Solving algebraic equations in terms of A-hypergeometric series

The roots of the general equation of degree n satisfy an A-hypergeometric system of diierential equations in the sense of Gel'fand, Kapranov and Zelevinsky. We construct the n distinct A-hypergeometric series solutions for each of the 2 n?1 triangulations of the Newton segment. This works over any eld whose characteristic is relatively prime to the lengths of the segments in the triangulation. A classical problem in mathematics is to nd a formula for the roots of the general equation of degree n in terms of its n + 1 coeecients. While there are formulas in terms of radicals for n 4, Galois theory teaches us that no such formula exists for the general quintic An alternative approach is to expand the roots into fractional power series (or Puiseux series). In 1757 Johann Lambert expressed the roots of the trinomial equation x p + x + r as a Gauss hypergeometric function in the parameter r. Series expansions of more general algebraic functions were subsequently given by Euler, Chebyshev and Eisenstein, among others. The poster \Solving the Quintic with Mathematica" 12] gives a nice introduction to these classical techniques and underlines their relevance for symbolic computation. The state of the art in the rst half of our century appears in works of Richard Birke-land 2] and Karl Mayr 10]. They proved that the roots are multivariate hypergeometric functions (in the sense of Horn) in all of the coeecients and they gave series expansions for the roots and their powers. The purpose of this note is to reene the these results. Our point of departure is the fact that the roots satisfy the A-hypergeometric diier-ential equations introduced by Gel'fand, Kapranov and Zelevinsky 6],,7]. Here A denotes the connguration of n + 1 equidistant points on the aane line. It follows from recent work 1