The Castelnuovo-Mumford regularity of a graded ring is an important invariant in computational commutative algebra, and there is increasing interest in multigraded generalizations. We study connections between two recent definitions of multigraded regularity with a view towards a better understanding of the multigraded Hilbert function of fat point schemes in P1 × · · · × Pk . MSC 2000: 13D02, ...

We show that the Castelnuovo–Mumford regularity of the binomial edge ideal of a graph is bounded below by the length of its longest induced path and bounded above by the number of its vertices.

The author continues the study of Frobenius amplitude, introduced in an earlier paper, and compares this to various other positivity notions. These are applied to deduce vanishing theorems. A refinement of Castelnuovo-Mumford regularity is also given.

Let I be a homogeneous ideal of the polynomial ring K[x0, . . . , xn], where K is an arbitrary field. Avoiding the construction of a minimal graded free resolution of I, we provide effective methods for computing the Castelnuovo-Mumford regularity of I that also compute other cohomological invariants of K[x0, . . . , xn]/I. We then apply our methods to the defining ideal I(V) of a projective mo...

We prove that for every integer \(k\ge 1\), there exists a connected graph \(H_k\) such \(v(H_k)={\text {reg}}(H_k)+k\), where v(G) and \({\text {reg}}(G)\) denote the v-number (Castelnuovo–Mumford) regularity of G respectively.

Let $L$ be a lattice in $ZZ^n$ of dimension $m$. We prove that there exist integer constants $D$ and $M$ which are basis-independent such that the total degree of any Graver element of $L$ is not greater than $m(n-m+1)MD$. The case $M=1$ occurs precisely when $L$ is saturated, and in this case the bound is a reformulation of a well-known bound given by several authors. As a corollary, we show t...

In this paper we study and relate several invariants connected to the solving degree of a polynomial system. This provides rigorous framework for estimating complexity system equations via Gröbner bases methods. Our main results include connection between last fall one regularity Castelnuovo–Mumford regularity.