Lecture on the combinatorial algebraic method for computing algebraic integrals

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.