Notes on three formulas of Abel

These notes explore three amazing formulas proved by Abel in his 1826 Paris memoir on what we now call Abelian integrals.We discuss the first two formulas from the point of view of symbolic computation and explain their connection to residues and partial fractions.The third formula arises from the first two and is related to the genus and lattice points in the Newton polygon.Contents 1. Abel's Paris memoir 481 2. Our approach to Abel's formulas 484 3. Abel's formula for dv 485 4. Abel's formula for v 499 5. Abel's formula for 507 6.Other versions of Abel's theorem 517 Appendix A. How Abel proved his formulas for dv and v 521 References 534 1. Abel's Paris memoirIn October of 1826, Niels Henrik Abel submitted Mmoire sur une proprit gnrale d'une classe trs tendue de fonctions transcendantes [1] to the Paris Academy. 1 This amazing paper contains a number of results that fall under the rubric "Abel's theorem."The purpose of these notes is to explore one of the more underappreciated versions of Abel's theorem and its connection to the genus of the associated Riemann surface.Abel starts with an equation written in the formThis is equation (1) in [1, p. 146]. 2 The p i = p i (x) are polynomials in x ("fonctions rationnelles et entires d'une mme quantit variable x").Abel does not specify a field.We will work over a computable 3 subfield K C, so that y = (x, y) K[x, y].We assume that (x, y) is irreducible over C[x, y].