Questions tagged [laurent-polynomials]
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19 questions
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Right maximal ideals in skew-Laurent rings over division Rings
Let $R$ be a Noetherian domain, and let $D$ denote its division ring. Define $S = D_q[x_1^{\pm 1}, x_2^{\pm 1}]$ as an iterated skew-Laurent polynomial ring with the relation $x_1 x_2 = q x_2 x_1$. Is ...
- 923
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Finitely generated module over a skew Laurent polynomial ring $\mathbb{K}[x_1^{\pm 1},x_2^{\pm1}]$
Let $A := \mathbb{K}_{\Lambda}[x_1^{\pm 1}, x_2^{\pm 1}, x_3^{\pm 1}, x_4^{\pm 1}]$, $B := \mathbb{K}_{\Lambda'}[x_1^{\pm 1}, x_2^{\pm 1}, x_3^{\pm 1}]$, and $C := \mathbb{K}_{\Lambda''}[x_1^{\pm 1}, ...
- 923
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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 ...
- 91
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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
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313
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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
$$
\Delta f(m,n) = f(m+1,n)+f(m-1,n)+f(m,n+1)+f(m,n-1)-4f(m,n) = 0~.
$$
...
- 277
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422
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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
$$f_n(x_1,\dots,x_r)=\left(1+\prod_{j=1}^r(1+x_j)+\prod_{j=...
- 43.1k
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387
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Automorphisms of the ring of Laurent polynomials
Is the group of automorphisms of the ring $\mathbb{F}[t,t^{-1}]$ of Laurent polynomials known? Here, $\mathbb{F}$ is an algebraically closed field of characteristic $0$ and I am considering not ...
- 353
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Idempotent Laurent polynomials (in noncommuting variables)
Let $K$ be a field and $R=K\langle X_1,\dots,X_n,X_1^{-1},\dots,X_n^{-1}\rangle$ the Laurent polynomial ring in $n$ noncommuting variables. Can $R$ have idempotents distinct from $0$ and $1$?
- 491
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" Laurent expansion" of quasi-periodic complex complex function
Suppose a complex function $f(z)$ depends only on $z$, and satisfies the quasi-periodicity in both directions:
$$f(z+ a_x)= e^{i \theta_{a_x}} f(z)$$
$$f(z+ i a_y)= e^{i \theta_{a_y}} f(z)$$
where $\...
- 487
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Integer but not Laurent sequences
Are there any sequence given by a recurrence relation:
$x_{n+t}=P(x_t,\cdots,x_{t+n-1})$, where $P$ is a positive Laurent Polynomial, satisfy:
if $x_0=\cdots=x_{n-1}=1$, then the sequence is only ...
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308
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Denominators of certain Laurent polynomials
Consider the following somos-like sequence
$$x_n=\frac{x_{n-1}^2+x_{n-2}^2}{x_{n-3}}.$$
It's known that $x_n$ is a Laurent polynomial in $x_0, x_1$ and $x_2$. I got interested in the denominators of ...
- 43.1k
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Cancellation problem for Laurent polynomial rings and power series rings
Throughout, let $k$ be an algebraically closed field. For two $k$-algebras $A,B$ let us write $A \cong_k B$ to mean that $A,B$ are isomorphic as $k$-algebras.
It is known that if $A$ is an integral ...
user111524
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Complex symmetric Matrices over the the field of Laurent series
Let $K=\mathbb C((z))$ be the field of Laurent series in the variable $z$, and consider the involution on $K$ that sends $f(z)$ to $f(-z)$. A complex symmetric matrix of size $r$ over $K$ is a matrix $...
- 1,891
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A monotonicity property related to Laurent polynomials
Let $L$ be a Laurent polynomial with real coefficients, i.e.,
$$L(z)=\sum_{j=-r}^{s}a_{j}z^{j},$$
where $r,s\in\mathbb{N}$ and $a_{j}\in\mathbb{R}$. Assume further that the set $L^{-1}(\mathbb{R})\...
- 2,188
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``Occasional'' Laurent phenomenon
This question is motivated by Richard Stanley's A question on the Laurent phenomenon (motivated by his answer to the question what is the probability that a scissor became the champion?).
He asked ...
- 12.3k
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389
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Laurent polynomials associated to partitions and a $Q$-deformation of $\sigma(d)$
Let $\alpha \vdash d$ be a partition of $d$, i.e. $\alpha = (\alpha_1 \geq \alpha_2 \geq …\geq \alpha_l)$, where $\sum_k \alpha_k = d$. Define a Laurent polynomial in $Q$ as follows:
$$
P_\alpha(Q) = ...
- 5,940
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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 ...
- 41
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633
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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
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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