We present lifting techniques for triangular decompositions of zero-dimensional varieties, that extend the range of the previous methods. We discuss complexity aspects, and report on a preliminary implementation. Our theoretical results are comforted by these experiments.

We study the triangular representation of zero-dimensional varieties defined over the rational field (resp. a rational function field). We prove polynomial bounds in terms of intrinsic quantities for the height (resp. degree) of the coefficients of such triangular sets, whereas previous bounds were exponential. We also introduce a rational form of triangular representation, for which our estimates become linear. Experiments show the practical interest of this new representation.

The RegularChains library. This library provides functionalities for computing modulo regular chains based on the algorithms of [7]. The operation PolynomialRing allows the user to define the polynomial ring in which the computations take place and the order of the variables. The field of coefficients can be Q, a prime field, or a field of multivariate rational functions over Q or a prime field. Let us summarize the most frequently used operations. First, triangularize decomposes the common roots of any polynomial set into regular chains. The operations NormalForm and Inverse compute the normal form and the inverse (when it exists) of a polynomial w.r.t. a regular chain. The operation RegularGcd computes the gcd of two polynomials modulo a regular chain and MatrixInverse computes the inverse (when it exists) of a polynomial matrix w.r.t. a regular chain. As we saw above, computations modulo a regular chain may lead to a case discussion. In fact, this is achieved by means of the D5 Principle [4]. It is natural to ask whether these cases can be re-combined. The operation MatrixCombine implements an algorithm that recombine these cases, when this is possible. It is a byproduct of the notion of the equiprojectable decomposition of a polynomial system introduced in [2] and provided by the operation EquiprojectableDecomposition.Conclusions. The RegularChains library provides routines for computing (polynomial GCDs, inverses, . . . ) modulo regular chains. In particular, this includes computing over towers of field extensions (algebraic or transcendental). In general, this allows computing modulo any radical polynomial ideal, since the operation triangularize can decompose any such ideal into regular chains. New developments will be included in the next release. Indeed, the work reported in [2] has led to a modular algorithm [2] for triangular decompositions of the simple roots of a polynomial system. A preliminary implementation of it shows significant improvements in running time and allows us to solve more difficult problems. AbstractThe 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.

For every integer d ≥ 10, we construct infinite families {G n } n∈N of d + 1-regular graphs which ave a large girth ≥ log d |G n |, and for d large enough ≥ 1, 33 · log d |G n |. These are Cayley graphs on P GL 2 (F q ) for a special set of d + 1 generators whose choice is related to the arithmetic of integral quaternions. These graphs are inspired by the Ramanujan graphs of Lubotzky-Philips-Sarnak and Margulis, with which they coincide when d is prime. When d is not equal to the power of an odd prime, this improves the previous construction of Imrich in 1984 where he obtained infinite families {I n } n∈N of d + 1-regular graphs, realized as Cayley graphs on SL 2 (F q ), and which are displaying a girth ≥ 0, 48 · log d |I n |. And when d is equal to a power of 2, this improves a construction by Morgenstern in 1994 where certain families {M n } n∈N of 2 k + 1-regular graphs were shown to have girth ≥ 2/3 · log 2 k |M n |.

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We discuss changing the variable order for a regular chain in positive dimension. This quite general question has applications going from implicitization problems to the symbolic resolution of some systems of differential algebraic equations.We propose a modular method, reducing the problem to computations in dimension zero and one. The problems raised by the choice of the specialization points and the lack of the (crucial) information of what are the free and algebraic variables for the new order are discussed. Strong (but not unusual) hypotheses for the initial regular chain are required; the main required subroutines are change of order in dimension zero and a formal Newton iteration.

We give bit-size estimates for the coefficients appearing in triangular sets describing positive-dimensional algebraic sets defined over Q. These estimates are worst case upper bounds; they depend only on the degree and height of the underlying algebraic sets. We illustrate the use of these results in the context of a modular algorithm.This extends results by the first and last author, which were confined to the case of dimension 0. Our strategy is to get back to dimension 0 by evaluation and interpolation techniques. Even though the main tool (height theory) remains the same, new difficulties arise to control the growth of the coefficients during the interpolation process.

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