Around general neron desingularization

It gives some new forms of General Neron Desingularization and new applications.Comment: It simplifies the proof of Theorem 1

A method to compute the General Neron Desingularization in the frame of one dimensional local domains

An algorithmic proof of General Neron Desingularization is given here for one dimensional local domains and it is implemented in \textsc{Singular}. Also a theorem recalling Greenberg' strong approximation theorem is presented for one dimensional Cohen-Macaulay local rings.Comment: In this version some examples were added and many misprints were corrected especially from the last sectio

Nested Artin approximation

A short proof of the linear nested Artin approximation property of the algebraic power series rings is given here.Comment: We simplified the statements of proposition

Stanley conjecture on intersections of four monomial prime ideals

We show that the Stanley's Conjecture holds for an intersection of four monomial prime ideals of a polynomial algebra $S$ over a field and for an arbitrary intersection of monomial prime ideals $(P_i)_{i\in [s]}$ of $S$ such that $P_i\not\subset \Sigma_{1=j\not =i}^s P_j$ for all $i\in [s]$.Comment: accepted to Communications in Algebr

Simple General Neron Desingularization in local Q{\bf Q}-algebras

In this form will appear in Communications in Algebra.Comment: Proposition 5 was extended and a sketh proof is include

Stanley depth of monomial ideals

Let $I\supsetneq J$ be two monomial ideals of a polynomial algebra over a field generated in degree $\geq d$, resp. $\geq d+1$ . We study when the Stanley Conjecture holds for $I/J$ using the recent result of \cite{IKM} concerning the polarization

On the Bass-Quillen Conjecture and Swan's question

We present a question which implies a complete positive answer for the Bass-Quillen Conjecture.Comment: In this form will appear in Combinatorial Structures in Algebra and Geometry, Editors Dumitru Stamate, Thomasz Szemberg, Springer Proceedings in Mathematics and Statistics Series, 202

Upper bounds of depth of monomial ideals

Let $J\subsetneq I$ be two ideals of a polynomial ring $S$ over a field, generated by square free monomials. We show that some inequalities among the numbers of square free monomials of $I\setminus J$ of different degrees give upper bounds of \depth_SI/J

Depth of factors of square free monomial ideals

Let $I$ be an ideal of a polynomial algebra over a field, generated by $r$ square free monomials of degree $d$. If $r$ is bigger (or equal, if $I$ is not principal) than the number of square free monomials of $I$ of degree $d+1$, then \depth_SI= d. Let $J\subsetneq I$, $J\not =0$ be generated by square free monomials of degree $\geq d+1$. If $r$ is bigger than the number of square free monomials of $I\setminus J$ of degree $d+1$, or more generally the Stanley depth of $I/J$ is $d$, then \depth_SI/J= d. In particular, Stanley's Conjecture holds in theses cases

Depth and minimal number of generators of square free monomial ideals

Let $I$ be an ideal of a polynomial algebra $S$ over a field generated by square free monomials of degree $\geq d$. If $I$ contains more monomials of degree $d$ than $(n-d)/(n-d+1)$ of the total number of square free monomials of $S$ of degree $d+1$ then \depth_SI\leq d, in particular the Stanley's Conjecture holds in this case