Questions tagged [laurent-polynomials]

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Perturbing pole of Laurent polynomial/series in a single summand

I am working with the ring of Laurent polynomials $\mathbb{F}[X,X^{-1}]$ over $\mathbb{F}$ for some algebraically closed field $\mathbb{F}$ of any characteristic. I encountered a problem emerging from ...
Jens Fischer's user avatar
9 votes
1 answer
628 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: $x^{-2}$, $...
Thien's user avatar
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2 votes
1 answer
296 views

Difference equation satisfied by discrete harmonic functions on square lattice

A function $f:\mathbb Z^2 \rightarrow \mathbb R$ is said to be discrete harmonic if it satisfies the discrete Laplacian equation $$ \Delta f(m,n) = f(m+1,n)+f(m-1,n)+f(m,n+1)+f(m,n-1)-4f(m,n) = 0~. $$ ...
Pranay Gorantla's user avatar
6 votes
1 answer
416 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=...
T. Amdeberhan's user avatar
1 vote
1 answer
294 views

Automorphisms of the ring of Laurent polynomials

Is the group of automorphisms of the ring $\mathbb{F}[t,t^{-1}]$ of Laurent polynomials known? Here, $\mathbb{F}$ is an algebraically closed field of characteristic $0$ and I am considering not ...
cl4y70n____'s user avatar
7 votes
2 answers
356 views

Idempotent Laurent polynomials (in noncommuting variables)

Let $K$ be a field and $R=K\langle X_1,\dots,X_n,X_1^{-1},\dots,X_n^{-1}\rangle$ the Laurent polynomial ring in $n$ noncommuting variables. Can $R$ have idempotents distinct from $0$ and $1$?
Ralle's user avatar
  • 471
3 votes
1 answer
125 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 $\...
user34104's user avatar
  • 487
13 votes
2 answers
575 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 ...
Sylvester W. Zhang's user avatar
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 ...
T. Amdeberhan's user avatar
4 votes
1 answer
409 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 ...
user avatar
7 votes
2 answers
222 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 $...
Z.A.Z.Z's user avatar
  • 1,871
3 votes
0 answers
101 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})\...
Twi's user avatar
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4 votes
0 answers
214 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 ...
Alexey Ustinov's user avatar
10 votes
1 answer
383 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) = ...
Jim Bryan's user avatar
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1 vote
0 answers
100 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 ...
Farhad's user avatar
  • 41
2 votes
0 answers
612 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 ...
John Jiang's user avatar
  • 4,354
8 votes
1 answer
930 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^...
Gjergji Zaimi's user avatar