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A unified approach for degree bound estimates of linear differential operators

Louis Gaillard

TL;DR

This work frames four central problems on D-finite functions—Hermite-based creative telescoping, differential resolvent, LCLM, and symmetric product—within a single linear-relations paradigm driven by a pseudo-linear map $\theta = \partial_x + T$. By representing $T$ through matrix fractions $T = XM^{-1}Y$ and employing state-space realisations, the authors derive precise degree bounds via determinantal-denominator theory, improving prior bounds in generic settings to $O(d_y^2 d_x)$ for several cases. Key contributions include a unified theorem bounding the degrees of the solution polynomials in $\eta$ in terms of $\det M$ and the input data, and tight analyses for the four concrete instances, along with direct Cramer-style bounds and a thorough treatment of matrices of rational functions. The results enable tighter output-size estimates and pave the way for output-sensitive, unified algorithms for manipulating and composing D-finite functions in symbolic computation.

Abstract

We identify a common scheme in several existing algorithms addressing computational problems on linear differential equations with polynomial coefficients. These algorithms reduce to computing a linear relation between vectors obtained as iterates of a simple differential operator known as pseudo-linear map. We focus on establishing precise degree bounds on the output of this class of algorithms. It turns out that in all known instances (least common left multiple, symmetric product,. . . ), the bounds that are derived from the linear algebra step using Cramer's rule are pessimistic. The gap with the behaviour observed in practice is often of one order of magnitude, and better bounds are sometimes known and derived from ad hoc methods and independent arguments. We propose a unified approach for proving output degree bounds for all instances of the class at once. The main technical tools come from the theory of realisations of matrices of rational functions and their determinantal denominators.

A unified approach for degree bound estimates of linear differential operators

TL;DR

This work frames four central problems on D-finite functions—Hermite-based creative telescoping, differential resolvent, LCLM, and symmetric product—within a single linear-relations paradigm driven by a pseudo-linear map . By representing through matrix fractions and employing state-space realisations, the authors derive precise degree bounds via determinantal-denominator theory, improving prior bounds in generic settings to for several cases. Key contributions include a unified theorem bounding the degrees of the solution polynomials in in terms of and the input data, and tight analyses for the four concrete instances, along with direct Cramer-style bounds and a thorough treatment of matrices of rational functions. The results enable tighter output-size estimates and pave the way for output-sensitive, unified algorithms for manipulating and composing D-finite functions in symbolic computation.

Abstract

We identify a common scheme in several existing algorithms addressing computational problems on linear differential equations with polynomial coefficients. These algorithms reduce to computing a linear relation between vectors obtained as iterates of a simple differential operator known as pseudo-linear map. We focus on establishing precise degree bounds on the output of this class of algorithms. It turns out that in all known instances (least common left multiple, symmetric product,. . . ), the bounds that are derived from the linear algebra step using Cramer's rule are pessimistic. The gap with the behaviour observed in practice is often of one order of magnitude, and better bounds are sometimes known and derived from ad hoc methods and independent arguments. We propose a unified approach for proving output degree bounds for all instances of the class at once. The main technical tools come from the theory of realisations of matrices of rational functions and their determinantal denominators.

Paper Structure

This paper contains 29 sections, 22 theorems, 23 equations, 1 table.

Table of Contents

  1. Introduction
  2. Results in specific instances
  3. Hermite reduction for creative telescoping
  4. Reduction to Problem \ref{['pb:lin_rel_pseudo_lin_map']}
  5. Realisation of $T$
  6. Strict properness
  7. Degree bound
  8. Differential resolvent
  9. Reduction to Problem \ref{['pb:lin_rel_pseudo_lin_map']}
  10. Realisation of $T$
  11. Strict properness
  12. Degree bound
  13. Least common left multiple
  14. Reduction to Problem \ref{['pb:lin_rel_pseudo_lin_map']} and realisation
  15. Strict properness
  16. ...and 14 more sections

Key Result

Theorem 1

Let $T \in k(x)^{n \times n}$ be a strictly proper rational matrix, $\theta = \partial_x + T$ and $a \in k[x]^n$. Suppose we have a realisation (eq:realisation) of $T$. Let $\Delta = \det M$, $\delta = \deg \Delta$, and $d_a = \deg a$. Then, there exists a solution $\eta = (\eta_0,\dots,\eta_\rho) \

Theorems & Definitions (22)

  • Theorem 1
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Theorem 2
  • Lemma 4
  • Theorem 3
  • Lemma 5
  • Theorem 4
  • Theorem 5
  • ...and 12 more