6
votes
0answers
295 views

On the number of polynomials that divide $x^{q-1}-1$ in some subspaces of $\mathbb F_q[x]$

Let $\mathbb F_q$ be a finite field (where $q$ is in general a power of a prime), and let $e, k$ be positive integers with $k \leq e < q-1$. Let $f_0(x), \ldots, f_k(x) \in \mathbb F_q[x]$ be ...
6
votes
3answers
310 views

On the rank of a matrix $S$ with coefficients in $\mathbb F_{2^m}$

Let $\mathbb F$ be a finite field with characteristic 2 and let $S \in M(2k, 2k, \mathbb F)$ be the matrix defined as follows $$ S=\left[\begin{array}{ccccccc} 0 & ...
2
votes
1answer
101 views

Ranks of higher incidence matrices of designs

In 1978 Doyen, Hubaut and Vandensavel proved that if $S$ is a Steiner triple system $S(2,3,v)$ then the $GF(2)$ rank of its incidence matrix $N$ is $$ Rk_{2}(N)=v-(d_{p}+1), $$ where $d_{p}$ is the ...
4
votes
1answer
178 views

Full-rank rectangular matrices over GF(2)

Given positive integers $k$, $m$, $n$, let $A$ be an $m \times n$ matrix over $GF(2)$ constructed as follows. Let $X_1, \ldots, X_m$ be independent random subsets of $\{1,\ldots,n\}$ with cardinality ...
0
votes
1answer
146 views

Number of Minimal left ideals in the full matrix ring over a finite commutative local ring

Inspired with another QUESTION I would like to know the number of minimal left ideals of $M_n(R)$ in terms of $n$ and $R$ where $R$ is a finite local commutative ring with identity ?
6
votes
2answers
317 views

Dimension of incomplete matrix over finite fields.

Hi, Suppose one has an incompletely specified $2^n \times 2^n$ matrix over some fixed finite field $\mathbb{F}_{p^k}$. In fact, one knows that the diagonal entries are zero and all other entries are ...
2
votes
2answers
157 views

linear independence of orbits via a set of transformations in char p

Let $T_1, \ldots, T_n \in GL(n,\mathbb{F}_p)$. Suppose for all $\vec{v} \in \mathbb{F}_p^n$ we have $\det (T_1 \vec{v}, T_2 \vec{v}, \ldots, T_n \vec{v}) = 0$. Now, let $k$ be a finite extension of ...
1
vote
2answers
246 views

When is PSU(2,q^2) = PSL(2,q) ?

The context for this question is from page 284 - 287 of Berger's paper: ...
2
votes
1answer
246 views

Finding a subspace disjoint from a union of subspaces

Let $k$ be a finite field (I care about $\mathbb F_p$, especially $\mathbb F_2$) and let $V_1,...,V_N\subset k^n$ be subspaces. I want to find a subspace $S\subset k^n$ such that $S\cap V_i=0$ for ...
0
votes
0answers
220 views

Combinatorial Interpretation of an Extension of Gaussian Polynomials

It is well-known that the Gaussian polynomial (or Gaussian coefficient, q-binomial coefficient) $\binom{n}{k}_q$ counts the number of $k$-dimensional subspaces of an $n$-dimensional vector space over ...
0
votes
2answers
148 views

Matrices whose range is equal to the column set [closed]

Is there such a thing? I suppose there are two cases in which they may exist: over a finite field or infinite matrices (but let's stick to countable matrices then).
4
votes
2answers
802 views

Vandermonde matrices and general position

I was wondering if it is known whether a Vandermonde matrix over a sufficiently large finite field is in general position with respect to intersections of subspaces spanned by subsets of columns, i.e. ...
4
votes
1answer
274 views

Aschbacher classes and $\mathbb{F}_p$-subspace stabilizers in classical linear groups

I am reading the Kleidman-Liebeck book ("The subgroup structure of the finite classical groups") which is about the Aschbacher classification of maximal subgroups of the classical almost simple ...
2
votes
2answers
163 views

Cases of almost-linear time solvable linear systems

Let a square $n\times n$ real matrix ${\bf A}$ and two vectors ${\bf x}$ and ${\bf b}$ of length $n$, such that $${\bf A}{\bf x}={\bf b}.$$ Solving for ${\bf x}$ through standard Gaussian Elimination ...
3
votes
1answer
2k views

Number of n-th roots of unity over finite fields [closed]

How many $n$-th roots of unity does a finite field $\mathbb{GF}\left(p^k\right)$ have? ($p$ is prime). And then kind of related to that, is there a finite field with exactly $n_1,\ldots,n_N$ ...
4
votes
0answers
328 views

Given two linear operators A and B over a finite field, is there a third operator C whose kernel is the intersection of kernels of A and B?

Let $V$ be a finite dimensional linear space over a finite field $k$. Let $A$ and $B$ be two endomorphisms of $V$. Question 1. Is there an endomorphism $C$ of $V$, which is expressed in terms of ...
2
votes
2answers
307 views

vectors with entries from a finite ring

I've been working recently with vectors over finite fields, but I was hoping to work in a more general setting and consider vectors over finite commutative rings. The question I had is as follows: if ...
1
vote
1answer
390 views

maximal number of mutually orthogonal vectors

Let $F$ be a field, $n$ be a positive integer. Denote by $h_{F}(n)$ the maximal dimension of a subspace $X\subset F^n$ such that $(x,y)=0$ for any two (not necessary distinct) vectors $x,y\in F^n$, ...
8
votes
0answers
858 views

roots of quadratic forms

This may be a very silly question, but I was wondering what is known about the roots of a quadratic form over variables $x_1,\ldots,x_n,y_1,\ldots,y_m$ in the finite field $\mathbb{F}_p$. I'm not ...
10
votes
3answers
403 views

Local-globalism for similar matrices?

My background on number theory is very weak, so please bear with me... Given two matrices $A$ and $B$ in $\mathbb{Z}^{n\times n}$. Assume that for every prime $p$, the images of $A$ and $B$ in ...