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Gröbner bases and hypergeometric functions. (English) Zbl 0918.33004
Buchberger, Bruno (ed.) et al., Gröbner bases and applications. Based on a course for young researchers, January 1998, and the conference “33 years of Gröbner bases”, Linz, Austria, February 2–4, 1998. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 251, 246-258 (1998).
This work illustrates the use of Gröbner bases and Buchberger’s algorithm in the algebraic study of linear partial differential equations. The \(A\)-hypergeometric system of Gel’fand, Kapranov and Zelevinsky (GKZ) for a function \(\varphi (x_1,x_2,x_3,x_4)\) in four complex variables is considered as a reference example. It is shown that, for every open ball \(U\) in \(\mathbb{C}^4\), the dimension of the \(\mathbb{C}\)-vector space of holomorphic functions \(\varphi\) on \(U\) is at most five. The creation operators that allow to obtain new solutions from known solutions are also obtained. Finally, series solutions of GKZ systems are studied and an \(A\)-hypergeometric series is constructed as illustrative example.
For the entire collection see [Zbl 0883.00014].
Reviewer: G.Zet (Iaşi)

33C20 Generalized hypergeometric series, \({}_pF_q\)
68W30 Symbolic computation and algebraic computation
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
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