers, hypergeometric products, q-hypergeometric products or their mixed versions. These linear systems are formulated in the setting of \(\varPi \varSigma \)-extensions and our goal is to find a denominator bound (also known as universal denominator) for the solutions; i.e., a non-zero polynomial d such that the denominator of every solution of the system divides d. This is the first step in computing all rational solutions of such a rather general recurrence system. Once the denominator bound is known, the problem of solving for rational solutions is reduced to the problem of solving for polynomial solutions.

Dedicated to Sergei A. Abramov on the occasion of his 70th birthday.

Supported by the Austrian Science Fund (FWF) grant SFB F50 (F5009-N15).

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Notes

  1. 1.

    Throughout this article, all fields contain the rational numbers \(\mathbb {Q}\) as subfield.

  2. 2.

    If z is the zero vector, then the assumption \(\gcd (z,d)=1\) implies \(d=1\).

  3. 3.

    Some authors would denote \(\mathbb {F}(t)[\sigma ]\) by the more precise \(\mathbb {F}(t)[\sigma ;\sigma ,0]\).

  4. 4.

    A more rigorous way would be to introduce a new symbol for the variable. However, a lot of authors simply use the same symbol and we decided to join them.

  5. 5.

    The other two rings do not admit determinants since they lack commutativity.

  6. 6.

    Note that the commutation rule \(\sigma ^{-1} a = \sigma ^{-1}(a)\sigma ^{-1}\) follows immediately from the rule \(\sigma a = \sigma (a) \sigma \).

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Acknowledgements

We would like to thank Michael Karr for his valuable remarks to improve the article.

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Authors and Affiliations

  1. Research Institute for Symbolic Computation (RISC), Johannes Kepler University, Altenbergerstraße 69, 4040, Linz, Austria

    Johannes Middeke & Carsten Schneider

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  1. Johannes Middeke
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Correspondence to Johannes Middeke .

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  1. Research Institute for Symbolic Computation, Johannes Kepler University, Linz, Austria

    Carsten Schneider

  2. Department of Physics and Computer Science, Wilfrid Laurier University, Waterloo, Ontario, Canada

    Eugene Zima

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Middeke, J., Schneider, C. (2018). Denominator Bounds for Systems of Recurrence Equations Using \(\varPi \varSigma \)-Extensions. In: Schneider, C., Zima, E. (eds) Advances in Computer Algebra. WWCA 2016. Springer Proceedings in Mathematics & Statistics, vol 226. Springer, Cham. https://doi.org/10.1007/978-3-319-73232-9_7

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