All Questions
Tagged with laurent-polynomials co.combinatorics
7 questions
9
votes
1
answer
687
views
Number of Laurent monomials of n variables with degree at most d
Introduction: We have a question of how to calculate the number of $n$-variables Laurent monomials of degree at most $d$.
For example: If $n=2$, $d=2$ then we have 19 monomials, which are:
$x^{-2}$, $...
- 93
6
votes
1
answer
422
views
Constant term extraction using combinatorial Nullstellensatz
$\DeclareMathOperator\CT{CT}$Given a Laurent polynomial $g$, let $\CT(g)$ denote its constant term.
Consider the specific Laurent polynomial
$$f_n(x_1,\dots,x_r)=\left(1+\prod_{j=1}^r(1+x_j)+\prod_{j=...
- 43.1k
15
votes
2
answers
600
views
Integer but not Laurent sequences
Are there any sequence given by a recurrence relation:
$x_{n+t}=P(x_t,\cdots,x_{t+n-1})$, where $P$ is a positive Laurent Polynomial, satisfy:
if $x_0=\cdots=x_{n-1}=1$, then the sequence is only ...
10
votes
2
answers
308
views
Denominators of certain Laurent polynomials
Consider the following somos-like sequence
$$x_n=\frac{x_{n-1}^2+x_{n-2}^2}{x_{n-3}}.$$
It's known that $x_n$ is a Laurent polynomial in $x_0, x_1$ and $x_2$. I got interested in the denominators of ...
- 43.1k
4
votes
0
answers
216
views
``Occasional'' Laurent phenomenon
This question is motivated by Richard Stanley's A question on the Laurent phenomenon (motivated by his answer to the question what is the probability that a scissor became the champion?).
He asked ...
- 12.3k
2
votes
0
answers
633
views
analogues of power sum polynomials for symmetric Laurent polynomials
To deal with root systems of type B C D, one needs to understand symmetric Laurent polynomials $\Lambda$. I am wondering if the naive definition of power sum symmetric Laurent polynomials form a basis ...
- 4,456
8
votes
1
answer
961
views
Vanishing constant term in powers of a Laurent polynomial
This is motivated by idle curiosity. I recently learned a result of Duistermaat and Van Der Kallen in "Constant terms of powers of a Laurent polynomial" which says that:
If the constant term of $f^...
- 85.5k