Minimal Monomial Reductions and the Reduced Fiber Ring of an Extremal Ideal

We characterize when the monomial maximal ideal of a simplicial affine semigroup ring has a monomial minimal reduction. When this is the case, we study the Cohen-Macaulay and Gorenstein properties of the associated graded ring and provide several bounds for the reduction number with respect to the monomial minimal reduction.

Foa a given local ring, we study the fiber cone of ideals with analytic spread one. In this case, the fiber cone has a strucure as a module over its Noether normalization which is a polynomial ring in one variable over the residue field. One may then apply the structure theorem for graded modules over a graded principal domain to get a complete descriptionof the fiber cone as a module. We analyze this structure in order to study and characterize in terms of the ideal itself t...

Given an ideal of forms in an algebra (polynomial ring, tensor algebra, exterior algebra, Lie algebra, bigraded polynomial ring), we consider the Hilbert series of the factor ring. We concentrate on the minimal Hilbert series, which is achieved when the forms are generic. In the polynomial ring we also consider the opposite case of maximal series. This is mainly a survey article, but we give a lot of problems and conjectures. The only novel results concern the maximal series ...

The proper basis formulated herein constitutes an improvement on the Gr\"obner basis for a zero-dimensional polynomial ideal. Let $K[\mathbf{x}]$ be a polynomial ring over a field $K$ with $\mathbf{x}:=(x_1,\dotsc,x_n)$. With $x_1$ being the least variable, a zero-dimensional polynomial ideal $I\subset K[\mathbf{x}]$ always has an eliminant $\chi\in K[x_1]\setminus K$ such that $I\cap K[x_1]=(\chi)$ after eliminating the other variables $\tilde{\mathbf{x}}:=(x_2,\dotsc,x_n)$....

Let $J\subset S=K[x_0,...,x_n]$ be a monomial strongly stable ideal. The collection $\Mf(J)$ of the homogeneous polynomial ideals $I$, such that the monomials outside $J$ form a $K$-vector basis of $S/I$, is called a {\em $J$-marked family}. It can be endowed with a structure of affine scheme, called a {\em $J$-marked scheme}. For special ideals $J$, $J$-marked schemes provide an open cover of the Hilbert scheme $\hilbp$, where $p(t)$ is the Hilbert polynomial of $S/J$. Those...

Let $R$ be a polynomial ring over a field $k$ with irrelevant ideal $\frak m$ and dimension $d$. Let $I$ be a homogeneous ideal in $R$. We study the asymptotic behavior of the length of the modules $H^{i}_{\frak m}(R/I^n)$ for $n\gg 0$. We show that for a fixed number $\alpha \in \mathbb Z$, $\limsup_{n\rightarrow \infty}\frac{\lambda(H^{i}_{\frak m}(R/I^n)_{\geq -\alpha n})}{n^d}<\infty.$ Combining this with recent strong vanishing results gives that $\limsup_{n\rightarrow \...

We survey classical and recent results on symbolic powers of ideals. We focus on properties and problems of symbolic powers over regular rings, on the comparison of symbolic and regular powers, and on the combinatorics of the symbolic powers of monomial ideals. In addition, we present some new results on these aspects of the subject.

When studying local properties of a polynomial ideal, one usually needs a theoretic technique called localization. For most cases, in spite of its importance, the computation in a localized ring cannot be algorithmically preformed. On the other hand, the standard basis method is very effective for the computation in a special kind of localized rings, but for a general semigroup order the geometry of the localization of a positive-dimensional ideal is difficult to interpret. ...

We give a new proof of Hilbert's Syzygy Theorem for monomial ideals. In addition, we prove the following. If S=k[x_1,...,x_n] is a polynomial ring over a field, M is a squarefree monomial ideal in S, and each minimal generator of M has degree larger than i, then the projective dimension of S/M is at most n-i.

The paper presents two algorithms for finding irreducible decomposition of monomial ideals. The first one is recursive, derived from staircase structures of monomial ideals. This algorithm has a good performance for highly non-generic monomial ideals. The second one is an incremental algorithm, which computes decompositions of ideals by adding one generator at a time. Our analysis shows that the second algorithm is more efficient than the first one for generic monomial ideals...