This article proposes an algorithm that merges pairwise coprime bivariate lexicographic Gröbner bases (lexGb) and an algorithm that splits a lexGb into smaller lexGbs without factorization, in a Chinese Remainder Theorem fashion.

Newton's method is an ubiquitous tool to solve equations, both in the archimedean and non-archimedean settings --- for which it does not really differ. Broyden was the instigator of what is called "quasi-Newton methods". These methods use an iteration step where one does not need to compute a complete Jacobian matrix nor its inverse. We provide an adaptation of Broyden's method in a general non-archimedean setting, compatible with the lack of inner product, and study its Q and R convergence. We prove that our adapted method converges at least Q-linearly and R-superlinearly with R-order [EQUATION] in dimension m. Numerical data are provided.

Computing gcd over a triangular set T is the core routine of the machinery of some triangular decomposition methods, in the realm of polynomial ideal theory. As such it has been studied intensively and is well-understood and implemented in several situations, especially in the case where coefficients are over a radical triangular set; It is not the case over a non-radical one. This paper introduces a gcd notion in this case, when additionally for simplicity T is assumed to be primary. It is built upon the Henselian property of the coefficient ring, and is natural in that it is linked with the subresultant sequence of a and b modulo T. A general algorithm still relies on some assumptions, except for the case of a triangular set of one variable.

This work is limited to the zero-dimensional, radical, and bivariate case. A lexicographical Gröbner basis can be simply viewed as Lagrange interpolation polynomials. In the same way the Chinese remaindering theorem generalizes Lagrange interpolation, we show how a triangular decomposition is linked to a specific Gröbner basis (not the reduced one). A bound on the size of the coefficients of this specific Gröbner basis is proved using height theory, then a bound is deduced for the reduced Gröbner basis. Besides, the link revealed between the Gröbner basis and the triangular decomposition gives straightforwardly a numerical estimate to help finding a lucky prime in the context of modular methods.

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. Categories and Subject Descriptors: I.I.2 [Computing

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.