Analytic continuation and Riemann surface determination of algebraic functions by computer

Abstract

Lety=f(x) be an algebraic function which is defined as a root of bivariate polynomialP(x, y). GivenP(x, y), we present a practical computation method for analytic continuation and Riemann surface determination off(x). First, we determine the branch points off(x) by calculating the roots of a univariate polynomial numerically. Next, we expandf(x) into the Puiseux series at each branch point by the extended Hensel construction method proposed by Sasaki and Kako recently. Then, using well-known Smith’s theorem, we determine which series are connected the each other for each pair of branch points, by numerically estimating the values of the series at some middle point between the branch points. This allows us to determine the Riemann surface off(x) completely. Analytic continuation is performed similarly. We employ floating-point number arithmetic to perform these calculations.