Fractional power series and the method of dominant balances

Presented by John Chapman (Keele University, UK)

Abstract: In this talk I'll give a general treatment of the method of dominant balances for a single polynomial equation, in which an arbitrary number of parameters is to be scaled in such a way that the maximum possible number of terms in the equation is in balance at leading order.  This leads in general to a fractional power series (a `Puiseux series'), in which, surprisingly, there can be large and irregular gaps (lacunae) in the fractional powers actually occurring.

A complete theory is given to determine the gaps, requiring the notion of a Frobenius set from number theory, and its complement, a Sylvester set.  The starting point is the Newton polytope in arbitrarily many dimensions, and key tools for obtaining precise results are Faà di Bruno’s formula for the high derivatives of a composite function, and Bell polynomials.  Full account is taken of repeated roots, of arbitrary multiplicity, in the dominant balance which launches a Puiseux series.  The fractional powers in these series can have remarkably large denominators, even for a polynomial of modest degree, and the nature of Frobenius sets is such that it can take hundreds of terms for long-run regularity to emerge.

The talk is applied in outlook, as the method of dominant balances is widely used in physics and engineering, where it gives results of extraordinary accuracy, far beyond the expected range.  The work has been conducted in a collaboration begun at the Isaac Newton Institute, Cambridge, with H. P. Wynn (London School of Economics).  We believe the results are new.  Despite hundreds of years of use of Puiseux series (since 1676), we are not aware of any previous attempt to give a complete quantitative account of their gaps.

The talk may also be of interest to number theorists and algebraic geometers, especially to anyone interested in computational algebraic geometry - currently a booming area at the interface of algorithmic methods and pure mathematics.