All Questions
31 questions
14
votes
0
answers
603
views
Is the Zariski density proof of Cayley-Hamilton circular?
This old MO thread and its comments contains a discussion of the Zariski density proof of Cayley-Hamilton (I have also asked a separate question about the proof Victor gives in the comments here). ...
- 118k
1
vote
0
answers
130
views
A basis of the weight space in the semi-invariant ring corresponding to the weight $\langle(2,3,2),\cdot\rangle$
I'm trying to understand Example 10.11.1 on page 225 of the book "An introduction to quiver representations" by Harm Derksen and Jerzy Weyman (see the attached screenshot below)
I want to ...
- 839
0
votes
1
answer
134
views
Existence of cyclic subspace decompositions for pairs of commuting matrices
Let $\mathbb{K}$ be an arbitrary field (possibly finite). Let $V$ be a finite-dimensional vector space over $\mathbb{K}$, and let $A,B$ be two linear endomorphisms of $V$ which commute.
For $v\in V$, ...
4
votes
0
answers
219
views
Map $\operatorname{Sym}^{mp}(V^*) \longrightarrow K^{q}$ defined by $q$ points in $\operatorname{Sym}^p(V)$
EDIT : I have edited the question and made it more specific with respect to the kind of answer I expect.
Let $V$ be a finite dimensional $K$-vector space and let $x_1, \dotsc, x_q \in V$ be $q$ points,...
- 7,300
3
votes
0
answers
249
views
Grothendieck schemes and the Sheffer differential op calculus (Rota, Roman, et al. finite operator calculus)
In "Left differential operators on non-commutative algebras" on p. 4, Michiel Hazewinkel displays "precisely the right definition of differential operator" as
$$D\; X^n = F(\tfrac{...
- 10.5k
5
votes
1
answer
523
views
Is there a non-split algebraic torus (over a finite field) satisfying the following properties?
Is there a non-split algebraic torus $T$ (over a finite field $\mathbb{F}_{\!q}$) satisfying the following properties?
$T$ is not $\mathbb{F}_{\!q}$-isomorphic to the direct product of algebraic tori ...
- 1,833
0
votes
0
answers
112
views
Existence of a subspace of having no isotropic 2-plane
Let $V$ be a vector space of dimension $n$ over the field $\mathbb {Q} $. A subspace $W$ is isotropic for a skew-bilinear form $\alpha$ on $V$ if $\alpha(x,y) = 0$ for all $x,y \in W$.
More ...
- 923
0
votes
1
answer
116
views
Order 2 matrices with entries in the polynomial ring over a field are diagonalisable
This is a variant on the question posed here, in which the OP asks for a characterisation of the diagonalisable involutions in $\operatorname{GL}_n(A)$, where $A$ is a $k$-algebra for some field $k$ ...
- 721
36
votes
3
answers
2k
views
Are large powers of polynomials linearly independent?
Let $P_1,\dots,P_k$ be polynomials over $\mathbf{C}$, no two of them being proportional.
Does there exist an integer $N$ such that $P_1^N,\dots,P_k^N$ are linearly independent?
- 4,653
2
votes
0
answers
115
views
Linear algebra. commutative algebra, a matrix having the maximal rank
I found a matrix that looks to have the maximal rank for my research.
I would really appreciate if some of you could give me any comments or suggestions.
Thanks in advance.
Sincerely, Yong-Su Shin
Let ...
- 21
1
vote
2
answers
406
views
When splitting of short exact sequence preserves the kernels
This is a problem that I thought at first was obvious but that became less clear the more I thought about it. Assume we have a finitely generated algebra $A$ over a field $k$, and a short exact ...
- 5,901
3
votes
0
answers
155
views
Bass theorem on non-affine scheme
A famous theorem of Bass tells that over a noetherian ring $A$, with $\operatorname{Spec}(A)$ connected, every projective module of infinite type is free.
Now, consider a connected noetherian scheme $...
- 3,472
3
votes
0
answers
56
views
Regular sequence of binary forms in a subspace
Let $A_d=\mathbb{C}[x_1,\dots,x_n]_d$ denote the vector space of homogeneous polynomials of degree d. And assume $U\subset A_d$ is a subspace of codimension $d-1$ such that there is no point in $\...
- 51
14
votes
1
answer
2k
views
Why does this matrix have zero determinant?
This curious identity arose from studying reductions of the maximal ideal in certain monomial algebra. It can be proved "by hand", (i.e, using Macaulay 2), but I am seeking a more conceptual ...
- 30.5k
1
vote
0
answers
246
views
Frobenius twist of a field
Let $k$ be a field of characteristic $p>0$ (not necessarily perfect). Consider the Frobenius endomorphism $F : k \to k$, $x \mapsto x^p$. I am curious about what happens when we take $k$ as a $k$-...
2
votes
0
answers
238
views
Computing the codimension of the variety defined by a system of quadratic forms
Suppose I have an $m \times n$ matrix $L$, where $m \leq n$ and each entry is $L_{i,j}(x_1, ..., x_s)$ which is a linear form over $\mathbb{C}$. Let $\mathbf{y} = (y_1, \ldots, y_n)$. Let us consider ...
- 3,625
2
votes
1
answer
440
views
Coordinate free expression for the determinant of a $2 \times 2$ matrix
Let $E\rightarrow X$ be a rank two vector bundle on a variety $X$. How can one write an abstract map $Sym^2(E)\rightarrow (\wedge^2 E)^{\otimes 2}$ that in local coordinates, when we fix a frame of $E$...
user61586
4
votes
1
answer
127
views
On the linear factors of a polynomial obtained from the determinant of a matrix whose entries are related to Binomial expansion
Consider the polynomial ring $R=\mathbb C[x,y]$.
Consider the matrix $A=\begin{pmatrix} x^5+y^5&5x^5&10x^5&10x^5&5x^5\\5y^5&x^5+y^5 &5x^5&10x^5&10x^5 \\10y^5&5y^5&...
- 1,209
8
votes
1
answer
604
views
Number of zeros of the derivatives of a polynomial
What is the maximum total number of zeroes a univariate polynomial $f\in\mathbb{C}[z]$ of degree $d$, together with all of its derivatives, can have at $k$ given points of $\mathbb{C}$?
I am ...
- 2,179
1
vote
1
answer
111
views
Sum of certain decomposable elements
Let $V$ be be a vector space of dimension $m$ over any field and $\ell\leq m$ be a positive integer. Let $\omega_1,\ldots,\omega_r \in\bigwedge^\ell V$ are linearly independent, completely ...
- 179
-1
votes
1
answer
121
views
On rank one torsion-free modules over local rings [closed]
Let $A$ be a local ring which is also an integral domain and $M$ be a rank one $A$-module. Denote by $k$ the residue field of $A$. Is $\dim M \otimes_A k \le 1$? If not, is there a known upper-bound ...
- 2,126
0
votes
0
answers
100
views
Solutions of the linear equation from K[[X_1,X_2,X_3]] to K[[X_1,X_2]]
Let $A_3 := K[[X_1,X_2,X_3]]$ be a three-variable formal power series ring over a field $K$. We consider a linear equation
$(\sharp) \phantom{aa} a_1(X_1,X_2,X_3)Y_1 + \ldots + a_n(X_1,X_2,X_3)Y_n = ...
- 563
2
votes
2
answers
3k
views
Solving a system of equations using Gröbner basis
In Sage (or any other package) when using Gröbner basis to solve a system of equations (some of which are non-linear equations) does computing the Gröbner basis for the ideal ID generated by the ...
- 23
3
votes
1
answer
891
views
Is every element of $\mathrm{SL}(n,R)$ of finite order diagonalizable?
Let $k>0$ be an integer, let $R$ be a ring (commutative, unital), which contains $\mathbb{Q}$ (i.e. with a ring homomorphism $\mathbb{Q}\to R$) and all $k$-roots of unity. The examples I have in ...
- 7,722
4
votes
0
answers
312
views
Dimension of a commuting nilpotent variety
Fix $k$ an algebraically closed field, $n$ a natural number, and $\lambda=(\lambda_1,\ldots,\lambda_m)$ a partition of $n$. Let $A$ be any $n\times n$ nilpotent matrix with entries in $k$ whose ...
- 768
1
vote
1
answer
367
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 ($\...
- 413
0
votes
0
answers
320
views
Invariants of the Determinant Form
Consider a form of degree $r$ in $n$, that is, a homogeneous polynomial
$$f(x_1, \ldots, x_n)=\sum_{i_1+\ldots i_n=r}\alpha_{i_1 ... i_n}x_1^{i_1} ... x_n^{i_n}
$$
After the linear change of ...
- 801
4
votes
2
answers
401
views
Variety determined by interior product of the determinant?
Let $\Lambda^k(V)$ be the space of alternating $k$-linear tensors on $V$. Consider the map $f: \left(\mathbb{R}^n\right)^{n-k} \to \Lambda^k(\mathbb{R}^n)$ given by $\left(v_1,v_2, ..., v_{n-k}\right)...
- 12.1k
3
votes
3
answers
461
views
Multiplicity of eigenvalues in 2-dim families of symmetric matrices
Say you have 2 symmetric matrices, $A$ and $B$, and you know that every linear combination $xA+yB$ ($x,\\,y\in \mathbb{R}$) has an eigenvalue of multiplicity at least $m>1$. Such a situation can of ...
- 1,452
6
votes
0
answers
267
views
Is there a straightforward way to solve unmixed, homogeneous systems of polynomials?
I came across this problem in my research. It might just be an easy algebraic geometry question, but I don't know much algebraic geometry.
Suppose we have a system of $k\leq n$ polynomials in $\...
9
votes
2
answers
2k
views
Jordan Form Over a Polynomial Ring
Let $X$ be the set of $k\times k$ matrix with entries in $\mathbb{C}$, and let $M\in X$. The group $GL(k,\mathbb{C})$ acts on $X$ by conjugation, and according to the Jordan decomposition theorem (...
- 2,097