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.
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