What Can (and Can't) we Do with Sparse Polynomials?

TL;DR

This paper reviews recent advances in algorithms for sparse polynomials, focusing on arithmetic, interpolation, and factorization, highlighting theoretical and practical progress and future research opportunities.

Contribution

It provides a comprehensive overview of current sparse polynomial algorithms, emphasizing recent theoretical developments and practical implementations.

Findings

Recent algorithms improve efficiency for sparse polynomial arithmetic.

Advances in sparse polynomial interpolation techniques.

Progress in sparse polynomial factorization methods.

Abstract

Simply put, a sparse polynomial is one whose zero coefficients are not explicitly stored. Such objects are ubiquitous in exact computing, and so naturally we would like to have efficient algorithms to handle them. However, with this compact storage comes new algorithmic challenges, as fast algorithms for dense polynomials may no longer be efficient. In this tutorial we examine the state of the art for sparse polynomial algorithms in three areas: arithmetic, interpolation, and factorization. The aim is to highlight recent progress both in theory and in practice, as well as opportunities for future work.