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 \emph{ad hoc} methods.

翻译：线性微分方程与递推关系揭示了其解的诸多性质。因此，这些方程非常适合用于表示特殊函数的解并进行计算。我们将一类现有的计算此类表示的算法识别为一种基本算子（称为\emph{伪线性映射}）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不同背景下的各种计算问题，包括线性微分或递推方程的有效闭包性质、代数函数所满足的微分方程的计算，以及许多其他问题。我们提出了一种统一的方法，用于为所有这些问题的解建立精确的次数界。该方法依赖于该类所有具体实例所共享的共通结构。对于每个问题，所获得的界是紧的。它要么改进，要么恢复了先前通过\emph{特设}方法推导出的最佳已知界。