# Questions tagged [monomial-ideals]

A monomial ideal in a polynomial ring is an ideal generated by monomials.

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### Gröbner bases of resultants and their monomial ideals

$\newcommand{QQ}{\mathbb{Q}}$ Consider the ring $R = \QQ[x, a_1,\ldots,a_m]$ for a certain integer $m$ and the homogeneous polynomial $$f = x^{m+1} + \sum_{i=1}^m a_i^i x^{m+1 - i}$$ Now let  ...
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### Cut ideal of two graphs?

Consider a connected graph $G$ and a connected graph $H$. Their graph ideals are their path ideals. The Alexander duality of the graph ideals give the cut ideals. $G$ and $H$ are not connected to each ...
### if $\Delta$ is pure, then what happens to betti-numbers of $I_{\Delta}$ or $I_{\Delta^v}$
Assume that $\Delta$ is a simplicial complex and $\Delta ^v$ is its Alexander dual. Let in addition $\Delta$ be pure, then what happens to betti-numbers of $I_{\Delta}$ or $I_{\Delta^v}$? Is there a ...
Exercise 8.8 in Monomial Ideals by Herzog and Hibi: Let $I\subset S=K[x_{1},...,x_{n}]$ be a graded ideal which has linear quotients with respect to a homogeneous system of generators \$f_{1},...,f_{m}...