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Symmetric Division of Linear Ordinary Differential Operators

Lixin Du, Manuel Kauers

TL;DR

The paper addresses the division problem for symmetric products of linear differential operators over $C(x)[D]$: given $L$ and a symmetric product $M=L\otimes Q$, it seeks the (maximal) symmetric quotient $Q$. It introduces colon spaces and local/global quasi-symmetric quotients, and presents an algorithm that reconstructs $Q$ from the solution spaces via truncations and linear-algebra, with a rigorous bound on the possible order of $Q$ guided by the colon-space dimension $\delta=\dim_C(V(M):V(L))$. It establishes several structural results: the maximal symmetric quotient exists and is unique up to left multiplication, several special cases (hyperexponential, C-finite, algebraic) yield global quotients with simplified coefficients, and the degree bounds for $Q$ are derived in both Fuchsian and general settings using generalized indicial polynomials and (generalized) Fuchs relations. The framework combines algebraic-analytic tools (colon spaces, indicial polynomials, generalized series) with truncation-based algorithms to produce practical procedures and concrete bounds, illustrated by extensive examples including large-order cases. The results have implications for symbolic computation with D-finite functions and demonstrate that symmetric-division techniques, despite initial cryptographic motivation, are algorithmically tractable and informative about the operator structure.

Abstract

The symmetric product of two ordinary linear differential operators $L_1,L_2$ is an operator whose solution set contains the product $f_1f_2$ of any solution $f_1$ of $L_1$ and any solution $f_2$ of~$L_2$. It is well known how to compute the symmetric product of two given operators $L_1,L_2$. In this paper we consider the corresponding division problem: given a symmetric product $L$ and one of its factors, what can we say about the other factors?

Symmetric Division of Linear Ordinary Differential Operators

TL;DR

The paper addresses the division problem for symmetric products of linear differential operators over : given and a symmetric product , it seeks the (maximal) symmetric quotient . It introduces colon spaces and local/global quasi-symmetric quotients, and presents an algorithm that reconstructs from the solution spaces via truncations and linear-algebra, with a rigorous bound on the possible order of guided by the colon-space dimension . It establishes several structural results: the maximal symmetric quotient exists and is unique up to left multiplication, several special cases (hyperexponential, C-finite, algebraic) yield global quotients with simplified coefficients, and the degree bounds for are derived in both Fuchsian and general settings using generalized indicial polynomials and (generalized) Fuchs relations. The framework combines algebraic-analytic tools (colon spaces, indicial polynomials, generalized series) with truncation-based algorithms to produce practical procedures and concrete bounds, illustrated by extensive examples including large-order cases. The results have implications for symbolic computation with D-finite functions and demonstrate that symmetric-division techniques, despite initial cryptographic motivation, are algorithmically tractable and informative about the operator structure.

Abstract

The symmetric product of two ordinary linear differential operators is an operator whose solution set contains the product of any solution of and any solution of~. It is well known how to compute the symmetric product of two given operators . In this paper we consider the corresponding division problem: given a symmetric product and one of its factors, what can we say about the other factors?

Paper Structure

This paper contains 21 sections, 38 theorems, 135 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Truncated Series
  4. The ring of linear differential operators
  5. Indicial polynomials
  6. Generalized series solutions
  7. The colon space
  8. Symmetric division algorithm
  9. The maximal symmetric quotient
  10. Overview of the algorithm
  11. Three special cases
  12. The hyperexponential case
  13. The C-finite case
  14. The algebraic case
  15. Truncation and intersection of power series subspaces
  16. ...and 6 more sections

Key Result

Lemma 2.1

Let $f=\sum_{i= \lambda_0}^{\lambda}a_ix^i+O(x^{\lambda+1})$ and $g=\sum_{i= \mu_0}^{\mu}b_ix^i+O(x^{\mu+1})$ be Laurent series in $C((x))$, where $\lambda_0,\lambda,\mu_0,\mu\in {\mathbb{Z}}$ with $\lambda\geq \lambda_0$, $\mu\geq \mu_0$. Then

Theorems & Definitions (80)

  • Lemma 2.1
  • Lemma 2.2
  • Corollary 2.3
  • proof
  • Lemma 2.4: Singer1996reducibility
  • Lemma 2.5
  • Lemma 2.6
  • Corollary 2.7
  • Lemma 2.8
  • proof
  • ...and 70 more