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There is a positive answer to this question now by Pham Hung Quy's post here and this paper, but one can also ask this question about other gradings. This is not an answer, but a long comment: $\m …

The answer follows from Theorem 1.5.8 of Bruns/Herzog "Cohen-Macaulay-Rings": The basic reason is that for any non-graded prime $p$ one has $\text{ht}(p/p') = 1$. To see this replace $R$ by $R/p'$ and …

This is not a complete answer, just some related thoughts. I don't really know about the size that you define; the following (and the Mayr-Meyer paper) are about the degree of the $g_i$ with coefficie …

When you restrict to special classes like monomial or binomial ideals (those generated by polynomials with one (monomial) or two (binomial) terms) then combinatorial characterizations exist. For insta …

The minimal primes (and sometimes also their intersection) can be computed relatively quickly (compared to primary decomposition) using Algorithm 4 of http://arxiv.org/pdf/0906.4873v3. I've looked at …

I found a precursor of the notion in the paper "On unmixedness theorem" by Chow (American Journal of Mathematics, Vol. 86, No. 4, Oct., 1964). He considers a Segre-type product of the form $\bigoplus_ …

$k[t]$ is a PID when $k$ is a field.

You should apply a generic linear coordinate transform to the ideal and then compute the initial ideal. The matrices for which the result is the generic initial ideal is a (Zariski) open subset (Lemm …

The question of algorithmically deciding if an ideal is binomial after a (suitable, e.g. linear) automorphism of affine space is decidable and various algorithms are discussed in "When is a polynomial …

Let $k$ be a field and $A\subset \mathbb{N}^d$ a vector configuration. Let $R,S$ be commutative $k$-algebras, both graded by the affine semigroup $\mathbb{N}A$. Is the 'multidgraded Segre product' $R …

Let $S=k[x_1,\dots,x_n]$ be a polynomial ring, and $A:=k[x^{u^{(1)}}, \dots x^{u^{(l)}}]$ a monomial subalgebra, generated by monomials $x^{u^{(i)}} = \prod_{j=1}^n x_j^{u^{(i)}_{j}}$ with $u^{(i)} \i …

Is there an algorithmic (or other) way to prove that a (projective) variety is not isomorphic to a toric variety? I'd be happy with an algebraic answer (for affine or projective varieties), using the …

Let $R$ be a commutative Noetherian ring and $I\subset R$ an ideal that is irreducible in the sense that if $I = J_1 \cap J_2$, then $I=J_1$ or $I=J_2$. Is (the ideal generated by) $I$ irreducible in …

There is certainly some structure in your example, so maybe also to other ideals that you have in mind? The first thing I would do is to make experiments and try to guess a formula. Here is Macaul …

In the graded situation the concept of "minimal generators" is well-defined. Just think about the minimal generators as part of the minimal free resolution. Their number is the first total Betti num …