Modular composition & polynomial GCD in the border of small, shallow circuits

TL;DR

The paper addresses the border complexity of two core algebraic problems—modular composition and univariate polynomial GCD—over infinite fields. It develops near-linear-size, polylog-depth algebraic circuits with division gates that compute these problems in the border sense, and shows these circuits can be constructed in near-linear time. A key technical advance is a border-analogue of the fundamental theorem of symmetric polynomials, enabling translation from root-based to coefficient-based inputs, together with Newton iteration and resultant techniques. The results illuminate the potential of border complexity as a framework for subquadratic, parallelizable algebraic algorithms, while also highlighting barriers to certain lower-bound proofs and motivating further exploration of border versus exact complexity. The work also extends to symmetric-polynomial questions and lays groundwork for parallel GCD and modular composition in broader algebraic settings.

Abstract

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.

Theorems & Definitions (72)

• theorem 2.1: Border complexity of modular composition
• theorem 2.2: Border complexity of GCD
• theorem 2.3: Border Complexity of symmetric polynomials
• theorem 2.4: Resultant of $y+f(x)$ and $g(x)$
• lemma 4.1: Polynomial Multiplication SS71CK91
• lemma 4.2: Polynomial Division with Remainder Sieveking1972Kun74
• lemma 4.3: Univariate Multipoint Evaluation BM74
• lemma 4.4: Univariate Interpolation BM74
• corollary 4.5: Multiplying linear forms
• corollary 4.5: Multiplying linear forms