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

TL;DR

This paper establishes bounds on the rational rank of supports of generalized power series solutions to differential and $q$-difference equations, linking it to the equations' supports and demonstrating convergence in maximum rank cases.

Contribution

It introduces bounds on the rational rank of solutions' supports and shows convergence when the support has maximum rational rank, using Newton polygon techniques.

Findings

Rational rank of solutions is bounded by that of the equation plus its order.

Maximum rational rank solutions are convergent.

Initial segments can be extended to full solutions in maximum rank cases.

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.