0
votes
0answers
74 views

Complexity of turning a d-degree polynomial to 2-degree polynomial

For a very simple example, $(1+x)^4=x^4+4x^3+6x^2+4x+1$ is a 4 degree polynomial, and I want to change it to a 2-degree polynomial by add more variables, for this example, we can simply let $y=x^2$, ...
4
votes
2answers
157 views

Finding roots of a recursively given polynomial sequence / eigenvalues of an almost Toeplitz matix

I would like to find the roots of the polynomial sequence given by a recurrence relation as follows: $V_0(x) = 1-a^2$ $V_1(x) = 1-a^2 - x$ $V_{k \geq 2}(x) = (1+a^2 - x)V_{k-1}(x) - a^2V_{k-2}(x)$ ...
0
votes
1answer
127 views

Irreducibility of a resultant of real and imaginary parts of a characteristic polynomial

The following question is motivated by the study of a stability border for a robust linear time-invariant control system. Let us we have an affine family of $n\times n$ matrices with indeterminate ...
2
votes
1answer
188 views

Powers of linear functions span the space of polynomial functions?

Let $P_n$ be the space of polynomials in $n$ variables over a field of characteristic 0. I'm very sure that the space spanned by powers of linear functions is the whole space $P_n$. Anyone can come ...
2
votes
1answer
103 views

Union of orthogonal complements of subspaces is not contained in a proper algebraic variety

Consider an $n \times n$-matrix $A$ and an $m \times n$-matrix $C$ where $m < n$. For each $t \ge 0$ the kernel $\ker Ce^{At}$ is a (say $k$-dimensional) subspace. Suppose the intersection of these ...
3
votes
3answers
537 views

A basis of the symmetric power consisting of powers

I have asked this question on math.se, but did not get an answer - I was quite surprised because I thought that lots of people must have though about this before: Let $V$ be a complex vector space ...
1
vote
0answers
200 views

Algebraic Independence of Polynomials in n Variables with Real Coefficients

I am considering the problem of determining the algebraic independence of $n$ polynomials in $m$ variables with real coefficients, where $m \geq n$. The variables will be denoted by $a_{1}, a_{2}, ... ...
8
votes
3answers
420 views

Relating a Polynomial equation to the characteristic equation of a Hermitian matrix

This question arose out of mere curiosity. Given a polynomial equation and I happen to know that its roots are real (but not the roots itself). Does it mean it is the characteristic equation of a ...
2
votes
2answers
345 views

The number of solutions of a matrix equation

Let $P(X) = a_nX^n + \cdots + a_1X + a_0$ be a polynomial, $a_i \in \mathbb{R}$ for all $i$. Set $$S = \lbrace A \in \mathbb{M}_n: P(A) = 0 \rbrace.$$ We consider the following relation $\sim$ on ...
14
votes
1answer
654 views

Reconstructing a word

Let $w(a,b)$ be a word in two letter alphabet. Let $$A=\left(\begin{array}{lll}x_1 & x_2 & x_3\\\ x_4 &x_5 & x_6\\\ x_7 & x_8 & x_9\end{array}\right), ...
3
votes
1answer
385 views

polynomial matrices and its spectrum

Hello, all! I have a polynomial non-singular square matrix over $\mathbf{F} _q[x]$, $$\underset{l \times l}{G(x)} = \left( \begin{matrix} g _{0,0}(x) & g _{0,1}(x) & \ldots & g ...
10
votes
3answers
961 views

Which polynomials are determinants of a symmetric matrix with linear entries?

Let $k$ be a field. Can each degree $n$ polynomial $P(t) \in k[t]$ be written as the determinant of the matrix $A + tB$, where $A$ and $B$ are two symmetric $(n \times n)$-matrices with entries in ...
2
votes
1answer
284 views

Symmetric polynomials preserving $-1,1$ matrices

If $A$ is an $n\times n$ integer matrix, then trivially $S=A+A^t$ and $P = AA^t$ where $t$ is ``transpose", are both symmetric. Assume that $A$ is also a "$\lbrace -1,1 \rbrace$" matrix, i.e., the ...
2
votes
2answers
368 views

on existence of matrices X, Y s.t. XAY is diagonal over non-commutative ring

Given $A\in Mat_{n\times n}(R)$ where $R$ is a non-commutative associative ring are there exist any (non-zero) matrices $X, Y\in Mat_{n\times n}(R)$ such that $XAY=diag(a_1, \ldots , a_n)$ for some ...
0
votes
1answer
176 views

Codimension of non-common condition is 2?

If we have n homogeneous polynomials (over algebraically closed field) $f_1\ldots , f_n$ on variables $x_0, \ldots , x_n$ $$ f_i(x_0, \ldots , x_n) = \sum_{j_0,\ldots , j_n} a_{i, j_0, \ldots , j_n} ...
1
vote
1answer
348 views

Rank-1 decomposition conjecture for matrix with linear function elements

Can Anyone prove the following conjecture? Consider $k$ rational function vectors $V_1(x_1,\cdots,x_n),\cdots,V_k(x_1,\cdots,x_n)$. They are called \textbf{linearly dependent} if there exists ...
4
votes
1answer
695 views

annihilator/common left multiple of matrix polynomials

Let $A_{n,d}$ be the space of polynomials of degree $d$ or less whose coefficients are real $n\times n$ matrices --- or, if you prefer, the space of matrices whose entries are degree-$d$ polynomials. ...
-3
votes
2answers
2k views

Factorizing polynomials of several variables (in a different perespective)

I am looking for factorization of polynomials of several variables in the way outlined below. Consider a second degree polynomial of two variables over the complex numbers. "P(x,y) = Ax^2 + Bxy + ...
7
votes
2answers
2k views

Quadratic forms over finite fields

I'm reading some very old papers (by Birch et al) on quadratic forms and I don't get the following point: If $f$ is a quadratic form in $X_1,X_2,\cdots,X_n$ over a finite field, then one can ...
22
votes
7answers
2k views

When is a monic integer polynomial the characteristic polynomial of a non-negative integer matrix?

Suppose $P(x)$ is a monic integer polynomial with roots $r_1, ... r_n$ such that $p_k = r_1^k + ... + r_n^k$ is a non-negative integer for all positive integers $k$. Is $P(x)$ necessarily the ...