Improved modular composition below n^{1.5}

Determine whether there exists an algorithm for modular composition—given polynomials f(x), g(x), h(x) over a commutative ring with h monic, compute f(g) mod h—that achieves strictly better asymptotic complexity than Brent and Kung’s algorithm and, in particular, runs in o(n^{1.5}) arithmetic operations for degree bound n.

Background

The modular composition problem asks, given three polynomials f(x), g(x), h(x) with h monic, to compute f(g) mod h. When h(x)=x^n, it specializes to power series composition. Brent and Kung’s classical algorithm achieves a complexity of O(n^{(ω+1)/2}) + O(mul(n)(n log n)^{1/2}), where ω is the matrix multiplication exponent.

Motivated by this, major references (Bürgisser–Clausen–Shokrollahi, open problem 2.4; von zur Gathen–Gerhard, research problem 12.19) posed whether one can surpass Brent and Kung’s approach, and even the n^{1.5} barrier. Subsequent work provided partial answers: Kedlaya and Umans achieved (n log q)^{1+o(1)} bit complexity over finite fields, and Neiger–Salvy–Schost–Villard gave an algebraic algorithm with complexity O(n^κ), where κ≤1.43, under field assumptions. The present paper contributes another partial answer by achieving near-linear time for power series composition over arbitrary commutative rings.