On the complexity of the D5 principle

Abstract

The standard approach for computing with an algebraic number is through the data of its irreducible minimal polynomial over some base field k. However, in typical tasks such as polynomial system solving, involving many algebraic numbers of high degree, following this approach will require using probably costly factorization algorithms. Della Dora, Dicrescenzo and Duval introduced "dynamic evaluation" techniques (also termed "D5 principle") [3] as a means to compute with algebraic numbers, while avoiding factorization. Roughly speaking, this approach leads one to compute over direct products of field extensions of k , instead of only field extensions. In this work, we address complexity issues for basic operations in such structures. Precisely, let [EQUATION] be a family of polynomials, called a <i>triangular set</i>, such that <i>k</i> ← <i>K</i> = <i>k</i>[<i>X</i><inf>1</inf>,...,<i>X<inf>n</inf></i>]/<b>T</b> is a direct product of field extensions. We write δ for the dimension of <i>K</i> over <i>k</i>, which we call the <i>degree</i> of <b>T.</b> Using fast polynomial multiplication and Newton iteration for power series inverse, it is a folklore result that for any ε > 0, the operations (+, X) in <i>K</i> can be performed in <i>c</i><sup><i>n</i></sup><inf>ε</inf>δ<sup>1+ε</sup> operations in <i>k</i>, for some constant <i>c</i><inf>ε</inf>. Using a fast Euclidean algorithm, a similar result easily carries over to inversion, <i>in the special case when K is a field.</i> Our main results are similar estimates for the general case, where <i>K</i> is merely a product of fields. Following the D5 philosophy, meeting zero-divisors in the computation will lead to <i>splitting</i> the triangular set <b>T</b> into a family thereof, defining the same extension. Inversion is then replaced by <i>quasi-inversion:</i> a quasi-inverse [6] of α ∈ <i>K</i> is a splitting of <b>T</b>, such that α is either zero or invertible in each component, together with the data of the corresponding inverses.