Tagged Questions

-1
votes
1answer
70 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
0answers
40 views

Example request: prime ideal that degenerates to a square free monomial ideal which is *not* Cohen-Macaulay

Let $P$ be a prime ideal in a polynomial ring which, with respect to some monomial order, has a squarefree initial ideal, $\textrm{in}(P)$. I would like to know an example where $\textrm{in}(P)$ is ...
0
votes
1answer
34 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
49 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
470 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
106 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
45 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
781 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 ...