Non-dimensional Newton-Puiseux Expansions

TL;DR

This paper advances the theory of Newton-Puiseux expansions by integrating algebraic geometry, toric ideals, and dimensional analysis to better understand fractional power series solutions of differential equations.

Contribution

It introduces a new approach to non-dimensionalization using toric ideals and connects classical dimensional analysis with algebraic geometry in the context of asymptotic expansions.

Findings

Non-dimensional parameters influence Newton-Puiseux expansions.

Toric ideals facilitate optimal non-dimensionalization.

Application to differential equations demonstrates the method's effectiveness.

Abstract

Recent results in the theory and application of Newton-Puiseux expansions, i.e. fractional power series solutions of equations, suggest further developments within a more abstract algebraic-geometric framework, involving in particular the theory of toric varieties and ideals. Here, we present a number of such developments, especially in relation to the equations of van der Pol, Riccati, and Schr\"{o}dinger. Some pure mathematical concepts we are led to are Graver, Gr\"{o}bner, lattice and circuit bases, combinatorial geometry and differential algebra, and algebraic-differential equations. Two techniques are coordinated: classical dimensional analysis (DA) in applied mathematics and science, and a polynomial and differential-polynomial formulation of asymptotic expansions, referred to here as Newton-Puiseux (NP) expansions. The latter leads to power series with rational exponents, which depend on the choice of the dominant part of the equations, often referred to as the method of "dominant balance". The paper shows, using a new approach to DA based on toric ideals, how dimensionally homogeneous equations may best be non-dimensionalised, and then, with examples involving differential equations, uses recent work on NP expansions to show how non-dimensional parameters affect the results. Our approach finds a natural home within computational algebraic geometry, i.e. at the interface of abstract and algorithmic mathematics.