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Sufficient set of integrability conditions of an orthonomic system

M. Marvan

TL;DR

This paper tackles the problem of constructing a provably irredundant, sufficient set of integrability conditions for orthonomic PDE systems. It introduces a reduction subsystem $\Sigma'$ of the infinitely prolonged system $\Sigma^\infty$ and a canonical framework for identifying substantial integrability conditions via monomial ideals and the principal subsets $\mathcal{X}^k_\mu$, yielding a finite, irredundant set. The authors prove that, when these conditions are satisfied, the reduced system is passive and equivalent to the full prolongation, endowing the jet space with a diffiety structure. They provide concrete algorithms to compute first- and second-kind integrability conditions and establish irredundancy across a broad class of systems, with practical advantages over previous syzygy-based approaches. The work also analyzes cross-derivatives and the poset of nontrivial cross-derivatives, offering combinatorial insight and guiding efficient implementations for large-scale, overdetermined PDE computations.

Abstract

Every orthonomic system of partial differential equations is known to possess a finite number of integrability conditions sufficient to ensure the validity of all. Herewith we offer an efficient algorithm to construct a sufficient set of integrability conditions free of redundancies.

Sufficient set of integrability conditions of an orthonomic system

TL;DR

This paper tackles the problem of constructing a provably irredundant, sufficient set of integrability conditions for orthonomic PDE systems. It introduces a reduction subsystem of the infinitely prolonged system and a canonical framework for identifying substantial integrability conditions via monomial ideals and the principal subsets , yielding a finite, irredundant set. The authors prove that, when these conditions are satisfied, the reduced system is passive and equivalent to the full prolongation, endowing the jet space with a diffiety structure. They provide concrete algorithms to compute first- and second-kind integrability conditions and establish irredundancy across a broad class of systems, with practical advantages over previous syzygy-based approaches. The work also analyzes cross-derivatives and the poset of nontrivial cross-derivatives, offering combinatorial insight and guiding efficient implementations for large-scale, overdetermined PDE computations.

Abstract

Every orthonomic system of partial differential equations is known to possess a finite number of integrability conditions sufficient to ensure the validity of all. Herewith we offer an efficient algorithm to construct a sufficient set of integrability conditions free of redundancies.

Paper Structure

This paper contains 12 sections, 9 theorems, 46 equations, 1 figure, 2 algorithms.

Table of Contents

  1. Introduction
  2. 1.
  3. 2.
  4. 3.
  5. Orthonomic systems
  6. Reduction subsystem
  7. Integrability conditions
  8. Cross-derivatives
  9. The algorithm
  10. Irredundancy
  11. Examples
  12. The poset of nontrivial cross-derivatives

Key Result

Lemma 4.7

Let $x$ be an independent variable, $\sigma \in \mathcal{X}^*$ a monomial, and $F$ a function of independent variables and parametric derivatives. Let $S D_\tau S D_x p = S D_{x \tau} p$ for every derivative $p \in 'var F$ and every monomial $\tau \le \sigma$. Then $S D_\sigma S D_x F = S D_{x \sigm

Figures (1)

  • Figure 1: A typical principal subset (Example \ref{['ex:1a']})

Theorems & Definitions (38)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Example 3.2
  • Definition 4.1
  • Definition 4.2
  • Definition 4.3
  • Definition 4.5
  • Remark 4.6
  • Lemma 4.7
  • ...and 28 more