All Questions
Tagged with co.combinatorics finite-fields
136 questions
67
votes
6
answers
7k
views
How to recognise that the polynomial method might work
A couple of days ago I was at a nice seminar given by Christian Reiher, during which he told us about a short proof of the following special case of a theorem of Olson.
Theorem. Let $(a_1,b_1),\dots,(...
- 29k
23
votes
4
answers
3k
views
Exhibit an explicit bijection between irreducible polynomials over finite fields and Lyndon words.
Let $q$ be a power of a prime. It's well-known that the function $B(n, q) = \frac{1}{n} \sum_{d | n} \mu \left( \frac{n}{d} \right) q^d$ counts both the number of irreducible polynomials of degree $n$...
- 118k
22
votes
1
answer
970
views
Grothendieck on polyhedra over finite fields
In Grothendieck's Sketch of a Programme he spends a few pages discussing polyhedra over arbitrary rings and concludes with some intriguing remarks on specializing polyhedra over their "most ...
- 528
20
votes
2
answers
2k
views
Sums of powers mod p
For prime $p > 7$ with $p-1=rs$, $r>1$, $s>1$, let $A=\{x^r|x \in \mathbb{Z}_p\}$ and $B = \{x^s|x \in \mathbb{Z}_p\}$. If $g$ is a primitive root mod $p$ then $A = \{0\} \cup \{g^{ir}|0 \leq ...
- 619
17
votes
4
answers
1k
views
A mixing property for finite fields of characteristic $2$
In connection with this MO post, here is a question somewhat implicitly contained in a joint paper of S. Kopparty, S. Saraf, M. Sudan, and myself.
Let ${\mathbb F}$ be a finite field, and suppose ...
- 23k
17
votes
1
answer
381
views
Do mutually dual finite vector spaces have the same orbit cardinalities under a linear group action?
Let $G$ be a finite group acting linearly on a finite dimensional vector space $V$ over a finite field. By Burnside's lemma,
$$
|V/G| = \frac 1{|G|} \sum_{g\in G} q^{\dim(ker(g - I))}.
$$
Since $g-I$ ...
- 5,654
16
votes
3
answers
2k
views
Periodic orbits and polynomials
There are two simple and classic enumerations that still I'm puzzled about. Let's start with a simple counting problem from a well-known dynamical system.
fact 1 Consider the "tent map" f:[0,1]→[...
- 60.5k
16
votes
2
answers
1k
views
Examples of problems where considering "discrete analogues" has provided insight or led to a solution of the original problem
The Kakeya conjecture posits that any Kakeya set in $\mathbb{R}^n$ has dimension $n$.
A discrete (finitized?) version of this problem is the Finite Field Kakeya conjecture, which was proved by Dvir ...
14
votes
2
answers
655
views
Number triangle
This question arose just out of curiosity. Note the triangle of 0-1's below, whose construction is as follows. Choose any number, say 53 as done here. The first line of the triangle is the binary ...
- 1,216
13
votes
2
answers
1k
views
Number of commuting pairs (triples, n-tuples) in GL_n(F_q) (and other groups)?
Question 1 What is the number of pairs of commuting elements in GL_n(F_q) ?
I am aware of many results concerning commuting elements in Mat_n(F_q), but I am interested in GL i.e. non-degenerate ...
- 24.9k
13
votes
2
answers
1k
views
An expander (?) graph
For a prime $p$, consider the graph on the vertex set ${\mathbb F}_p$, in
which every vertex $z$ is adjacent to $z\pm 1$ and also to $z^{-1}$ (unless
$z=0$). I was told that this graph is known to be ...
- 23k
13
votes
1
answer
468
views
Near-linear mappings from $\mathbb F_p$ to $\mathbb R$
$\newcommand{\F}{{\mathbb F}}$
$\newcommand{\R}{{\mathbb R}}$
$\renewcommand{\phi}{\varphi}$
Let $p\ge 5$ be a prime.
If the functions $\phi_1,\phi_2,\phi_3\colon\F_p\to\R$ satisfy $\phi_1(x)+\...
- 23k
13
votes
0
answers
188
views
Why is $ULU=NU$ (a refinement of $|N|=q^{n^2-n}$)?
Let $G=GL_n(\mathbb{F}_q)$, $U$, $L$, $N$ the subsets of upper-triangular unipotent, lower-triangular unipotent, all unipotent matrices respectively. Then $ULU=NU$ means that for any $g\in G$ the ...
- 3,772
11
votes
2
answers
387
views
Bounds for the difference in the number of ones in $M$ and $M^{-1}$
If $M$ is a full rank $n$ by $n$ binary matrix over $\mathbb{F}_2$, how much larger or smaller can the number of $1$s in $M^{-1}$ be, compared to the number of $1$s in $M$?
Clearly the identity matrix ...
- 3,377
11
votes
2
answers
788
views
Blocking sets in three dimensional finite affine spaces
What is the smallest possible size of a set of points in $\mathbb{F}_q^3$ which intersects (blocks) every line?
Clearly the union of three affine hyperplanes that intersect in a singleton, say $x = 0,...
- 1,197
11
votes
2
answers
604
views
Does $q$-Catalan number count subspaces?
Consider the $n$-element subsets $\{a_1<a_2<\cdots <a_n\}$ of $\{1,\ldots ,2n\}$ satisfying $a_i\geq 2i$ for all $i=1,\ldots ,n$. The number of such subsets is given by $${2n\choose n}-{2n\...
- 1,305
11
votes
1
answer
258
views
Counting nonzero hyperdeterminants over $\mathbb{F}_q$
The hyperdeterminant $D(A)$ is a multidimensional generalization of the
determinant. It is a polynomial in the entries of a $(k_1+1)\times
(k_2+1)\times\cdots \times (k_n+1)$ array $A$. The ...
- 50.8k
10
votes
2
answers
882
views
The maximal subset of a finite field where the sum of any subset is non-zero
Given a finite field $\mathbb{F}_q$ with $q=p^m$ where $p$ is the characteristic.
For any subset $S=\{a_1,\dots,a_n\}$ of $\mathbb{F}_q$, if any partial sum (i.e. the sum of elements in a non-empty ...
- 441
10
votes
1
answer
807
views
How many Lie and associative algebras over a finite field are there?
This question is related to the following general question:
Given a variety of (non-associative) algebras $\mathcal V$, a finite field $\mathbb{F}_q$, with $q$ elements, and a positive integer $n$, ...
- 398
10
votes
1
answer
561
views
How many rich directions does a set in $\mathbb F_p^2$ determine?
$\newcommand{\F}{\mathbb F}$
A subset $P$ of the affine plane $\F_p^2$ is said to determine a direction if there is a line in this direction containing at least two points of $P$.
A set of size $|P|&...
- 23k
10
votes
0
answers
436
views
Commuting matrix variety $[A,B]=0$ - can one geometrically explain divisibility of $F_ q$ point count by high powers of $q$?
$\DeclareMathOperator\Comm{Comm}\DeclareMathOperator\Id{Id}$Consider the variety $\Comm$ of commuting matrices $[A,B]=0$ over some field $K$. It is much studied, and interesting for various reasons.
...
- 24.9k
9
votes
2
answers
543
views
A mixing property of linear map over finite fields
Let $F$ be a finite field of odd size $q$, and $\phi_0 : F \mapsto F$ be any map from $F$ to itself. For each $a \in F$, set $\phi_a : x \in F \mapsto \phi_0 (x) + ax $.
When $\phi_0 : x \mapsto x^2 ...
- 235
9
votes
1
answer
425
views
Are there functions $\mathbb{F}_2^n \to \mathbb{F}_2$ satisfying these special relations?
Let $\mathbb{F}_2=\{0,1\}$ be the field with two elements, and let
$u:\mathbb{F}_2^n\rightarrow \mathbb{F}_2$. Suppose that $n$ is odd.
Is it possible that
$$
\sum_{x \in \mathbb{F}_2^n}(-1)^{u(x)+u(...
- 6,741
9
votes
1
answer
543
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 ...
- 307
9
votes
0
answers
270
views
The number of non-singular $n\times n$ matrices over $\mathbb{F}_2$ with exactly $k$ non-zero entries
Suppose $M_{n}^{k}$ is the number of non-singular $n\times n$ matrices over $\mathbb{F}_2$, that have exactly $k$ non-zero entries.
Is there some sort of formula to calculate $M_n^k$?
If $k < n$ ...
- 2,618
9
votes
0
answers
245
views
Almost blocking sets in $\mathbb F_q^2$
$\newcommand{\F}{{\mathbb F}}$
Let $q$ be an odd prime power. A blocking set in the affine plane $\F_q^2$ is a set blocking (meeting) every line.
A union of two non-parallel lines is a blocking set ...
- 23k
8
votes
3
answers
385
views
Self-reciprocal polynomials over finite fields
Let $SRMI_q(2n)$ denote the number of self-reciprocal
irreducible monic polynomials of even degree $2n$ over the finite
field $\mathbf{F}_q$ with $q$ elements. Recall that
a polynomial $p(x) \in \...
- 481
8
votes
1
answer
375
views
A Balog-Szemeredi-Gowers-type question
A nice Lemma due to Konyagin asserts that for any subset $B \subset \mathbb{F}_p$ holds
$$
|B.B - B.B + B.B - B.B + B.B - B.B| \geq \frac{1}{2}\min\{p, |B|^2 \},
$$
where the standard notation for the ...
- 960
8
votes
1
answer
531
views
How large can the dimension of a 'Span of powers of a finite field basis' be?
Let $p$ be a prime. For finite field $\mathbb{F}_{p^k}$ and $d\in\mathbb{Z}^+$, I am considering the following quantity, where we interpret the field $\mathbb{F}_{p^k}$ also as a $\mathbb{F}_p$-vector ...
- 89
8
votes
0
answers
304
views
A strong sum-product "for translates" in finite fields
In the course of some recent research, I've sketched out a proof of the following result. My basis question is: is the result interesting?
Proposition There exists an absolute constant $c$ such ...
- 11.2k
7
votes
2
answers
843
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 ...
- 1,291
7
votes
2
answers
589
views
On a matrix problem in the field $\mathbb F_2$
Given $M$ a symmetric matrix in $\mathbb F_2^{n\times n}$ having $\mathsf{det}_\mathbb R(M)\neq0$ (non-singular in reals) and satisfying $PMP'=(M+J+I)$ or $P(M+J+I)P'=M$ where $P$ is a permutation ...
- 13.9k
7
votes
1
answer
253
views
Enumerating subspaces of $\mathbb{F}_q^n$ in terms of words and inversions
When $q$ is a prime power, then on the one hand the $q$-binomial coefficient $\binom{n}{k}_q$ equals the number of $k$-dimensional subspaces of $\mathbb{F}_q^n$, and on the other hand it is the ...
- 63.1k
7
votes
2
answers
441
views
How to count the number of tensors over a finite field of tensor rank $r$?
For simplicity, work over $\mathbb F_2$ and only consider order-$3$ equilateral tensors. For $r\in\mathbb Z_{>0}$, how many tensors $\mathfrak{T}\in\mathsf{Ten}_{n}^{\otimes 3}(\mathbb F_2)$ are ...
- 161
7
votes
2
answers
476
views
A quadratic form
Let $q$ be a power of 2. Let $P$ be the set of polynomials in
$F_q [x]$ of degree d or less.
Let $\mathbb{Z}$ be the ring of integers. For any $f \in P$, let $\psi(f)$ be the
number of distinct ...
- 503
7
votes
0
answers
300
views
Arrangement of subspaces over finite fields
I'm trying to find out what is already known about the following setup.
Let $V$ be an $n$-dimensional vector space over a finite field $F_q$ (I'm mostly interested in the case where $q$ is prime), and ...
- 1,062
7
votes
0
answers
276
views
Cyclic shift acting on finite Grassmannian
The (twisted) cyclic shift $(v_1,v_2,\ldots,v_n) \mapsto (v_2,v_3,\ldots,v_n,(-1)^{k-1}v_1)$ acting on the Grassmannian $\mathrm{Gr}(\mathbb{C};k,n)$ of $k$-planes in $\mathbb{C}^n$ is an important ...
- 24.2k
6
votes
1
answer
180
views
Smallest set of nonzero vectors in $\mathbb F_2^n$ which intersects every 2-dimensional subspace
What is the smallest set of nonzero vectors in $\mathbb F_2^n$ which intersects every 2-dimensional subspace?
For example, for n = 3, the set {001, 010, 011} does the job, and is minimal. For n = 4, {...
- 61
6
votes
1
answer
640
views
Upperbounding a sum of Legendre-Symbols
Let $p$ be a prime with $p\equiv 3 \mod 4$, for any $\mathcal{I} \subset \lbrace 0,...,p-1 \rbrace $ and any $\mathcal{J} \subset \lbrace 0,...,p-1 \rbrace $ with $\vert\mathcal{I}\vert \leq \sqrt{p} $...
- 93
6
votes
1
answer
457
views
Vector with many non-zero coordinates
Given finite field $\mathbb{F}_q$, positive integers $n$ and $k<n$. Given $k$-dimensional subspace $X$ of $\mathbb{F}_q^n$, for which $m=m(q,k,n)$ may we find for sure a vector in $X$ with at least ...
- 109k
6
votes
1
answer
372
views
A parity counting problem for subsets over finite fields
Let ${\mathbb F}_p$ be the prime field of $p$ elements and $b$ be an element in ${\mathbb F}_p$.
For a subset $T\subseteq {\mathbb F}_p$, define
$$Bias(T)=|N_e( {\mathbb F}_p,b)-N_o( {\mathbb F}_p,b)|...
- 121
6
votes
1
answer
297
views
Covering the finite plane with lines
This is, essentially, a geometrically rendered version of the question I asked a week ago, with the emphases slightly shifted; it seems more natural and appealing (to me, at least) in this form.
Let ...
- 23k
6
votes
1
answer
458
views
Applications of small Kakeya sets over finite fields
It was proved by Dvir that a Kakeya set in $\mathbb{F}_q^n$ has size at least $q^n/n!$, a bound which was later improved to $q^n/2^n$.
For $n = 2$ and $q$ odd the exact bound is $q(q+1)/2 + (q-1)/2$ ...
- 1,197
6
votes
0
answers
315
views
Number of square-free polynomials over a finite field - a combinatorial interpretation?
Cross-posted from MSE. The question has remained unanswered for six years but I still like it!
One can show using zeta functions that the number of (monic)square-free polynomials of degree $n$ over a ...
- 7,746
6
votes
0
answers
173
views
When are Hamming codes cyclic?
I've asked this question on math.stackexchange before, but it has not been solved.
The following statement appears to be true:
The $q$-ary Hamming code of codimension $r$ over $\mathbb{F}_q$ is ...
- 253
5
votes
2
answers
139
views
Sets blocking every $2$-flat in $AG(n,2)$
The following may be well-known $-$ but not known to me:
What is the smallest possible size of a set in ${\mathbb F}_2^n$ that blocks every $2$-flat?
Here "blocks" means "have a non-empty ...
- 23k
5
votes
1
answer
447
views
More expanders?
Having received several exhausting answers to my recent question about
the expansion properties of a certain graph, I now wonder whether anything is
known on the following graphs of a similar nature:
...
- 23k
5
votes
1
answer
459
views
$(n-2)$-blocking sets in $AG(n,2)$
Let's define $k$-blocking set in affine space $AG(n,q)$ a set that meets every coset (translate of subspace) of dimension $k$.
I have seen a lot work related to minimal $(n-1)$-blockings set.
...
- 337
5
votes
1
answer
163
views
Polynomials vanishing on prescribed layers
Given a prime $p$ and an integer $n\ge p$, what is the smallest possible degree of a polynomial $Q\in\mathbb F_p[x_1,\dotsc, x_n]$ such that $Q$ vanishes on every vector $x\in\{0,1\}^n$ of weight $w(x)...
- 23k
5
votes
1
answer
287
views
Given a symmetric polynomial in F_q, write it in terms as elementary symmetric polynomials. How to find out the coefficient?
Consider the finite field $F_q$, where $q$ is a power of an odd prime and $N$ is a power of $q$. We have a homogeneous symmetric polynomial
$$
E_{l,s}(x) = \sum_{\substack{l_1+l_2+\cdots +l_s=l \\ l_i\...
