# Questions tagged [laurent-polynomials]

The laurent-polynomials tag has no usage guidance.

11
questions

**3**

votes

**1**answer

76 views

### “ Laurent expansion” of quasi-periodic complex complex function

Suppose a complex function $f(z)$ depends only on $z$, and satisfies the quasi-periodicity in both directions:
$$f(z+ a_x)= e^{i \theta_{a_x}} f(z)$$
$$f(z+ i a_y)= e^{i \theta_{a_y}} f(z)$$
where $\...

**13**

votes

**2**answers

525 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

286 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 ...

**3**

votes

**1**answer

205 views

### Cancellation problem for Laurent polynomial rings and power series rings

Throughout, let $k$ be an algebraically closed field. For two $k$-algebras $A,B$ let us write $A \cong_k B$ to mean that $A,B$ are isomorphic as $k$-algebras.
It is known that if $A$ is an integral ...

**7**

votes

**2**answers

195 views

### Complex symmetric Matrices over the the field of Laurent series

Let $K=\mathbb C((z))$ be the field of Laurent series in the variable $z$, and consider the involution on $K$ that sends $f(z)$ to $f(-z)$. A complex symmetric matrix of size $r$ over $K$ is a matrix $...

**3**

votes

**0**answers

99 views

### A monotonicity property related to Laurent polynomials

Let $L$ be a Laurent polynomial with real coefficients, i.e.,
$$L(z)=\sum_{j=-r}^{s}a_{j}z^{j},$$
where $r,s\in\mathbb{N}$ and $a_{j}\in\mathbb{R}$. Assume further that the set $L^{-1}(\mathbb{R})\...

**4**

votes

**0**answers

174 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 ...

**10**

votes

**1**answer

355 views

### Laurent polynomials associated to partitions and a $Q$-deformation of $\sigma(d)$

Let $\alpha \vdash d$ be a partition of $d$, i.e. $\alpha = (\alpha_1 \geq \alpha_2 \geq …\geq \alpha_l)$, where $\sum_k \alpha_k = d$. Define a Laurent polynomial in $Q$ as follows:
$$
P_\alpha(Q) = ...

**1**

vote

**0**answers

91 views

### Notion of transversality over the field of Puiseux series.

To a given a Laurent polynomial $f$ over the field of Puiseux seris with parameter $t$, $f \in \mathbb{C} \lbrace\lbrace t \rbrace\rbrace[z_1^{\pm1},...,z_n^{\pm 1}]$, one can associate the ...

**2**

votes

**0**answers

493 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 ...

**8**

votes

**1**answer

766 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^...