The Multivariate Hensel Lemma for local rings is usually proved as a consequence of the Grothendieck version of Zariski's Main Theorem. This version deals with a more general situation that is a priori much more difficult. In this paper, we give a direct proof of the Multivariate Hensel Lemma for ultrametric fields, in the framework of constructive mathematics and without using~ZMT. In the framework of classical mathematics, our result entails the Lemma for rank-one valued fields.

We compare two Henselizations of a residually discrete valuation domain. Our constructive proof that a certain natural morphism is an isomorphism is also a proof in classical mathematics. Although this isomorphism is implicitly accepted as obvious in the literature, it seems that no proof was previously available.

Let R be a local domain, v a valuation of its quotient field centred in R at its maximal ideal. We investigate the relationship between R-h, the henselisation of R as local ring, and (v) over tilde, the henselisation of the valuation v, by focussing on the recent result by de Felipe and Teissier referred to in the title. We give a new proof that simplifies the original one by using purely algebraic arguments. This proof is moreover constructive in the sense of Bishop and previous work of the authors, and allows us to obtain as a by-product a (slight) generalisation of the theorem by de Felipe and Teissier. (C) 2020 Published by Elsevier Inc.

This paper gives an elementary proof of a theorem by de Felipe and Teissier in the paper Valuations and henselization (arXiv:1903.10793v1), to appear in Math. Annalen. The theorem compares two henselizations of a local domain dominated by a valuation domain. Our proofs are written in the constructive Bishop style.

The aim of this note is to discuss the following quite queer problem: to compute the reduced Grobner basis of an ideal I w.r.t. a term-ordering. without knowing neither the ideal nor the term-ordering but only a degree bound of the required Grobner basis, being allowed to pose a finite number of queries to an oracle which, given a term tau is an element of T, returns its canonical form Can(tau, I, less than or similar to) w.r.t. the unknown ideal I and term-ordering.. This problem was suggested to us by the desire to definitely dispose of a very weak paper wrongly claiming a cryptographic application of (non commutative) Grobner bases. The commutative reformulation is instead a non-obvious challenge and we consider it an helpful tool for understanding and visually describe the structure of the Grobner escalier of an ideal; moreover it allows to describe (and compute) the corner set, an helpful tool for computing Macaulay decomposition of a (non-necessarily 0-dimensional) algebra.

The division algorithm for ideals of algebraic power series satisfying Hironaka’s box condition is shown to be finite when expressed suitably in terms of the defining polynomial codes of the series. In particular, the codes of the reduced standard basis of the ideal can be constructed effectively.

In this paper we prove what we call Local Bézout Theorem (Theorem 3.7). It is a formal abstract algebraic version, in the frame of Henselian rings and \(\mathfrak {m}\)-adic topology, of a well known theorem in the analytic complex case. This classical theorem says that, given an isolated point of multiplicity r as a zero of a local complete intersection, after deforming the coefficients of these equations we find in a sufficiently small neighborhood of this point exactly r isolated zeroes counted with multiplicities. Our main tools are, first the border bases [11], which turned out to be an efficient computational tool to deal with deformations of algebras. Second we use an important result of de Smit and Lenstra [7], for which there exists a constructive proof in [13]. Using these tools we find a very simple proof of our theorem, which seems new in the classical literature.

This paper deals with the Peskine version of Zariski Main Theorem published in 1965 and discusses some applications. It is written in the style of Bishop's constructive mathematics. Being constructive, each proof in this paper can be interpreted as an algorithm for constructing explicitly the conclusion from the hypothesis. The main non-constructive argument in the proof of Peskine is the use of minimal prime ideals. Essentially we substitute this point by two dynamical arguments; one about gcd's, using subresultants, and another using our notion of strong transcendence. In particular we obtain algorithmic versions for the Multivariate Hensel Lemma and the structure theorem of quasi-finite algebras.

We consider a univariate polynomial f with real coefficients having a high degree N but a rather small number d + 1 of monomials, with d ≪ N . Such a sparse polynomial has a number of real root smaller or equal to d . Our target is to find for each real root of f an interval isolating this root from the others. The usual subdivision methods, relying either on Sturm sequences or Moebius transform followed by Descartes's rule of sign, destruct the sparse structure. Our approach relies on the generalized Budan-Fourier theorem of Coste, Lajous, Lombardi, Roy [8] and the techniques developed in Galligo [12]. To such a f is associated a set of d + 1 F-derivatives. The Budan-Fourier function V f ( x ) counts the sign changes in the sequence of F-derivatives of the f evaluated at x . The values at which this function jumps are called the F-virtual roots of f , these include the real roots of f . We also consider the augmented F-virtual roots of f and introduce a genericity property which eases our study. We present a real root isolation method and an algorithm which has been implemented in Maple. We rely on an improved generalized Budan-Fourier count applied to both the input polynomial and its reciprocal, together with Newton like approximation steps. The paper is illustrated with examples and pictures.

We give an elementary proof of what we call the Local Bézout Theorem. Given a system of n polynomials in n indeterminates with coefficients in a Henselian local domain, (V,m,k), which residually defines an isolated point in kn of multiplicity r, we prove (under some additional hypothesis on V) that there are finitely many zeroes of the system above the residual zero (i.e., with coordinates in m), and the sum of their multiplicities is r. Our proof is based on techniques of computational algebra.

We give an elementary proof of what we call Local B ezout Theorem. Given a system of n polynomials in n indeterminates with coecients in a henselian local domain, ( V;m;k), which residually denes an isolated point in k n of multiplicity r, we prove (under some additional hypothesis on V) that there are nitely many zeroes of the system above the residual zero (i.e., with coordinates in m), and the sum of their multiplicities is r. Our proof is based on techniques of computational algebra.

The aim of this note is to discuss the following quite queer Problem: \noindent GIVEN \noindent i) the free non-commutative polynomial ring, ${\Cal P} := {\Bbb F}\langle X_1,\ldots,X_n\rangle$ {\em (public)}, \noindent ii) a bilateral ideal ${\sf I}\subset {\Bbb F}\langle X_1,\ldots,X_n\rangle$ {\em (private)}, \noindent iii) a finite set $G := \{g_1,\ldots,g_l\}\subset{\sf I}$ of elements of the ideal ${\sf I}$ {\em (public)}, \noindent a noetherian semigroup term-ordering $\prec,$ {\rm (private)}, on the word semigroup ${\Cal T} := $, \noindent COMPUTE \noindent --a finite subset $H\subset\Gamma({\sf I})$ of the Gr\obner basis $\Gamma({\sf I})$ of ${\sf I}$ w.r.t. $\prec$ s.t., for each $g_i\in G$ its {\em normal form} $NF(g_i,H)$ w.r.t. $H$ is zero, \noindent means of a finite number of queries to an oracle, which, \noindent given a term $\tau\in{\Cal T}$ returns its {\em canonical form} $\Can(\tau,{\sf I},\prec)$ w.r.t. the ideal ${\sf I}$ and the term-ordering $\prec$. \qed This queer problem has been suggested to us by Bulygin (2005) where a similar problem, but with stronger assumptions, is faced in order to set up a chosen-cyphertext attack against the cryptographic system proposed in Rai (2004).

In his last ten years of mathematical activity, i.e., from 1990 to 2000, Federico Gaeta focused on Effective Methods in Algebraic Geometry (mainly Computational Invariant Theory), Combinatorics and Commutative Algebra, although his contributions in these fields were never published in journals. At the age of 70, while he was emeritus professor at the UCM, he participated in the spring semester at the IHP on Surfaces de Riemann et Fibrés Vectoriels where, after discussions with D. Eisenbud he wrote the paper A fully explicit resolution of the ideal defining N generic points in the plane. It is profusely quoted by many authors as a preprint of 1995. This paper was in fact a more explicit and computational version of an old publication of his, Sur la distribution des degrés de formes appartenant à la matrice de l'ideal homogène attaché à un groupe de N points génériques du plan, appeared in CR. Acad. Sci. Paris in 1951.

We give an elementary theory of Henselian local rings and construct the Henselization of a local ring. All our theorems have an algorithmic content.

We consider ramified (Galois) covers of the upper half plane in the category of Klein surfaces. We study the connection between the group theoretical ramification data of the cover and its geometrical properties, such as the number of the connected components of the boundary and orientability of the surface.

We present some interesting computational applications of Macaulay’s notion of inverse systems and Noether equations. In particular, we discuss an algorithm by Macualay which computes the forgotten notion (introduced by Emmy Noether) of reduced irreducible decomposition for ideals of the polynomial ring.

The division algorithm for ideals of algebraic power series satisfying Hironaka’s box condition is shown to be finite when expressed suitably in terms of the defining polynomial codes of the series.

Let K be a field of characteristic O, and let R denote K[X] or K[[X]]. It is well known that the roots of a polynomial FisinR[Z] are fractional powers series in K[[X 1d/]], where Kmacr is a finite extension of K and disin N, and they can be obtained by applying the Newton Puiseux algorithm. Although this is not true for polynomials in more than one variable, there is an important class of polynomials FisinR[Z] (R=K[[X 1, . . ., X n]]=K[[Xlowbar]]), called quasi-ordinary (QO) polynomials, for which the same property holds (i.e. their roots are fractional power series in Kmacr[[Xlowbar 1d/]]). The goal of the paper is to give an algorithm to compute these fractional power series for K (a computable field) and n=2