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

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. KEYWORDSsparse polynomial, interpolation, arithmetic, factorization ACM Reference Format: Daniel S. Roche. 2018. What Can (and Can't) we Do with Sparse Polynomials?.

High‐performance SIMD modular arithmetic for polynomial evaluation

“…doing a computation with the bivariate images, and (iii) recovering the factors of f using sparse interpolation techniques. 15,16,18 See Roche 19 for a recent discussion on sparse polynomial interpolation methods and an extensive bibliography. If f = ∏ r i=1 f i is the factorization of f over Z then usually the factors f i have a lot fewer terms than their product f.…”

On sparse interpolation of rational functions and gcds

“…In this paper, we will use this polynomial case as a basic building block, with an abstract specification. We refer to [1,4,8,9,13,17] for state of the art algorithms for polynomial sparse interpolation and further historical references.…”

Implementing the Tangent Graeffe Root Finding Method

“…This approach goes back to work of Prony in the eighteen's century [15] and became well known after Ben-Or and Tiwari's seminal paper [2]. It has widely been used in computer algebra, both in theory and in practice; see [16] for a nice survey.…”