0
votes
1answer
71 views

Do we need Gröbner bases to study factor rings of polynomials?

I'm trying to understand how we can systematically study the factor rings of polynomials over a ring K. For example imagine that we're working in $R=K[x_1,...,x_n]$ and we have the ideal ...
1
vote
1answer
78 views

Product of (strongly) stable ideals and lexsegment ideals

(1) Is the product of lexsegment ideals again a lexsegment ideal? (2) Is the product of (strongly) stable ideals again (strongly) stable? I know that both of them are false and I can find examples ...
-1
votes
1answer
95 views

The h-vector of a simplicial complex

Let $S$ be a polynomial ring over a field. I want to find an ideal $ I\subseteq S$ such that $(1,2,3,1,1,1)$ is the $h$-vector of $S/I$. We have a relation between $f$-vector and $h$-vector and ...
1
vote
0answers
189 views

An Algorithm to Find the Generators of the Radical of a Monomial Ideal

Working over $R=\mathbb{C}[x_1,...,x_n]$, I'm given a ring homomorphism with $i\in{1,...,n}$ and $t\in \mathbb{C}$. $\phi_{i,t}(x_j)=x_j$ for $j\neq i$ to themselves. From this I've proven that an ...
3
votes
1answer
264 views

Dimension of a certain quotient ring of $\mathbb{C}[x_0,\ldots,x_{m-1}]$.

Let $A=\mathbb C[x_0,\dots,x_{m-1}]$ be the polynomial ring on $m$ variables. Define $X(u)=\sum_{i=0}^{m-1} x_i u^{i+1}$ and denote by $(X(u)^r)_n$ the coefficient of $u^n$ in the expansion of the ...
2
votes
0answers
71 views

First order logic in polynomial equations

Have you ever wondered which points on a conic are the intersections of tangent lines of another surface through the origin? More generally, which points on a shape hold some specified relation to all ...
2
votes
0answers
82 views

How to tell if an ideal is absolutely prime

Consider the ideal $I=(ag-ec-1,ah+bg-cf-de)$ of $R=K[a,b,c,d,e,f,g,h]$. Is $I$ prime when $K=\overline{\mathbb{F}}_2$ is the algebraic closure of a field of 2 elements? Can computers answer this ...