Univariate Polynomials: Nearly Optimal Algorithms for Numerical Factorization and Root-finding

Accelerated approximation of the complex roots and factors of a univariate polynomial

The known algorithms approximate the roots of a univariate polynomial in nearly optimal arithmetic and Boolean time. They are, however, quite involved and require a high precision of computing when the degree of the input polynomial is large, which causes numerical stability problems. We observe that these difficulties do not appear at the initial stages of the algorithms, and in our present pa...

Nearly Optimal Algorithms for Univariate Polynomial Factorization and Rootfinding II: Computing a Basic Annulus for Splitting

Character Sums and Deterministic Polynomial Root Finding in Finite Fields

Let Fq be a finite field of q elements of characteristic p. The classical algorithm of Berlekamp [1] reduces the problem of factoring polynomials of degree n over Fq to the problem of factoring squarefree polynomials of degree n over Fp that fully split in Fp, see also [8, Chapter 14]. Shoup [15, Theorem 3.1] has given a deterministic algorithm that fully factors any polynomial of degree n over...

Deterministically Factoring Sparse Polynomials into Multilinear Factors and Sums of Univariate Polynomials

We present the first efficient deterministic algorithm for factoring sparse polynomials that split into multilinear factors and sums of univariate polynomials. Our result makes partial progress towards the resolution of the classical question posed by von zur Gathen and Kaltofen in [6] to devise an efficient deterministic algorithm for factoring (general) sparse polynomials. We achieve our goal...

Numerical factorization of multivariate complex polynomials

One can consider the problem of factoring multivariate complex polynomials as a special case of the decomposition of a pure dimensional solution set of a polynomial system into irreducible components. The importance and nature of this problem however justify a special treatment. We exploit the reduction to the univariate root finding problem as a way to sample the polynomial more efficiently, c...