# zbMATH — the first resource for mathematics

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)

##### MSC:
 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)