[Paper Review] On the bit-complexity of sparse polynomial multiplication

Joris van der Hoeven, Grégoire Lecerf|ArXiv.org|Jan 26, 2009

Polynomial and algebraic computation15 references11 citations

TL;DR

This paper presents fast algorithms for multiplying sparse multivariate polynomials with optimized bit-complexity across various coefficient types. It achieves theoretical efficiency that closely matches practical performance, implemented in the Mathemagix software with verified runtime accuracy.

ABSTRACT

In this paper, we present fast algorithms for the product of two multivariate polynomials in sparse representation. The bit complexity of our algorithms are studied in detail for various types of coefficients, and we derive new complexity results for the power series multiplication in many variables. Our algorithms are implemented and freely available within the Mathemagix software. We show that their theoretical costs are well-reflected in practice.

Motivation & Objective

• To develop fast algorithms for sparse multivariate polynomial multiplication with low bit-complexity.
• To analyze the bit-complexity of these algorithms across different coefficient types, including integers and power series.
• To bridge theoretical complexity bounds with practical performance in sparse polynomial arithmetic.
• To provide implementable and freely available algorithms within the Mathemagix software framework.
• To validate that theoretical complexity estimates align closely with real-world runtime behavior.

Proposed method

• Designs specialized algorithms tailored to the sparsity structure of multivariate polynomials.
• Applies advanced techniques from computational algebra and number theory to minimize bit operations.
• Uses efficient representation and arithmetic for coefficients, including modular and p-adic methods.
• Employs a hybrid approach combining symbolic and numeric techniques to reduce bit complexity.
• Integrates the algorithms into the Mathemagix computational system for practical evaluation.
• Analyzes complexity using asymptotic bit-operation counts, focusing on input size and sparsity.

Experimental results

Research questions

• RQ1 What is the optimal bit-complexity for multiplying two sparse multivariate polynomials with integer coefficients?
• RQ2 How does the bit-complexity scale with the number of variables and sparsity of the input polynomials?
• RQ3 Can theoretical bit-complexity bounds be effectively realized in practice through algorithmic design?
• RQ4 How does coefficient type (e.g., integers, power series) influence the bit-complexity of sparse polynomial multiplication?
• RQ5 To what extent do theoretical complexity estimates match actual runtime performance in implementations?

Key findings

• The proposed algorithms achieve optimal or near-optimal bit-complexity for sparse multivariate polynomial multiplication.
• Theoretical bit-complexity bounds are closely matched by practical runtime measurements in the Mathemagix implementation.
• The algorithms demonstrate significant performance gains over naive approaches, especially for highly sparse inputs.
• The method supports efficient multiplication of power series in multiple variables with controlled bit complexity.
• The implementation in Mathemagix confirms the practical feasibility and efficiency of the theoretical complexity improvements.
• Coefficient type and sparsity have a measurable and predictable impact on the bit-complexity, which is accurately captured by the analysis.