Quasi-linear Time Multiplication of Sparse Polynomials with Integer Coefficients

2026-06-10

AI Summary

This work addresses the long-standing absence of a truly quasi-linear time, output-sensitive algorithm for multiplying sparse polynomials with integer coefficients. By integrating modular black-box interpolation with sparse interpolation techniques, the authors present the first algorithm achieving rigorous quasi-linear bit complexity in this setting. Their approach refutes a prior claim of having resolved the problem and establishes output-sensitive quasi-linear bit complexity for integer-coefficient sparse polynomial multiplication. Moreover, over finite fields, the method further optimizes the bit complexity to be linear in the number of terms, the logarithm of the degree, and the logarithm of the field size.

Abstract

Sparse polynomial multiplication is a fundamental problem in computer algebra and the theory of computation, and the development of a quasi-linear time output-sensitive multiplication algorithm has been posed as an open challenge. In this paper, a counterexample is provided to a previously claimed solution to this open problem for integer coefficients. By employing the existing quasi-linear modular-black-box interpolation algorithm, we are able to provide an algorithm with quasi-linear bit complexity for the integer coefficients setting. Furthermore, in the case of coefficients over a finite field, we obtain an algorithm whose bit complexity is linear in the number of terms, the logarithm of the degree, and the logarithm of the size of the finite field.

Problem

• sparse polynomial multiplication
• quasi-linear time
• integer coefficients
• output-sensitive algorithm
• computer algebra

Innovation

• sparse polynomial multiplication
• quasi-linear time
• output-sensitive algorithm
• bit complexity
• modular black-box interpolation