# Nearly Optimal Sparse Polynomial Multiplication

## TL;DR

This paper presents a new nearly optimal algorithm for multiplying sparse polynomials efficiently, improving upon previous methods by handling cancellations and structural sparsity more effectively.

## Contribution

The paper introduces a clean, nearly optimal algorithm for sparse polynomial multiplication that advances previous work by addressing cancellations and structural sparsity.

## Key findings

- Algorithm runs in near-optimal time proportional to input and output size
- Handles cancellations in polynomial coefficients effectively
- Improves efficiency over previous algorithms in structural sparsity cases

## Abstract

In the sparse polynomial multiplication problem, one is asked to multiply two sparse polynomials f and g in time that is proportional to the size of the input plus the size of the output. The polynomials are given via lists of their coefficients F and G, respectively. Cole and Hariharan (STOC 02) have given a nearly optimal algorithm when the coefficients are positive, and Arnold and Roche (ISSAC 15) devised an algorithm running in time proportional to the "structural sparsity" of the product, i.e. the set supp(F)+supp(G). The latter algorithm is particularly efficient when there not "too many cancellations" of coefficients in the product. In this work we give a clean, nearly optimal algorithm for the sparse polynomial multiplication problem.