0
votes
0answers
23 views

What is the purpose of the generalised definition of a cluster algebra?

The seeds of a cluster algebra are normally of the form $(\textbf{x},B)$ where $\textbf{x}$ is a cluster and $B$ is a skew-symmetrizable matrix. However then I have come across a more general ...
0
votes
2answers
86 views

Hilbert series of the polynomial ring $K[X_1, \dots, X_s]$

Let $K$ be a field and $a_1, \dots, a_s \in \mathbb{N} \setminus \{0\}$. How can I compute the Hilbert series for $K[X_1, \dots, X_s]$, where $\deg(X_i)=a_i$?
2
votes
1answer
103 views

How to prove that the following morphism is surjective?

Let $R = k[x,y]$, where $k$ is a field. Let $g_0$ upto $g_n$ be homogeneous elements in $R$ of degree $d$ such that the $g_i$ have no common factors. Let $\phi: R(-d)^{n+1}_m \to R_m$ given by ...
0
votes
1answer
129 views

Monomial ideals: isomorphism problem for commutative algebras?

Let $I,J\unlhd K[x_1,\ldots,x_n]=K[x]$ be monomial ideals and $f\!: K[x]\to K[x]$ a graded isomorphism (given by a matrix $A=[\alpha_{i,j}]\in K^{n\times n}$, i.e. $x_i\mapsto\sum_j\alpha_{i,j}x_j$ is ...
1
vote
2answers
107 views

Simple Combinatorics in finite rings

Let $g = [g_{1} g_{2} \dots g_{r}] \in \Bbb Z_{q}^{*r}$ be a given vector with each $g_{i} \in \Bbb Z_{q}^{*}$ where $\Bbb Z_{q}^{*}$ is $\Bbb Z_{q} \backslash \{0\}$ and $q > 6$ is odd. How many ...
3
votes
0answers
95 views

Castelnuovo-Mumford regularity

Let $R=K[x_1,\dots,x_{10}]$, where $K$ is a field. Consider $$I=(x_1x_7,x_1x_{10},x_2x_8,x_3x_9,x_4x_{10},x_1x_5x_9,x_2x_6x_{10},x_1x_4x_5x_8,x_2x_5x_6x_9,x_3x_6x_7x_{10})$$ which is a squarefree ...
-1
votes
1answer
119 views

How to form a simplicial complex

reference from book Combinatorial Commutative Algebra by Miller and Sturmfels. i feel difficult to start learning this, when i read first few pages as i do not know where do simplicial complex come ...
0
votes
1answer
42 views

Binomial/Tensor Identity

Let $k$ be a a field and consider the space $k[x] \otimes_k k[x]$. I would like to verify the equation $$ \sum_{k=0}^{m+n} {m+n \choose k} x^k \otimes x^{(n+m)-k}= \sum_{i=0}^n \sum_{j=0}^m{n \choose ...
3
votes
1answer
56 views

Coefficients in products and powers of large polynomials

Let $f\in \mathbb{Z}[x_1,\dots,x_n]$ be a polynomial. I want to show that a certain monomial $m$ shows up with non-zero coefficient in the $r^{th}$ power of $f$. If you're lucky, you can do this as ...
0
votes
1answer
665 views

Dimension of the vector space of homogeneous polynomials.

Let $R$ be a polynomial ring with $n_k$ variables of degree $k$, for $1\leq k\leq m$. Is there a writeable formula to express the dimension of the vector space $R_l$ of degree $l$ homogeneous ...
2
votes
1answer
163 views

An identity on symmetric polynomial

In the polynomial algebra $\mathbb{C}[X_1, X_2,\ldots, X_n]$, we define a set of symmetric polynomials as follows $h_i(X_k, X_{k+1}, \ldots, X_n)$ = sum of all monomials of total degree $i$ in the set ...
0
votes
1answer
50 views

Applications of the Formal Laurent Lattice

Attach a (Laurent) monomial weight $x_1^{i_1} \cdots x_n^{i_n}$ to each point $(i_1, \dots, i_n)$ of $\mathbb{Z}^{n}$ and call it $\mathbb{Z}^{n}[x_1, x_{1}^{-1}, \dots, x_n, x_n^{-1}]$. Does this ...
7
votes
3answers
1k views

Finding the power series of a rational function

In many combinatorial enumeration problems it is possible to find a rational generating function (i.e. the quotient of two polynomials) for the sequence in question. The question is - given the ...