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Symbolic Integration in Weierstrass-like Extensions

Shaoshi Chen, Manuel Kauers, Wenqiao Li, Xiuyun Li, David Masser

TL;DR

This work addresses symbolic integration for functions defined by nonlinear first-order differential equations, exemplified by Weierstrass-like extensions generated by a transcendental $t$ with $t'$ satisfying a polynomial relation. It develops a theoretical framework that extends special polynomials to these extensions and couples it with a Hermite-style reduction, plus special and polynomial reductions, to decompose integrands and decide integrability within the field. The paper applies these tools to integrals of powers of the Weierstrass function $\wp$, deriving a linear recurrence for $J_n(t)$ and explicit formulas for low orders $I_3$ and $I_4$, while revealing non-elementarity for certain basic integrals and expressing others through $\zeta$ and $\sigma$. Collectively, the results broaden differential-algebraic integration to Weierstrass-like elements and provide concrete algorithms for computing and simplifying such integrals, with direct ties to elliptic function theory.

Abstract

This paper studies the integration problem in differential fields that may involve quantities reminiscent of the Weierstrass $\wp$ function, which are defined by a first-order nonlinear differential equation. We extend the classical notion of special polynomials to elements of Weierstrass-like extensions and present algorithms for reduction in such extensions. As an application of these results, we derive some new formulae for integrals of powers of $\wp$.

Symbolic Integration in Weierstrass-like Extensions

TL;DR

This work addresses symbolic integration for functions defined by nonlinear first-order differential equations, exemplified by Weierstrass-like extensions generated by a transcendental with satisfying a polynomial relation. It develops a theoretical framework that extends special polynomials to these extensions and couples it with a Hermite-style reduction, plus special and polynomial reductions, to decompose integrands and decide integrability within the field. The paper applies these tools to integrals of powers of the Weierstrass function , deriving a linear recurrence for and explicit formulas for low orders and , while revealing non-elementarity for certain basic integrals and expressing others through and . Collectively, the results broaden differential-algebraic integration to Weierstrass-like elements and provide concrete algorithms for computing and simplifying such integrals, with direct ties to elliptic function theory.

Abstract

This paper studies the integration problem in differential fields that may involve quantities reminiscent of the Weierstrass function, which are defined by a first-order nonlinear differential equation. We extend the classical notion of special polynomials to elements of Weierstrass-like extensions and present algorithms for reduction in such extensions. As an application of these results, we derive some new formulae for integrals of powers of .
Paper Structure (8 sections, 17 theorems, 56 equations, 1 algorithm)

This paper contains 8 sections, 17 theorems, 56 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. An appetizer
  3. Algebraic Functions
  4. Special Polynomials and Places
  5. Hermite Reduction
  6. Special and Polynomial Reductions
  7. The Appetizer Revisited
  8. Splitting Factorization

Key Result

Theorem 4

Let $p \in k[t]$ be irreducible. Then $p$ is special if and only if there exists a place $P$ of $K$ lying above $p$ such that $\nu_{P}(p') > 0$.

Theorems & Definitions (43)

  • Example 1
  • Definition 2
  • Definition 3
  • Theorem 4
  • proof
  • Definition 5
  • Proposition 6
  • proof
  • Definition 7
  • Proposition 8
  • ...and 33 more