Symbolic Integration in Weierstrass-like Extensions

2026/02/06 by Shaoshi Chen, Manuel Kauers, Wenqiao Li +2 · 1 voice
Computer Science · Mathematics · #cs.SC #math.AG #math.CA

paper · pdf · doi:10.1145/3815436.3815449

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.

Citations

• Complete Reduction for Derivatives in a Primitive Tower
• Liouville's Theorem on integration in finite terms for \mathrm D_∞, SL₂ and Weierstrass field extensions
• Hermite Reduction for D-finite Functions via Integral Bases
• D-Algebraic Functions
• Ax-Schanuel and exceptional integrability
• Lazy Hermite Reduction and Creative Telescoping for Algebraic Functions
• Elliptic polylogarithms and iterated integrals on elliptic curves I: general formalism
• Reduction-Based Creative Telescoping for Fuchsian D-finite Functions
• Reduction-Based Creative Telescoping for Algebraic Functions
• Liouvillian Solutions of First Order Non Linear Differential Equations
• Hermite Reduction and Creative Telescoping for Hyperexponential Functions
• Complexity of Creative Telescoping for Bivariate Rational Functions
• Integration of elementary functions
• The solution of the problem of integration in finite terms

Discussions

• Shaoshi Chen, Manuel Kauers, Wenqiao Li, Xiuyun Li, David Masser: Symbolic Integration in Weierstrass-like Extensions https://arxiv.org/abs/2602.06873 https://arxiv.org/pdf/2602.06873 https://arxiv.or [bsky, 0 points, 0 comments]

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• Weierstrass integrability of differential equations
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