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D-Algebraic Functions

Rida Ait El Manssour, Anna-Laura Sattelberger, Bertrand Teguia Tabuguia

TL;DR

This work develops constructive, algorithmic tools for D-$algebraic$ functions, i.e., zeros of differential polynomials, and proves closure properties under arithmetic and composition. It introduces two complementary strategies: a jet-based elimination method (Method I) and an order-bounded approach (Method II) to compute ADEs for rational expressions, antiderivatives, and compositions of univariate D-$algebraic$ functions, with provable bounds on the resulting differential order. Implementations in Macaulay2 and the Maple NLDE package enable practical computation of ADEs for complex expressions arising in physics, epidemiology, and applied mathematics, and the paper demonstrates applications to Painlevé transcendents, elliptic functions, and dynamical systems. The methods facilitate robust, symbolic handling of D-$algebraic$ expressions, expanding the toolkit for analyzing, verifying, and manipulating these functions in scientific computing. Overall, the paper advances both the theory and practice of algorithmic differential algebra, providing concrete procedures and bounds that improve on existing software capabilities and support broader applications.

Abstract

Differentially-algebraic (D-algebraic) functions are solutions of polynomial equations in the function, its derivatives, and the independent variables. We revisit closure properties of these functions by providing constructive proofs. We present algorithms to compute algebraic differential equations for compositions and arithmetic manipulations of univariate D-algebraic functions and derive bounds for the order of the resulting differential equations. We apply our methods to examples in the sciences.

D-Algebraic Functions

TL;DR

This work develops constructive, algorithmic tools for D- functions, i.e., zeros of differential polynomials, and proves closure properties under arithmetic and composition. It introduces two complementary strategies: a jet-based elimination method (Method I) and an order-bounded approach (Method II) to compute ADEs for rational expressions, antiderivatives, and compositions of univariate D- functions, with provable bounds on the resulting differential order. Implementations in Macaulay2 and the Maple NLDE package enable practical computation of ADEs for complex expressions arising in physics, epidemiology, and applied mathematics, and the paper demonstrates applications to Painlevé transcendents, elliptic functions, and dynamical systems. The methods facilitate robust, symbolic handling of D- expressions, expanding the toolkit for analyzing, verifying, and manipulating these functions in scientific computing. Overall, the paper advances both the theory and practice of algorithmic differential algebra, providing concrete procedures and bounds that improve on existing software capabilities and support broader applications.

Abstract

Differentially-algebraic (D-algebraic) functions are solutions of polynomial equations in the function, its derivatives, and the independent variables. We revisit closure properties of these functions by providing constructive proofs. We present algorithms to compute algebraic differential equations for compositions and arithmetic manipulations of univariate D-algebraic functions and derive bounds for the order of the resulting differential equations. We apply our methods to examples in the sciences.
Paper Structure (17 sections, 16 theorems, 97 equations, 1 figure, 1 table)

This paper contains 17 sections, 16 theorems, 97 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Algebraic aspects of differential equations
  3. The Weyl algebra and holonomic functions
  4. Differential algebra
  5. D-algebraic functions and their closure properties
  6. Definitions
  7. Closure properties of D-algebraic functions
  8. Multivariate D-algebraic functions
  9. Arithmetic with D-algebraic functions
  10. Method I
  11. Method II
  12. Algorithms and applications
  13. Algorithms
  14. Applications
  15. Algebraic varieties and jet schemes
  16. ...and 2 more sections

Key Result

Proposition 2.2

Let $f$ be holonomic. Its reciprocal $1/f$ is holonomic iff $f'/f$ is algebraic.

Figures (1)

  • Figure 1: The varieties $V(x_1^2 - 2x_1x_2 + 9x_2^2 - 1)$, $V(x_3-x_1^2-x_2^2)$, and $V(x_1^2x_2 - x_2^3 - x_1x_3^2)$, plotted with Mathematica Mathematica.

Theorems & Definitions (58)

  • Definition 2.1
  • Proposition 2.2
  • Example 2.3
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.6
  • Proposition 2.7
  • Remark 2.8
  • Example 2.9
  • Definition 3.1
  • ...and 48 more