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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 …

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 …

I am interested in polynomials with few terms ("short polynomials", "fewnomials") in ideals. A simple to state question is Given an ideal $I\subset k[x_1,\dots,x_n]$, what is the shortest polynomial …

One general fact that comes to mind: If an ideal $I\subset \mathbb{k}[x_1,\dots,x_n]$ contains an element of the form $f = gx_1 + h$ where $g,h$ don't use $x_1$, and $g$ is a nonzerodivisor mod $I$, …

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 …

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 …

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 …

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 …

Let $I\subset k[x_1,\dots,x_n]$ be an ideal in a polynomial ring in commuting variables. Is there a procedure to decide if $I$ contains a monomial and possibly to find one? Gröbner bases come to min …

To expand on Michael's comment, the Greuel, Pfister book Section 6.4 is about standard bases in formal power series rings. Quoting, The main result is that they can be computed, if the ideal is …

Assume we are given a set of ideals $I_1, \dots, I_s$ in a commutative polynomial ring. Let's define a subset indexed by $A\subseteq [s] = \{ 1,2,\dots, s\}$ as dependent if there exists an $a\in A$ s …

Many introductory texts on algebraic geometry set up some sort of algebra-geometry dictionary in which radical ideals correspond to varieties, and so on. I am wondering if there is a geometric way to …

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 …