Lecture on the combinatorial algebraic method for computing algebraic integrals

TL;DR

This paper introduces an efficient algebraic and combinatorial method based on Newton's polygon to compute integrals over algebraic curves defined by bivariate polynomials, with practical applications demonstrated.

Contribution

The paper presents a novel combinatorial algebraic approach for calculating integrals on algebraic curves, leveraging Newton's polygon, which improves computational efficiency.

Findings

Method effectively computes integrals on algebraic curves.

Applicable to rational functions and various Jordan arcs.

Demonstrated through multiple practical examples.

Abstract

Consider an algebraic equation $P(x,y)=0$ where $P\in \mathbb C[x,y] $ (or $\mathbb F[x,y]$ with $\mathbb F\subset \mathbb C$ a subfield) is a bivariate polynomial, it defines a plane algebraic curve. We provide an efficient method for computing integrals of the type $ \int_\gamma R(x,y)dx $ where $R(x,y)\in \mathbb C(x,y) $ is any rational fraction, and $y$ is solution of $P(x,y)=0$, and $\gamma$ any Jordan arc open or closed on the plane algebraic curve. The method uses only algebraic and combinatorial manipulations, it rests on the combinatorics of the Newton's polygon. We illustrate it with many practical examples.