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Symbolic Integration of Differential Forms: From Abel to Zeilberger

Shaoshi Chen, David A. Cox, Yisen Wang

TL;DR

The paper advances symbolic integration of differential forms by unifying historical Abelian-integral ideas with modern creative telescoping. It develops Hermite-style reductions for closed rational $1$-forms and extends them to general closed $p$-forms, obtaining explicit decompositions that separate exact parts from logarithmic components with controlled algebraic coefficients. Through Griffiths–Dwork reduction and Picard's problem, it clarifies when differential forms are rationally integrable and how to compute obstructions, while Liouville-type theorems for $1$-forms provide criteria for elementary antiderivatives. Finally, it presents a reduction-based creative telescoping framework for differential forms with one parameter, guaranteeing telescopers for closed algebraic $1$-forms and giving practical algorithms for their computation. Together, these results yield algorithmic tools for handling both algebraic and transcendental integrals of differential forms in multivariate settings with and without parameters.

Abstract

This paper focuses on symbolic integration of differential forms, with a particular emphasis on historical and modern developments, from Abel's addition theorems for Abelian integrals to Zeilberger's creative telescoping for parameterized integrals. It explores closed rational $p$-forms and provides algorithmic approaches for their integration, extending classical results like Hermite reduction and Liouville's theorem. The integration of closed differential forms with parameters is further examined through telescopers, offering a unified framework for handling both algebraic and transcendental cases.

Symbolic Integration of Differential Forms: From Abel to Zeilberger

TL;DR

The paper advances symbolic integration of differential forms by unifying historical Abelian-integral ideas with modern creative telescoping. It develops Hermite-style reductions for closed rational -forms and extends them to general closed -forms, obtaining explicit decompositions that separate exact parts from logarithmic components with controlled algebraic coefficients. Through Griffiths–Dwork reduction and Picard's problem, it clarifies when differential forms are rationally integrable and how to compute obstructions, while Liouville-type theorems for -forms provide criteria for elementary antiderivatives. Finally, it presents a reduction-based creative telescoping framework for differential forms with one parameter, guaranteeing telescopers for closed algebraic -forms and giving practical algorithms for their computation. Together, these results yield algorithmic tools for handling both algebraic and transcendental integrals of differential forms in multivariate settings with and without parameters.

Abstract

This paper focuses on symbolic integration of differential forms, with a particular emphasis on historical and modern developments, from Abel's addition theorems for Abelian integrals to Zeilberger's creative telescoping for parameterized integrals. It explores closed rational -forms and provides algorithmic approaches for their integration, extending classical results like Hermite reduction and Liouville's theorem. The integration of closed differential forms with parameters is further examined through telescopers, offering a unified framework for handling both algebraic and transcendental cases.
Paper Structure (7 sections, 11 theorems, 130 equations)

This paper contains 7 sections, 11 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Integration of Closed Rational $1$-Forms
  4. Integration of Closed Rational $p$-Forms
  5. Picard's Problem and Griffiths-Dwork Reduction
  6. Liouville's Theorem for Differential 1-forms
  7. Zeilberger's Creative Telescoping for Differential Forms with One Parameter

Key Result

Lemma 2.2

For a field $F$ of characteristic zero, let $F(y)$ be the field of rational functions in $y$ over $F$. Let $D_y$ denote the derivative $d/dy$ on $F(y)$ satisfying $D_y(y) = 1$ and $D_y(c)=0$ for all $c\in F$. Given pairwise coprime polynomials $u_1, \ldots, u_n\in F[y]\setminus F$, constants $c_1, \ Then $c_1 = \cdots = c_n = 0$ and $v\in F$.

Theorems & Definitions (34)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Lemma 2.4
  • Theorem 3.1
  • Remark 3.2
  • Remark 3.3
  • Example 3.4
  • Example 4.1
  • ...and 24 more