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Dorin Popescu

Publications and source records attributed to Dorin Popescu.

At least 37 records · Page 2Linked to original sources
arXiv2017

Nested Artin Strong Approximation Property

We study the Artin Approximation property with constraints in a different frame. As a consequence we give a nested Artin Strong Approximation property for algebraic power series rings over a field.

math.AC↗
arXiv2016

Size and Stanley depth of monomial ideals

The Lyubeznik size of a monomial ideal $I$ of a polynomial ring $S$ is a lower bound for the Stanley depth of $I$ decreased by $1$. A proof given by Herzog-Popescu-Vladoiu had a gap which is solved here.

math.AC↗
arXiv2016

Nested Artin approximation

A short proof of the linear nested Artin approximation property of the algebraic power series rings is given here.

math.AC↗
arXiv2015

Depth in a pathological case

Let $I$ be a squarefree monomial ideal of a polynomial algebra over a field minimally generated by $f_1,...,f_r$ of degree $ d\geq 1$, and a set $E$ of monomials of degree $\geq d+1$. Let $J\subsetneq I$ be a squarefree monomial ideal generated in degree $\geq d+1$. Suppose that all squarefree monomials of $I\setminus (J\cup E)$ of degree $d+1$ are some least common multiples of $f_i$. If $J$ contains all least common multiples of two of $(f_i)$ of degree $d+2$ then $\depth_SI/J\leq d+1$ and Stanley's Conjecture holds for $I/J$.

math.AC↗
arXiv2015

Stanley depth on five generated, squarefree, monomial ideals

Let $I\supsetneq J$ be two squarefree monomial ideals of a polynomial algebra over a field generated in degree $\geq d$, resp. $\geq d+1$ . Suppose that $I$ is either generated by four squarefree monomials of degrees $d$ and others of degrees $\geq d+1$, or by five special monomials of degrees $d$. If the Stanley depth of $I/J$ is $\leq d+1$ then the usual depth of $I/J$ is $\leq d+1$ too.

math.AC↗
arXiv2014

Four generated, squarefree, monomial ideals

Let $I\supsetneq J$ be two squarefree monomial ideals of a polynomial algebra over a field generated in degree $\geq d$, resp. $\geq d+1$ . Suppose that $I$ is either generated by three monomials of degrees $d$ and a set of monomials of degrees $\geq d+1$, or by four special monomials of degrees $d$. If the Stanley depth of $I/J$ is $\leq d+1$ then the usual depth of $I/J$ is $\leq d+1$ too.

math.AC↗
arXiv2014

Three generated, squarefree, monomial ideals

Let $I\supsetneq J$ be two squarefree monomial ideals of a polynomial algebra over a field generated in degree $\geq d$, resp. $\geq d+1$ . Suppose that $I$ is generated by three monomials of degrees $d$. If the Stanley depth of $I/J$ is $\leq d+1$ then the usual depth of $I/J$ is $\leq d+1$ too.

math.AC↗
arXiv2014

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.

math.AC↗
arXiv2014

Hilbert series and Lefschetz properties of dimension one almost complete intersections

We generalize some properties related to Hilbert series and Lefschetz properties of Milnor algebras of projective hypersurfaces with isolated singularities to the more general case of an almost complete intersection ideal $J$ of dimension one. When the saturation $I$ of $J$ is a complete intersection, we get explicit formulas for a number of related invariants. New examples of hypersurfaces $V:f=0$ in $P^n$ whose Jacobian ideal $J_f$ satisfies this property and with explicit nontrivial Alexander polynomials are given in any dimension. A Lefschetz type property for the graded quotient $I/J$ is proved for $n=2$ and a counterexample due to A. Conca is given for such a property when $n=3$. Two conjectures are also stated in the paper.

math.AG↗
arXiv2013

Depth of some special monomial ideals

Let $I\supsetneq J$ be two squarefree monomial ideals of a polynomial algebra over a field. Suppose that $I$ is generated by one squarefree monomial of degree $ d>0$, and other squarefree monomials of degrees $\geq d+1$. If the Stanley depth of $I/J$ is $\leq d+1$ then almost always the usual depth of $I/J$ is $\leq d+1$ too.

math.AC↗
arXiv2012

Depth of some square free monomial ideals

Let $I\supsetneq J$ be two square free monomial ideals of a polynomial algebra over a field generated in degree $\geq 1$, resp. $\geq 2$ . Almost always when $I$ contains precisely one variable, the other generators having degrees $\geq 2$, if the Stanley depth of $I/J$ is $\leq 2$ then the usual depth of $I/J$ is $\leq 2$ too, that is the Stanley Conjecture holds in these cases.

math.AC↗
arXiv2012

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 $\depth_SI/J$.

math.AC↗