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On integrals of non-autonomous dynamical systems in finite characteristic

Nalini Joshi, Pieter Roffelsen

TL;DR

This work constructs simultaneous polynomial integrals of motion for the difference-differential Painlevé system formed by dP_{II} and P_{IV} over fields of finite characteristic p>0 using a difference Lax pair. The central object is 𝓘_p = Tr[A(p−1)⋯A(0)], which is a polynomial integral with total degree 3p and bi-degree (2p,2p); for p≠3 a corrected invariant 𝒢_p yields complete Weyl-group invariance. The authors relate fibre reducibility to Riccati reductions, produce sporadic algebraic solutions from fibre singularities, and establish a projective reduction to dP_I under a hyperplane reduction α_2=1/2. They further show that the integrals descend to dP_I under this projective reduction, and provide explicit reductions and factorisations in low primes, highlighting rich arithmetic structure on finite fields. The results open paths to arithmetic studies of Painlevé equations and potential cryptographic applications over finite fields.

Abstract

We use a difference Lax form to construct simultaneous integrals of motion of the fourth Painlevé equation and the difference second Painlevé equation over fields with finite characteristic $p>0$. For $p\neq 3$, we show that the integrals can be normalised to be completely invariant under the corresponding extended affine Weyl group action. We show that components of reducible fibres of integrals correspond to reductions to Riccatti equations. We further describe a method to construct non-rational algebraic solutions in a given positive characteristic. We also discuss a projective reduction of the integrals.

On integrals of non-autonomous dynamical systems in finite characteristic

TL;DR

This work constructs simultaneous polynomial integrals of motion for the difference-differential Painlevé system formed by dP_{II} and P_{IV} over fields of finite characteristic p>0 using a difference Lax pair. The central object is 𝓘_p = Tr[A(p−1)⋯A(0)], which is a polynomial integral with total degree 3p and bi-degree (2p,2p); for p≠3 a corrected invariant 𝒢_p yields complete Weyl-group invariance. The authors relate fibre reducibility to Riccati reductions, produce sporadic algebraic solutions from fibre singularities, and establish a projective reduction to dP_I under a hyperplane reduction α_2=1/2. They further show that the integrals descend to dP_I under this projective reduction, and provide explicit reductions and factorisations in low primes, highlighting rich arithmetic structure on finite fields. The results open paths to arithmetic studies of Painlevé equations and potential cryptographic applications over finite fields.

Abstract

We use a difference Lax form to construct simultaneous integrals of motion of the fourth Painlevé equation and the difference second Painlevé equation over fields with finite characteristic . For , we show that the integrals can be normalised to be completely invariant under the corresponding extended affine Weyl group action. We show that components of reducible fibres of integrals correspond to reductions to Riccatti equations. We further describe a method to construct non-rational algebraic solutions in a given positive characteristic. We also discuss a projective reduction of the integrals.
Paper Structure (20 sections, 9 theorems, 161 equations, 1 figure, 5 tables)

This paper contains 20 sections, 9 theorems, 161 equations, 1 figure, 5 tables.

Table of Contents

  1. Introduction
  2. Overview paper
  3. Acknowledgements
  4. Main Results
  5. Analysis of integrals of motion
  6. Lax form
  7. Proof of Theorem \ref{['thm:main_theorem']}
  8. Symmetry group action on integrals of motion
  9. Fibres and special solutions
  10. Regularity of integrals of motion
  11. Reducible fibres and reductions to Riccati equations
  12. Rational solutions
  13. Singularities on fibres and sporadic algebraic solutions
  14. Projective reduction of integrals to $d\textrm{P}_{\textrm{I}}$
  15. A projective reduction to $d\textrm{P}_{\textrm{I}}$
  16. ...and 5 more sections

Key Result

Theorem 2.2

For any prime $p\geq 2$, the polynomial $\mathcal{I}_p$ is a simultaneous integral of motion for $\textrm{d}\textrm{P}_{\textrm{II}}$ and $\textrm{P}_{\textrm{IV}}$ over fields of characteristic $p$, that is, of total degree $3p$ and bi-degree $(2p,2p)$ in $(f,g)$ with for a polynomial $O(f,g)\in \mathbb{F}_p[f,g,\alpha_1,\alpha_2,t]$ of degree less than $2p$ in $f$ and degree less than $2p$ in

Figures (1)

  • Figure 4.1: Representation of blow ups leading to the initial value space $\mathcal{X}=\widehat{\mathcal{X}}\setminus D$, with in blue the components of the effective anti-canonical divisor $D$.

Theorems & Definitions (25)

  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Remark 2.4
  • Remark 3.1
  • Lemma 3.2
  • proof
  • proof : Proof of Theorem \ref{['thm:main_theorem']}
  • Remark 3.3
  • Lemma 3.4
  • ...and 15 more