微分方程与 q-差分方程广义幂级数解支集的有理秩
The rational rank of the support of generalized power series solutions of differential and $q$-difference equations
AI总结：
本文研究 n 阶微分与 q-差分方程广义幂级数解，证明解支集的有理秩不超过方程支集有理秩加 n，并在最大有理秩时证明解收敛且初始段可补全为真解。
摘要：
给定一个 n 阶微分或 q-差分方程 P，我们证明广义幂级数解的指数集合的有理秩以 P 的支集的有理秩加上 n 为界。我们还证明当解的支集具有最大有理秩时，该解是收敛的。利用牛顿多边形技巧，我们还证明在最大有理秩情形下，一个初始段总能被补全为真正的解。微分情形与 q-差分情形所用技巧相同。
英文摘要：
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.
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