Modular composition is the problem of computing the coefficient vector of the polynomial $f(g(x)) \bmod h(x)$, given as input the coefficient vectors of univariate polynomials $f$, $g$, and $h$ over an underlying field $\mathbb{F}$. While this problem is known to be solvable in nearly-linear time over finite fields due to work of Kedlaya & Umans, no such near-linear-time algorithms are known over infinite fields, with the fastest known algorithm being from a recent work of Neiger, Salvy, Schost & Villard that takes $O(n^{1.43})$ field operations on inputs of degree $n$. In this work, we show that for any infinite field $\mathbb{F}$, modular composition is in the border of algebraic circuits with division gates of nearly-linear size and polylogarithmic depth. Moreover, this circuit family can itself be constructed in near-linear time. Our techniques also extend to other algebraic problems, most notably to the problem of computing greatest common divisors of univariate polynomials. We show that over any infinite field $\mathbb{F}$, the GCD of two univariate polynomials can be computed (piecewise) in the border sense by nearly-linear-size and polylogarithmic-depth algebraic circuits with division gates, where the circuits themselves can be constructed in near-linear time. While univariate polynomial GCD is known to be computable in near-linear time by the Knuth--Schönhage algorithm, or by constant-depth algebraic circuits from a recent result of Andrews & Wigderson, obtaining a parallel algorithm that simultaneously achieves polylogarithmic depth and near-linear work remains an open problem of great interest. Our result shows such an upper bound in the setting of border complexity.

翻译：模组合问题旨在计算多项式$f(g(x)) \bmod h(x)$的系数向量，其中输入为定义在底层域$\mathbb{F}$上的单变量多项式$f$、$g$和$h$的系数向量。尽管Kedlaya与Umans的工作表明该问题在有限域上可在近线性时间内求解，但在无限域上尚未发现类似的近线性时间算法；目前最快的算法来自Neiger、Salvy、Schost与Villard近期的工作，其对于次数为$n$的输入需要$O(n^{1.43})$次域运算。本文证明，对于任意无限域$\mathbb{F}$，模组合问题属于具有除法门的近线性规模与多对数深度的代数电路的边界复杂度类。此外，该电路族本身可在近线性时间内构造。我们的技术还可扩展至其他代数问题，最显著的是单变量多项式最大公因式的计算问题。我们证明，在任意无限域$\mathbb{F}$上，两个单变量多项式的GCD可在边界意义下（分段地）通过具有除法门的近线性规模与多对数深度的代数电路计算，且这些电路本身可在近线性时间内构造。尽管已知Knuth–Schönhage算法或Andrews与Wigderson近期结果中的常数深度代数电路可在近线性时间内计算单变量多项式GCD，但如何设计一个同时实现多对数深度与近线性工作量的并行算法，仍然是备受关注且悬而未决的问题。我们的结果表明，在边界复杂度的框架下，存在这样的上界。