代数函数微分方程
Differential Equations for Algebraic Functions
摘要：
一元代数函数满足多项式系数的线性微分方程是一个经典问题。线性递归遵循其幂级数展开式的系数。证明了最低阶线性微分方程的系数在函数的次数上是三次的。我们还证明了存在系数仅为二次的阶线性度的线性微分方程。此外，我们证明了存在的递归的顺序和程度接近最优。我们研究计算这些微分方程和递归的复杂性。本文给出了代数级数展开式的一个快速算法。
英文摘要：
It is classical that univariate algebraic functions satisfy linear differential equations with polynomial coefficients. Linear recurrences follow for the coefficients of their power series expansions. We show that the linear differential equation of minimal order has coefficients whose degree is cubic in the degree of the function. We also show that there exists a linear differential equation of order linear in the degree whose coefficients are only of quadratic degree. Furthermore, we prove the existence of recurrences of order and degree close to optimal. We study the complexity of computing these differential equations and recurrences. We deduce a fast algorithm for the expansion of algebraic series.
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