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The rational rank of the support of generalized power series solutions of differential and $q$-difference equations

J. Cano, P. Fortuny Ayuso

Abstract

Given a differential or $q$-difference equation $P$ of order $n$, we prove that the set of exponents of a generalized power series solution has its rational rank bounded by the rational rank of the support of $P$ plus $n$. We also prove that when the support of the solution has maximum rational rank, it is convergent. Using the Newton polygon technique, we show also that in the maximum rational rank case, an initial segment can always be completed to a true solution. The techniques are the same for the differential and the $q$-difference case.

The rational rank of the support of generalized power series solutions of differential and $q$-difference equations

Abstract

Given a differential or -difference equation of order , we prove that the set of exponents of a generalized power series solution has its rational rank bounded by the rational rank of the support of plus . We also prove that when the support of the solution has maximum rational rank, it is convergent. Using the Newton polygon technique, we show also that in the maximum rational rank case, an initial segment can always be completed to a true solution. The techniques are the same for the differential and the -difference case.
Paper Structure (10 sections, 16 theorems, 63 equations, 2 figures)

This paper contains 10 sections, 16 theorems, 63 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation and a Technical Lemma
  3. Solutions and Rational Rank
  4. The Newton-Puiseux Polygon
  5. Admissible polynomials with maximal rational rank and characte-ristic polynomials
  6. On the completion of admissible polynomials to solutions
  7. Autonomous ordinary differential equations of first order
  8. Example
  9. Acknowledgments
  10. Competing Interests

Key Result

Lemma 1

With the above notation and conditions, for any $j=1,\ldots,m$, there exists derivation $\mathcal{D}_j$ on $\mathbb{C}((x^{\Gamma'}))$ such that

Figures (2)

  • Figure 1: Newton Polygon of $Q=P[r(x)]$ and some elements that appear in the proof of Lemma \ref{['le:sum-zero']}.
  • Figure 2: Newton polygons of $P_0,P_{1},P_2$ and $P_3$ following the admissible generalized polynomial $x+x^{\tau}+x^2+x^3$ up to it is reached the stabilization step.

Theorems & Definitions (31)

  • Lemma 1
  • proof
  • Theorem 1
  • Corollary 1
  • Corollary 2
  • Theorem 2
  • Corollary 3
  • proof
  • proof : End of proof of Theorem \ref{['the:main1']}
  • proof : End of proof of Theorem \ref{['the:main2']}
  • ...and 21 more