All Questions
Tagged with reference-request polynomials
194 questions
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Proper Pisot n-tuples
Recall that x is a Pisot number if it is real and x>1, while all of its conjugates have magnitude less than 1. Then $\{(x)^k\}$ (where $\{\cdot\}$ is the fractional part of x) approaches 0 ...
- 680
9
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0
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274
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What is known about vector subspaces of polynomial rings closed under factors?
Let $R$ be a commutative ring. Call a nonempty subset $F$ of $R$ a factroid if it is closed under sums and factors. That is:
If $a,b \in F$, then $a+b \in F$, and
If $a,b \in R$ with $a\in R$ ...
- 1,802
4
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0
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180
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Subgroups that conjugate-cover the ambient group
Let $G$ be a finite group, and suppose that a set of proper subgroups $H_1,\dotsc,H_n$ satisfy $G=\bigcup_{g\in G}\bigcup_{i=1}^nH_i^g$, where $H_i^g$ is the conjugate of $H_i$ by $g$. In this case, ...
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84
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Beyond the Bring Radical: What is known about "generating radicals" for roots of polynomials of a given degree?
Famously, there is no general solution by radicals to find roots of polynomials (real, say) with degree $d\geq 5$. Somewhat less famously, there is a general solution[?] in degree $5$ using the so-...
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46
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Reference on polynomial attached to permutation group
Let $G$ be a permutation group acting on some set. Let $C(g)$ be the set of associated cycles of an element $g\in G$, and define $l(c)$ to be the length of a cycle $c$. Now set
$$T(G) = \sum_{g\in G}\...
4
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165
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Szegő's inequality
I know Erdős-Lax's inequality and a couple of proofs. It states that:
If $P(z)=\sum_{v=0}^{n} a_{v} z^{v}$ is a complex polynomial of degree $n$ having no zeros in $|z|<1$, then
$$
\max _{|z|=1}\...
- 2,829
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108
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Primitive element theorem for algebraic functions
Given a function $f(x) : \mathbb{R}^n \to \mathbb{R}$, we call it algebraic if it satisfies a polynomial equality $g(y, x) = 0$.
This is analogous to an algebraic number being the root of a univariate ...
- 326
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141
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Historical background of finding the roots of cubic equations using continued fractions
I came across an algebra problem book written in 1899 for students of Dar al-Fonun ([dɒːɾolfʊˈnuːn], meaning, "polytechnic college",) the only modern educational institute in Iran at the ...
- 2,437
1
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2
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156
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Numerical evaluation of monomial divided differences
Suppose $f(x)=x^{n+1}$ for some $n\in\mathbb{N}$, and define the divided difference $$f[a,b]=\frac{a^{n+1}-b^{n+1}}{a-b}.$$
I am wondering about the best way to numerically evaluate $f[a,b]$ to high ...
- 45
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124
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Do Zernike polynomials form an orthogonal basis of $L^2 ( \mathbb{D} )$?
The family of Zernike polynomials is defined as follow over the unit disc $\mathbb{D}= \{ x \in \mathbb{R}^2, \ \lVert x \rVert \leq 1 \}$.
For $n \geq 0$ and $0 \leq m \leq n$ such that $n-m$ is even,...
- 2,306
5
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2
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754
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A version of Hilbert's Nullstellensatz for real zeros
$\newcommand\R{\Bbb R}$Let $Q(x_1,\dots,x_n)\in\R[x_1,\dots,x_n]$ be an irreducible polynomial such that the dimension of the set $Z:=\{(x_1,\dots,x_n)\in\R^n\colon Q(x_1,\dots,x_n)=0\}$ (defined, say,...
- 128k
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644
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On the derivative of the Bernstein polynomial
$\newcommand\Z{\Bbb Z}\newcommand\De{\Delta}$For a natural $n$ and a function $g\colon\Z\to\Bbb R$, let $B_n g$ be the corresponding Bernstein polynomial, so that
$$(B_n g)(x)=\sum_{k\in\Z} g(k)\binom ...
- 128k
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65
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Reference Request: Factorization method for polynomials whose maximum absolute value of coefficient is 1
So, today I came up with a method for factoring polynomials whose coefficients are either $-1$ or $1.$
Let me explain with examples.
Example No. 1. Factorize $P(x)=x^8+x^7+1$
Solution. It is known ...
- 161
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63
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Factorization of the symmetric function identity $E(t)=1/H(t)$ with the refined Euler characteristic polynomials of associahedra / Lagrange inversion
I've come across two matrix identities, flagged with daggers below, relating the two sets of elementary and complete homogeneous symmetric polynomials/functions via the two sets of refined Lah and ...
- 10.5k
2
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73
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An iterative formula for the Kreweras-Voiculescu polynomials (reference request)
Let
$$N(x) = 1 + \sum_{k \ge 1} N_k(h_1,h_2,...,h_k) \;x^k$$
$$ = 1 + h_1 x + (h_1^2 + h_2) x^2 + (h_1^3 + 3h_1h_2 + h_3)x^3 + (h_1^4 + 6 h_2 h_1^2 + 4 h_3 h_1 + 2 h_2^2 + h_4) x^4 + \cdots$$
be the ...
- 10.5k
5
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268
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Enumerating monomials in Schur polynomials
Let $s_{\lambda}(x_1,\dots,x_k)$ be the Schur polynomial associated to the partition $\lambda=(\lambda_1\geq\lambda_2\geq\cdots\geq\lambda_k>0)$.
Among the many things involved with these ...
- 43.2k
7
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175
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Jacobi Conjecture in characteristic $p$, results of Pekka Nousiainen
Pekka Nousiainen proved in his PhD thesis "On the Jacobian problem in positive characteristic" at Pennsylvania State University, 1981, a version of the Jacobian Conjecture mod $p$. The ...
- 25.5k
6
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1
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588
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A numerical matrix of power sum polynomials
Let $p_i=x_1^i+x_2^i+\cdots+x_m^i=\sum_{k=1}^mx_k^i$ be the power sum polynomials. Then, the determinant of the $m\times m$ Hankel matrix $M_m=(p_{i+j-2})$, for $1\leq i,j\leq m$, has a neat ...
- 43.2k
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109
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Applications of Jack polynomials
I developed four libraries (Julia, R, Python, Haskell) for the computation of Jack polynomials. I developed them for fun because I found this was programmatically interesting. But now I'd like them to ...
- 2,560
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1
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205
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Catalan and path pairs in polynomials
Define $\mathbf{K}_n$ to be the set of all $(2n+1)$-tuple sequences $\mathbf{a}=(a_0,a_1,\dots,a_{2n})\in\{-1,1\}^{n+1}$ satisfying: (a) there are $n$ occurrences of $-1$ and $n+1$ of $+1$; (b) all ...
- 43.2k
2
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358
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q-polynomials in terms of a basis
Consider the polynomials
$$f_n(q)=\prod_{j=1}^n(1+q^j) \qquad \text{and} \qquad g_m(q)=1+q+q^2+\cdots+q^m.$$
I'll list a few examples to motivate my question. Direct calculations show that
$$f_1=g_1, \...
- 43.2k
3
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60
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Closed formula for $\sum\limits^\infty_{k=0}\frac1{(k+a)(k+b)} L^1_k(x)L^1_k(y) $
Let $ L^{\alpha}_{n}(x)=\sum^{n}_{k=0} \binom{n+\alpha}{n-k}\big(-1\big)^{k}\frac{x^{k}}{k!},\alpha>-1$ be Laguerre polynomials of type $ n$.
Is there a closed formula for $$\sum^{\infty}_{k=0}\...
- 501
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462
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Closed formula for Hermite polynomials
Hermite polynomials $H_k(x), x \in \mathbb{R}, k \in \mathbb{N}$ are defined by the formula
$$
H_k(x)=(-1)^k e^{x^2} \frac{d^k}{d x^k}\left(e^{-x^2}\right) .
$$
Each $H_k(x)$ is a polynomial of exact ...
- 501
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334
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Given $P\in \mathbb{Z}[x]$ there is a nonzero $Q\in \mathbb{Z}[x]$ such that $H(PQ)\leq M(P)$
I am looking for a proof (or references) for the following result:
If $P\in \mathbb{Z}[x]$ then there exists a nonzero polynomial $Q\in \mathbb{Z}[x]$
such that
$$H(PQ)\leq M(P)$$
where
$H(R)=\max\{|...
- 343
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1
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181
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Effective bound on "Jacobian rank" for (regular) planar algebraic curves
Let an irreducible, square-free complex polynomial $f\in \mathbb C[x,y]$ be given. It is well known that the curve $\mathcal C:=\{f=0\}$ is nonsingular if and only if $\mathbb C[x,y]=<f,\partial_xf,...
- 5,428
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349
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Log associahedra and log noncrossing partitions--raising ops and symmetric function theory for $A_n$ (references)
Where do the following three sets $[LA]$, $[ILA]$, and $[LN]$ of partition polynomials appear in the literature?
There are two sets of partition polynomials, not in the OEIS, that serve as the ...
- 10.5k
3
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2
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459
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Short sequence beats long sequence
I have encountered some comparison between two binomial sums. It was amusing how the one with "fewer" summands exceeds (in value) than the other which consists of many more terms. In fact, ...
- 43.2k
3
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151
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Extension of work by Novelli and Thibon on noncommutative symmetric functions and Lagrange inversion
(Edit May 12, 2023: I just put up a brief summary of some of my notes on the partition polynomials described below in my WordPress mini-arXiv at "As Above, So Below". It contains multinomial ...
- 10.5k
3
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70
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Degree of an even/odd part of a formal power series over a polynomial ring
Let $K$ be a field with $\operatorname{char}K\ne 2$ (say, $K=\mathbb{R}$ or $\mathbb{C}$) and consider a formal power series $f=f(x)\in K[[x]]$ such that $[K[x,f]:K[x]\,]=d$. Suppose $f_e,f_o\in K[[x]]...
- 1,190
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77
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Flexagons and noncrossing partitions
Turns out a couple of series related to the faces of flexagons
popped up in my explorations of combinatorial reciprocities in a group algebra for sets of partition polynomial (ParPs) related to the ...
- 10.5k
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6
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Can the positive root of this polynomial be expressed elementarily?
For each real $A>0$, let $x_A$ denote the positive root $x$ of the polynomial $x^5-3x-A$. Is the function $(0,\infty)\ni A\mapsto x_A$ elementary?
[I am using this definition of elementary ...
- 128k
3
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0
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116
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A theory of refined h- and f-polynomials for the permutahedra, associahedra, noncrossing partitions, and tropical Grassmannians (references)
Looking for references (insights) on a theory encompassing a notion of refined face polynomials and their associated refined h-polynomials that are generalizations of the relation between ordinary f-...
- 10.5k
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The solutions of $\sum_{i=1}^n w_i x^{1/i} = \alpha$
Let $n \in \mathbb N^* , \alpha \in \mathbb R$, and $w_i \in \mathbb R$ for all $i=1, \ldots, n$. Consider the map $f:\mathbb R_{\ge 0} \to \mathbb R$ defined by
$$
f(x) := \sum_{i=1}^n w_i x^{\color{...
- 657
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Largest part and length of a partition in play
If $\lambda=(\lambda_1\geq\lambda_2\geq\dots\geq\lambda_k)\vdash n$ is an integer partition of $n$ then $\lambda_1$ is its largest part and $k$ is its length, $\ell(\lambda)$.
Define the statistic $...
- 43.2k
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884
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And, yet, another evaluation to Catalan numbers
Construct the $n$-tuple Cartesian product of the ternary set $X_n=\{0,1,2\}\times\cdots\times\{0,1,2\}=\{0,1,2\}^n$. Define its subset $W_n$ according to the rule (here $y=(y_1,\dots,y_n)$ is made use ...
- 43.2k
2
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173
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Matching polynomial, but $K_2$ is replaced by $K_3$. Have these been studied?
Given a simple graph $G=(V,E)$, we can consider matchings, $M\subseteq E$,
where $M$ is a matching iff no vertex is shared between different edges.
The number of edges in $M$ is denoted $|M|$.
The ...
- 15.8k
3
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2
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255
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Inequality for Gaussian polynomials III
Recall the constructions $[n]_q=\frac{1-q^n}{1-q}, [n]!_q=[1]_q[2]_q\cdots[n]_q$ with $[0]!_q:=1$ and the $q$-binomials (Gaussian polynomials)
$$\binom{n}k_q=\frac{[n]!_q}{[k]!_q[n-k]!_q}.$$
Given two ...
- 43.2k
4
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128
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Inequality for $q$-binomials
Recall the constructions $[n]_q=\frac{1-q^n}{1-q}, [n]!_q=[1]_q[2]_q\cdots[n]_q$ with $[0]!_q:=1$ and the $q$-binomials (Gaussian polynomials)
$$\binom{n}k_q=\frac{[n]!_q}{[k]!_q[n-k]!_q}.$$
Given two ...
- 43.2k
2
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80
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Inequality on polynomials
Recall $[n]_q=\frac{1-q^n}{1-q}, [n]!_q=[1]_q[2]_q\cdots[n]_q$ and the Gaussian polynomial $\binom{a}{b}_q=\frac{[a]!_q}{[b]!_q[a-b]!_q}$ with $[0]!_q:=1$.
Given two polynomials $U(q)=\sum_k\alpha_kq^...
- 43.2k
2
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0
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117
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A multi-variable "Fibonacci polynomial"?
There is a tremendous literature on the Fibonacci sequence, including its polynomial analogue $F_{-1}=0, F_0=1$ and
$$F_n(x)=xF_{n-1}(x)+F_{n-2} \qquad \text{for $n\geq1$}.$$
What I have found is the ...
- 43.2k
8
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1
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269
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MacMahon Master Theorem for non-matching coefficients
Let $ A$ be a complex $ n$ by $ n$ matrix and $ x_1, \dots, x_n$ be a set of commuting variables. Let $ X_i = \sum_i a_{ij}x_j$. MacMahon's Master Theorem (MMT) states that
\begin{align}
[x_1^{...
- 1,608
1
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0
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60
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Index-decomposition identity for Chebyshev polynomials
Let $n$ be a positive integer. Denote $ \left[ n \right] \equiv \{1, \ldots , n \}$. Denote $ T_{n} \left( x \right) $ as the $n$-th Chebyshev polynomials of the first kind. Let $ m_{1}, \ldots , m_{n}...
- 149
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0
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151
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determine degree of boolean polynomial given as black box
I searched a lot but couldn't find a good resource that addresses this question. Given a boolean polynomial with $n$ boolean variables as a black box, what is the most efficient way to compute its ...
- 103
4
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1
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271
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The highest power of $2$ dividing a polynomial evaluated at $x=3$
Let $\nu_2(a)$ be the $2$-adic valuation of an integer $x$, i.e. the largest power $t$ such that $2^t$ divides $x$.
Define the operator $D=x\frac{d}{dx}$ and the polynomial $\Phi_k(x)=\frac{x^{k+1}-1}{...
- 43.2k
5
votes
1
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396
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Characteristic polynomial of a simple matrix: Chebyshev?
In my recent MO question, Darij Grinberg mentioned a closely related (structure-wise) determinant, that is,
$$\det\left(x_{\min\{i,j\}}\right)_{i,j}^{1,m}=x_1(x_2-x_1)(x_3-x_2)\cdots(x_m-x_{m-1}).$$
...
- 43.2k
4
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1
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313
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A determinant of perfect square polynomials
Usually, I like working with determinants related to the Vandermonde matrix, i.e.
$$\det(x_j^{i-1})=\prod_{i<j}(x_j-x_i).$$
However, I run into some unusual matrix and its determinant. Define the $(...
- 43.2k
3
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1
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165
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The inverse of a symbolic matrix (with reciprocal binomials) has Laurent entries
Recalling the $q$-binomials (Gaussian polynomials). Let $[n]_q!=\prod_{j=1}^n\frac{1-q^j}{1-q}$ and
$\binom{n}k_q=\frac{[n]_q!}{[k]_q!\cdot[n-k]_q!}$.
Now, consider the $n\times n$ matrix $\mathbf{M}...
- 43.2k
3
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0
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207
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On a variation of the Vandermonde matrix
The ubiquitous Vandermonde matrix, of entries $(x_i^{j-1})_{i,j}^{1,n}$, and its determinant
$$\prod_{i<j}^{1,n}(x_j-x_i)$$
have found many utilities in Combinatorics and Physics, among other ...
- 43.2k
4
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0
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186
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A problem in the spirit of P. Borwein's polynomials
A well-known conjecture (now a theorem) of P. Borwein (see Wang and Krattenthaler - An asymptotic approach to Borwein-type sign pattern theorems) states:
For all positive integers $n$, the sign ...
- 43.2k
1
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1
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204
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Interpret this matrix and its determinant
Let $n\geq1$ be an integer. Take the matrix $M(n)$, with entries, $M_{i,j}(n)=\sin\left(\frac{(i+j)\pi}2\right)$ if $i\neq j$ and $M_{i,i}(n)=x_i$.
I wish to ask (this question has been modified from ...
- 43.2k