An algorithm for polynomial multiplication that does not depend on the ring constants
Journal of Algorithms
Algebraic complexities and algebraic curves over finite fields
Journal of Complexity
On fast multiplication of polynomials over arbitrary algebras
Acta Informatica
Simple multivariate polynomial multiplication
Journal of Symbolic Computation
Algorithmic number theory
Modern Computer Algebra
Lower bounds on the bounded coefficient complexity of bilinear maps
Journal of the ACM (JACM)
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Fast integer multiplication using modular arithmetic
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Algebraic Complexity Theory
Fast fourier transforms over poor fields
Proceedings of the 36th international symposium on Symbolic and algebraic computation
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We study the complexity of polynomial multiplication over arbitrary fields. We present a unified approach that generalizes all known asymptotically fastest algorithms for this problem and obtain faster algorithms for polynomial multiplication over certain fields which do not support DFTs of large smooth orders. We prove that the famous Schönhage-Strassen's upper bound cannot be improved over the field of rational numbers if we consider only algorithms based on consecutive applications of DFT, as all known fastest algorithms are. This work is inspired by the recent improvement for the closely related problem of complexity of integer multiplication by Fürer and its consequent modular arithmetic treatment due to De, Kurur et al. We explore the barriers in transferring the techniques for solutions of one problem to a solution of the other.