线性微分方程和递推的度数界
Degree bounds for linear differential equations and recurrences
AI总结：
提出统一方法，为伪线性映射迭代的线性关系问题建立精确度数界，改进或恢复已知最优界。
摘要：
线性微分方程和递推揭示了其解的许多性质。因此，这些方程非常适合表示解以及用特殊函数进行计算。我们识别了一大类现有算法，这些算法将此类表示计算为称为伪线性映射的基本算子的迭代之间的线性关系。这种形式的算法已被设计并用于解决各种计算问题，在不同背景下，包括线性微分或递推方程的有效闭包性质、计算代数函数满足的微分方程等。我们提出了一种统一的方法，为所有这些问题的解建立精确的度数界。该方法依赖于该类所有具体实例共享的公共结构。对于每个问题，得到的界是紧的。它要么改进要么恢复了先前通过特设方法推导出的已知最优界。
英文摘要：
Linear differential equations and recurrences reveal many properties about their solutions. Therefore, these equations are well-suited for representing solutions and computing with special functions. We identify a large class of existing algorithms that compute such representations as a linear relation between the iterates of an elementary operator known as a \emph{pseudo-linear map}. Algorithms of this form have been designed and used for solving various computational problems, in different contexts, including effective closure properties for linear differential or recurrence equations, the computation of a differential equation satisfied by an algebraic function, and many others. We propose a unified approach for establishing precise degree bounds on the solutions of all these problems. This approach relies on a common structure shared by all the specific instances of the class. For each problem, the obtained bound is tight. It either improves or recovers the previous best known bound that was derived by ad hoc methods.
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