All Questions
Tagged with linear-algebra polynomials
153 questions
0
votes
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118
views
Uncomplete argument in Nishioka book
In Nishioka book "Mahler functions and transcendence" in the proof of Theorem 4.2.1, Nishioka asserts the following: For a matrix $A=(a_{i,j})_{1\le i\le m}$ with coefficients in $K[z]$ ($K$ ...
- 3,998
0
votes
1
answer
48
views
How to efficiently replace the Paterson-Stockmeyer algorithm and reduce non-scalar multiplications
Paterson-Stockmeyer algorithm
If we need to compute a high-degree polynomial expression, such as:
$$
P(y) = \sum_{k=0}^{B} a_k y^k
$$
the Paterson-Stockmeyer algorithm can process the powers in ...
- 3
0
votes
1
answer
98
views
Only special permutations result in a constant expression when permuting coefficients in a sum involving binomials?
Fix $n\geq 1$ and let $p_k(x) := x^k(x-1)^{n-k}$.
Suppose $\pi$ is a permutation on $\{0,1,\dotsc,n\}$, such that
$$
\sum_{k=0}^n (-1)^k \binom{n}{k} p_{\pi(k)}(x) \text{ is a constant}.
$$
Must it be ...
- 15.8k
2
votes
0
answers
89
views
The linear independence and linear elimination of non-crossing matching polynomials
Consider the polynomial set:
$$
f_{ij} = (t_i - t_j)x_i x_j + x_i - x_j, \quad (1 \leq j < i \leq 2n)
$$
where $ t_1, t_2, \dots, t_{2n} $ are pairwise distinct.
Let's look at the non-crossing ...
- 21
0
votes
0
answers
121
views
Closed form of coefficients of a finite field polynomial
I want to find a valid polynomial for a finite field $\mathbb{Z}_p[x]_{f(x)}$ with $d=deg(f(x))$. For this definition to hold, it can be deduced that $p$ must be prime and the polynomial $f(x)$ ...
- 47
3
votes
1
answer
102
views
Literature containing basic knowledge of homogeneous functions
Let $D$ be a nonempty open subset of $\mathbb{R}\times\mathbb{R}$ and $f:D\to\mathbb{R}$ be a function of two variables. For all $(x,y)\in D$ and $t>0$ such that $(tx,ty)\in D$, if the equality $f(...
- 1,091
1
vote
0
answers
60
views
Bounding the length in a module of evaluated skew polynomials
Let $R$ be a finite principal ideal ring, $S$ a Galois extension of $R$ of degree $m$ (so in particular $S$ is a free $R$-module of rank $m$, and we have an $R$-module isomorphism $S^n \cong \...
- 223
1
vote
0
answers
45
views
Rank of Hadamard product of column-wise polynomial evaluations and row-wise exponential evaluations
Consider the Hadamard product $A \odot B$ between two special matrices $A,B \in \mathbb{R}^{n \times m}$. The columns of $A$ are evaluations of polynomials, while the rows of $B$ are evaluations of ...
- 21
3
votes
2
answers
453
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Guess the next polynoms in the sequence (MO vs. AI :), count anticommuting $F_p$-matrices, P. Hrubeš conjecture
Here is a sequence of polynoms - (presumably) counting N-tuples of ANTI-commuting 2x2 matrices over $F_p, p>2$. (That is just the case of 2x2 matrices, and (surprisingly) it is not so easy to see a ...
- 24.9k
3
votes
1
answer
547
views
Why is this polynomial factorizable? [closed]
I met a curious problem on factorizing a homogenerous polynomial of degree 9.
Problem: Show that the following polynomial can be divided by $(a_1+a_2+a_3)$:
\begin{align}
&\quad\left|
\begin{array}...
- 357
3
votes
1
answer
153
views
Solving a recursion for polynomials defined by a matrix product
Define the polynomial $p_n(X) \in \mathbb{Z}[X_1,...,X_n]$ as the top left entry in $A^n$ for the $(d \times d)$ matrix
\begin{align*}
& A = \left(\begin{matrix}
X_1 & \dots & \...
- 1,295
0
votes
0
answers
79
views
Quick calculation of a symmetric product with two indices
Say I have a product $\prod_{1\le i \le N-1}\prod_{i<j\le N-1} (1+t_i t_j a_{ij})$, where $a_{ij}$s are real number. I want to calculate the coefficient of $\prod_{0 \le i < N} t_i$. Is there an ...
- 397
0
votes
1
answer
115
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Loss of degree for polynomials
Let $q$ be a power of a prime $p$, $m$ be an integer greater than $0$. Does there exist polynomials $P_0,\cdots,P_m$ of $\mathbb F_q[T]$ not all in $\mathbb F_q$ such that there exist polynomials $Q_0,...
- 3,998
4
votes
0
answers
140
views
Does an instance of this generalisation of the determinant exist?
Let $n$ be composite, $d$ a divisor greater than $1$ and $m=n/d$. Does anybody know if there is a general mapping $T$ from $n×n$ matrices to $m×m$ matrices that preserves the determinant? Over a field ...
- 1,245
1
vote
1
answer
71
views
Can non-periodic discrete auto-correlation be inversed?
I'm trying to understand whether discrete auto-correlation can be reversed.
That is, we are given $t_0, \dots, t_n \in \mathbb C$ and a set of equations
$$
t_{k} = \sum\limits_{i=0}^{n-k} b_i b_{i+k},
...
- 1,171
2
votes
0
answers
83
views
Closed form solutions to polynomial operator equations
To the best of my knowledge the problem at hand is a generalisation of monic matrix polynomials. Can a closed form solution to the following equation be found,
$$u_3A_3X^3B_3 + u_2A_2X^2B_2 + ...
1
vote
0
answers
72
views
Eigenvalues of a subset of matrix semigroup
My apologies for slightly longer post but I wanted to explain lower dimensional cases and their proofs before asking the actual question, which starts after the phrase The general case below.
A two-...
- 111
6
votes
2
answers
623
views
Can this system of equations about Newton's formula have concrete result?
Try to solve this system of equations:
$$
S_1=x_1+\dots+x_n=a;\\
S_2=x_1^2+\dots+x_n^2=a;\\
{}\cdots\\
S_n=x_1^n+\dots+x_n^n=a;
$$
And find the value of $S_{n+1}=x_1^{n+1}+\dots+x_n^{n+1},a\in\mathbb{...
- 75
2
votes
0
answers
424
views
Functional continuity of eigenvalues?
We have the following theorems!
Corollary VI.1.6 [Bhatia: Matrix Analysis]: Let $a_j(t)$, where $1\leq j \leq n$ be continuous complex valued functions defined on an interval $I$. Then there exists ...
- 233
1
vote
1
answer
141
views
Minimal number of linearly dependent rank-1 projectors
What is the minimal number of linearly dependent rank-1 projectors $\vec v \vec v^t$ in dimension n, under the condition that every set of n column vectors $\vec v$ is linearly independent.
PS: the ...
- 1,207
7
votes
1
answer
329
views
Monomials of linearly independent polynomials remains independent?
I met with a very elementary question on linear algebra which turns out to be quite resistant when I work on it. Any comments or references on either solutions or the possible value of the question ...
- 357
0
votes
0
answers
83
views
When is the sum of matrices (circulant + [super upper triangular]) not diagonalizable?
By the circulant matrix $C \in M_n(\mathbb{R})$, we mean that
$$ C = \left[\begin{array}{c|c|c|c} e_n & e_1 & \cdots & e_{n-1} \end{array}\right] $$
where $e_1,\dots,e_n$ are the standard ...
- 4,058
1
vote
0
answers
69
views
Convolutions of (m)-associahedra and (m)-noncrossing partition polynomials--combinatorial proofs?
I'm looking for combinatorial proofs of the convolutional identity COP below and its specializations I) and II).
(Edit 6/2/2023: A combinatorial proof is sketched in a blog post by Mike Spivey of a ...
- 10.5k
2
votes
0
answers
164
views
Can the equation $1+z^p+z^q+z^r=z^n$ have multiple complex roots $z$?
The math overflow post asks whether the equation $1+z^p+z^q=z^n$ can have multiple complex roots where $p<q<n$ (On the irreducibility of certain trinomials and quadrinomials).
Q. Let us ...
- 4,058
3
votes
1
answer
190
views
Can the equation $1+z+z^q=z^n$ have multiple complex roots $z$?
It is proved here that the equation $1+z+z^2=z^n$ have no multiple complex roots.
Q. Let us consider the equation $1+z+z^q=z^n$ where $q$ and $n$ are natural numbers with $1<q<n$. Any ...
- 4,058
6
votes
1
answer
588
views
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
2
votes
1
answer
358
views
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
votes
0
answers
67
views
Efficiently finding solutions to the Rainbow cryptosystem using quotient spaces
Repost of a mathematics stackexchange question here as this concerns my research and it went unanswered on there.
In this paper, Ward Beullens gives another way to look at the Rainbow cryptosystem. In ...
- 173
4
votes
0
answers
134
views
Irreducibility of polynomials associated to binomial coefficients
Let $n \geq 2$.
Let $M_n$ be the $(n+1) \times (n+1)$ matrix with entries $\binom{l}{k}$ for $0 \leq l,k \leq n$ and $U_n=M_n + M_n^T$ and let $f_n(x)$ denote the characteristic polynomial of $U_n$.
...
- 26.5k
0
votes
1
answer
142
views
Vandermonde matrix with polynomials
Let us consider the simple Vandermonde matrix $V_n$ with $V_{ij} = \omega^{(i - 1)(j - 1)}$ where $\omega = e^{2i\pi/n}$. Its well known that for a column vector $A$, $VA$ is equivalent to evaluating ...
3
votes
0
answers
295
views
Decomposition of a determinant
Let $M$ be a $4\times 4$ symmetric matrix whose entries $m_{i,j}$ for $i,j =1,\dots,4$ are homogeneous polynomials of degree $2$ in $3$ variables. Assume that $m_{1,1} = 0$.
Does there exist a ...
- 8,998
2
votes
0
answers
97
views
How to decompose a matrix over a ring $F[X_1,\ldots,X_k]$ as a product of two matrices
Let $F$ be a field. Assume any reasonable conditions if needed, such as $F=\mathbb R$, $F=\mathbb C$, $F$ is a finite field, or $F$ has a specific characteristic, etc. Let $C$ be an $n\times1$ matrix ...
- 131
2
votes
1
answer
302
views
Dimension of a general partial derivative of a linear subspace of polynomials
Let $U \subseteq \mathbb{C}[x_1,\dots, x_n]_d$ be a linear subspace of the vector space of homogeneous degree-$d$ polynomials (including zero). I would like a proof or counterexample of the claim that ...
- 980
1
vote
0
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45
views
Lifting module homomorphisms imposing conditions on characteristic polynomials
Suppose that we are in the setting described in the first two paragraphs of this MSE post. My question wants to deal with an instance of the study of the amount of freedom that the choice of the ...
- 573
6
votes
1
answer
913
views
Resultant of linear combinations of Chebyshev polynomials of the second kind
The Chebyshev polynomial $U_n(x)$ of the second kind is characterized by
$$
U_n(\cos\theta)=\frac{\sin(n+1)\theta}{\sin(\theta)}.
$$
It seems that
$$\operatorname*{Res}_x \left( U_n(x)+tU_{n-1}(x),\...
- 437
3
votes
1
answer
135
views
On well separated circular regions in the Riemann sphere and complex polynomials
It started with a conjecture I had, see A statement on complex polynomials, which was false for $n \geq 3$, as shown by Noam D. Elkies in his answer there. The present post is an attempt to salvage ...
- 5,215
5
votes
1
answer
833
views
A statement on complex polynomials
I have a feeling the following is true.
Assume that there are $n$ mutually disjoint closed disks $D_i$ in the complex plane and $n$ complex polynomials $p_i(z)$ of degree $n - 1$, with both types of ...
- 5,215
6
votes
1
answer
402
views
Values of the determinants $\det[(j-k)^m+\delta_{jk}]_{1\le j,k\le n}\ (m=1,2,3,\ldots)$
For positive integers $m$ and $n$, let $D_m(n)$ denote the determinant $\det[(j-k)^m+\delta_{jk}]_{1\le j,k\le n}$, where the Kronecker delta $\delta_{jk}$ is $1$ or $0$ according as $j=k$ or not.
...
- 15.6k
4
votes
2
answers
319
views
Invertible linear transformation on the space of polynomials
Let $\mathbb{F}[x]$ be the polynomial ring over the field $\mathbb{F}$. Let us assume $\operatorname{char}\mathbb{F}$ is not $2$. Then the $\mathbb{F}$-linear transformation
$$\mathscr{A}:\mathbb{F}[x]...
- 671
4
votes
0
answers
148
views
Derivative of characteristic polynomial of a graph and derivative of characteristic polynomial of a vertex-deleted subgraph have a common root
Let $G$ be a simple graph and $G-i$ be one of its vertex-deleted subgraphs. Let $\phi(G,x)$ and $\phi(G-i,x)$ be the characteristic polynomials of $G$ and of $G-i$ respectively, with respect to their ...
- 327
22
votes
2
answers
2k
views
Can one deduce the fundamental theorem of algebra from real calculus and linear algebra?
Motivation: let $A\in\mathbf{R}^{n\times n}$ be symmetric. Then by the method of Lagrange multipliers, a maximum of $x\mapsto x^tAx$ on the compact unit sphere $\mathbf{S}^{n-1}$ must be an ...
- 337
4
votes
1
answer
259
views
Why do these polynomials split almost in the middle?
Start with a palindromic sequence of integers $(a_0, a_1, \ldots, a_{n+1})$, i.e. $a_j=a_{n+1-j}$, and put $a_j:=0$ for $j<0$ and $j>n+1$. You may readily guess that the choice of the binomial ...
- 13.4k
5
votes
1
answer
396
views
Characteristic polynomial of a simple matrix: Chebyshev?
In my recent MO question, Darij Grinberg mentioned a closely related (structure-wise) determinant, that is,
$$\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}).$$
...
- 43.2k
4
votes
1
answer
313
views
A determinant of perfect square polynomials
Usually, I like working with determinants related to the Vandermonde matrix, i.e.
$$\det(x_j^{i-1})=\prod_{i<j}(x_j-x_i).$$
However, I run into some unusual matrix and its determinant. Define the $(...
- 43.2k
3
votes
1
answer
165
views
The inverse of a symbolic matrix (with reciprocal binomials) has Laurent entries
Recalling the $q$-binomials (Gaussian polynomials). Let $[n]_q!=\prod_{j=1}^n\frac{1-q^j}{1-q}$ and
$\binom{n}k_q=\frac{[n]_q!}{[k]_q!\cdot[n-k]_q!}$.
Now, consider the $n\times n$ matrix $\mathbf{M}...
- 43.2k
3
votes
0
answers
207
views
On a variation of the Vandermonde matrix
The ubiquitous Vandermonde matrix, of entries $(x_i^{j-1})_{i,j}^{1,n}$, and its determinant
$$\prod_{i<j}^{1,n}(x_j-x_i)$$
have found many utilities in Combinatorics and Physics, among other ...
- 43.2k
1
vote
1
answer
204
views
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
1
vote
0
answers
370
views
Combinatorial proof of a matrix equation
I'm looking for combinatorial proofs (using, e.g., trees) of the following particular matrix equation $(I)$ and also combinatoric operational analogs of its solution via matrix inversion and/or Cramer'...
- 10.5k
0
votes
0
answers
44
views
Polynomial representation with shared root
Let $R$ be a polynomial quotient ring of the form $F_3[x_1,x_2,...,x_n]/\langle \{x_i^3-x_i\}_{1\le i\le n} \rangle$ and $\{f_i\}_{1\le i \le m}$ be elements of $R$, where $m > n$. We know that the ...
13
votes
0
answers
447
views
Unit polynomial vector fields on the sphere
Let $\mathbb{S}^3 \subset \mathbb{R}^4$ be the unit $3$-sphere. Is there a classification available for $3$-homogeneous polynomial, unit norm, vector fields on $\mathbb{S}^3$?
More explicitly, a $3$-...
- 501