On the complexity of the D5 principle

• Authors: X. Dahan;É. Schost;M. Moreno Maza;W. Wu;Y. Xie
• Venue: ACM SIGSAM Bulletin
• Year: 2005

The standard approach for computing with an algebraic number isthrough the data of its irreducible minimal polynomial over somebase field k. However, in typical tasks such as polynomialsystem solving, involving many algebraic numbers of high degree,following this approach will require using probably costlyfactorization algorithms. Della Dora, Dicrescenzo and Duvalintroduced "dynamic evaluation" techniques (also termed "D5principle") [3] as a means to compute with algebraic numbers, whileavoiding factorization. Roughly speaking, this approach leads oneto compute over direct products of field extensions of k,instead of only field extensions.In this work, we address complexity issues for basic operationsin such structures. Precisely, let[EQUATION]be a family of polynomials, called a triangularset, such that k ←K =k[X1,...,Xn]/Tis a direct product of field extensions. We write δ forthe dimension of K over k,which we call the degree ofT. Using fast polynomial multiplication andNewton iteration for power series inverse, it is a folklore resultthat for any ε > 0, the operations (+, X) inK can be performed incnεδ1+εoperations in k, for some constantcε. Using afast Euclidean algorithm, a similar result easily carries over toinversion, in the special case when K is afield.Our main results are similar estimates for the general case,where K is merely a product of fields. Followingthe D5 philosophy, meeting zero-divisors in the computation willlead to splitting the triangular setT into a family thereof, defining the sameextension. Inversion is then replaced byquasi-inversion: a quasi-inverse [6] ofα ∈ K is a splitting ofT, such that α is either zero orinvertible in each component, together with the data of thecorresponding inverses.