Solving algebraic equations in terms of \(\mathcal A\)-hypergeometric series
type: Item
instance of: scholarly article
title: Solving algebraic equations in terms of \(\mathcal A\)-hypergeometric series
zbMATH Open document ID: 0963.33010
DOI: 10.1016/S0012-365X(99)00126-0
author: Bernd Sturmfels
published in: Discrete Mathematics
publication date: 2001-06-28T00:00:00Z
review text: It has long been realized that the solution of certain algebraic equations, as a function of their parameters, gives rise to hypergeometric type series. For example, as noted in the beginning paragraph of the present work, in 1757 Johann Lambert expressed the roots of the trinomial equation \(x^p+x+r\) as a Gauss hypergeometric function in the parameter \(r\). Continuing this theme, the system of linear partial differential equations satisfied by the roots of the general algebraic equation of degree \(n\), due to \textit{K. Mayr} [Monatsh. Math. Phys. 45, 280-313 (1937; Zbl 0016.35702)], are identified as a special case of the type \(A\) hypergeometric system of Gel'fand and collaborators. This allows (Theorem 3.2) an explicit series solution of each of the \(n\) roots of the general \(n\)th degree equation to be presented. The convergence of these series are ensured when certain explicit inequalities on the coefficients are satisfied. However, even though there are \(2^{n-1}\) distinct forms of the \(n\) series, each labelled by a subset of \(\{1,2,\dots,n-1\}\), there are concrete examples for which the inequalities do not hold in any case.
Mathematics Subject Classification ID: 33F99
Mathematics Subject Classification ID: 33C70
zbMATH DE Number: 1389517
reviewed by: Peter J. Forrester
MaRDI profile type: Publication
Wikidata QID: Q126643245
full work available at URL: s0012-365x(99)00126-0
OpenAlex ID: W1986398328
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description: scientific article; zbMATH DE number 1389517