Exact Computation of Polynomial Zeros Expressible by Square Roots

Abstract

In this paper we give an efficient algorithm to find symbolically correct zeros of a polynomial f ∈ R[X] which can be represented by square roots. R can be any domain if a factorization algorithm over R[X] is given, including finite rings or fields, integers, rational numbers, and finite algebraic or transcendental extensions of those. Asymptotically, the algorithm needs O(Tf(d2)) operations in R, where Tf(d) are the operations for the factorization algorithm over R[X] for a polynomial of degree d. Thus, the algorithm has polynomial running time for instance for polynomials over finite fields or the rationals. We also present a quick test for deciding whether a given polynomial has zeros expressible by square roots and describe some additional methods for special cases.