# Tagged Questions

**3**

votes

**0**answers

174 views

+100

### On weight enumerators of codes

Are there $[n,k]_q$ constant rate $\frac kn$ and constant alphabet linear code families with automorphism group of size $\Omega((n-n^\beta)!)$ that have minimum distance $d=O(n^\alpha)$ and number of ...

**0**

votes

**0**answers

58 views

### A matrix representation problem over finite field

Given a matrix $H\in\Bbb F_p^{n\times n}$ and a list of matrices $A_i,B_i\in\Bbb F_p^{n\times n}$ at each $i\in\{1,\dots,m\}$ where $m\leq n^{1+\beta}$ for some $\beta\in[0,1)$ with $\|H\|_0>n^{1+\...

**1**

vote

**2**answers

112 views

### Number of Skew Symmetric Matrices of fixed rank

The number of symmetric matrices of order $n$ and rank $r$ over finite fields has been counted e.g.
http://www.math.clemson.edu/~kevja/REU/2004/SymmetricRankRMatrices.pdf
Is the number of skew-...

**7**

votes

**4**answers

2k views

### Equivalent Statements of Riemann Hypothesis in the Weil Conjectures

In the cohomological incarnation, the Riemann hypothesis part of the Weil conjectures for a smooth proper scheme of finite type over a finite field with q elements says that: the eigenvalues of ...

**1**

vote

**0**answers

96 views

### Efficient deterministic algorithms of factorizing

My question is about efficient deterministic algorithms of factorizing polynomials of degree $n$ over $\mathbb{F}_q$.
Are there such algorithms that use poly$(n, \log q)$ bit operations?
I know ...

**11**

votes

**1**answer

421 views

### An efficient isomorphism between finite fields

Let $p$ be a prime number. Let $f$ and $g$ be irreducible polynomials over $\mathbb{F}_p$, both of degree $n$. We know that factor-rings $\mathbb{F}_p[x]/(f)$ and $\mathbb{F}_p[x]/(g)$ are isomorphic ...

**13**

votes

**3**answers

620 views

### Collatz-like properties of finite fields

I was wondering what an equivalent of the Collatz conjecture might be for finite fields. In a Collatz sequence a number is moved down within a set $\{2^k n : k \in \mathbb{Z}^* \}$ for some odd $n$ or ...

**4**

votes

**2**answers

212 views

### Exceptional isomorphisms between finite simple Chevalley groups

Steinberg's "Lectures on Chevalley Groups"
https://math.depaul.edu/cdrupies/research/papers/chevalleygroups.pdf
contain ``a complete list of isomorphisms" among the various finite simple Chevalley ...

**14**

votes

**2**answers

368 views

### Can you use Chevalley‒Warning to prove existence of a solution?

Recall the Chevalley‒Warning theorem:
Theorem. Let $f_1, \ldots, f_r \in \mathbb F_q[x_1,\ldots,x_n]$ be polynomials of degrees $d_1, \ldots, d_r$. If
$$d_1 + \ldots + d_r < n,$$
then the ...

**6**

votes

**0**answers

131 views

### What is known about the $q$-analogue of the simplex?

I am interested in the field with one element. I am thus interested in combinatorial interpretations of the Gaussian binomial coefficients. Richard Stanley's "Enumerative combinatorics" mentions ...

**6**

votes

**1**answer

374 views

### Finite field “contour” sum

Let $\Bbb{F}_q$ be a finite field. Choose a non-square $\delta \in \Bbb{F}_q^*$
and form the quadratic extension $\Bbb{F}_q\big( \sqrt{\delta} \, \big)$. For
an element $z \in \Bbb{F}_q\big( \sqrt{\...

**2**

votes

**1**answer

176 views

### Spin structure for varieties, especially finite field

I wonder about the notion of a spin structure for varieties over any field and results in this direction. For example, I wonder if there is something like a spin-bundle for the sphere $x^2+y^2+z^2=R^2$...

**2**

votes

**0**answers

335 views

### Counting points on an algebraic set over a finite field

Let $q=p^n$, for $p$ a prime. Let $C$ be an Artin–Schreier curve defined by $y^p-y=f(x)$ where $f(x) \in \mathbb{F}_{q}[x]$.
Let $C_g$ be $y^p-y=f(x)$ where $x \in g(\mathbb{F}_{p^{n}})$ for some $...

**1**

vote

**2**answers

166 views

### How to prove that $A$ is supersingular iff the Picard number $\rho(A)$ is equal to the second $l$-adic Betti number $b_2(A) = 6$?

Let $A$ be an abelian surface over algebraically closed field $k$ of characteristic $p > 2$. How to prove that $A$ is supersingular (in other words, there is an isogeny between $A$ and $E^2$, where ...

**3**

votes

**2**answers

143 views

### Do the irreducible modules of this finite group preserve a tensor product structure?

I am interested in a particular group $G$, where
$$ (A_4\times C_\ell) \lhd G \lhd S_4 \times D_\ell$$
Here, $C_\ell$ is cyclic, $D_\ell$ is dihedral of order $2\ell$, and the two inclusions both have ...

**50**

votes

**6**answers

4k 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,(...

**5**

votes

**0**answers

96 views

### Is a Kummer surface over an finite field $\mathbb{F}_q$ supersingular iff $\mathbb{F}_q$-unirational?

Let $A$ be an abelian surface over an finite field $\mathbb{F}_q$. In particular, I am interested in the case when $A$ is a Jacobian variety. Is the Kummer surface $K_A/\mathbb{F}_q$ Shioda-...

**3**

votes

**2**answers

227 views

### Gcd of polynomials over a finite field [closed]

Let $\mathbb{F}_q[X]$ be the polynomial ring over the finite field with $q$ elements. Let $f$ be a polynomial of the form $x^n-a$ and let $g$ be a polynomial of the form $x^m-b$. Is it known whether $\...

**4**

votes

**3**answers

703 views

### Hypersurface missing just one point

Let $\mathbb F_q$ be a finite field and $n$ an integer.
What is the minimal degree $d = d(q,n)$ of a polynomial $f \in \mathbb F_q[X_1,\dots,X_n]$ such that the set $Z(f)$ of zeros of $f$ in the ...

**5**

votes

**1**answer

268 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 ...

**3**

votes

**2**answers

406 views

### probability of having linearly independent sparse vectors over finite fields

Suppose that there are $i< N$ linearly independent $N$-dimensional vectors $\mathbf{v}_1,\mathbf{v}_2,\ldots,\mathbf{v}_i$ over a finite field $\mathbb{F}_q$, whose elements are denoted as $\{0,1,2,...

**2**

votes

**2**answers

301 views

### Irreducible algebraic sets via irreducible polynomials

There are many results about irreducible polynomials over finite fields:
we know a cardinality of all irreducible polynomials with given degree, we know explicit examples of irreducible polynomials, ...

**1**

vote

**0**answers

68 views

### Calculation of cardinality of Jacobians

The problem of calculation the number of rational points on curves over finite fields is $\#P$-complete - "Counting curves and their projections".
Is it true for calculation of number of rational ...

**13**

votes

**3**answers

1k 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]→[...

**4**

votes

**1**answer

156 views

### Schwartz-Zippel lemma for an algebraic variety

Let $X $ be a smooth affine subvariety of $(\overline{\mathbb{F}_q})^n$ defined by a prime ideal $I$. Let $f$ $\in \mathbb{F}_q[x_1,\ldots,x_n]$ be a polynomial such that $f \notin I$.
Let $r_1, \...

**8**

votes

**4**answers

1k views

### computer algebra system for polynomial algebras over finite fields

Is there a computer algebra system that can do arithmetic over polynomial algebras over finite fields where I can specify the extension?
Exempli gratia, if $f(x), g(x) \in \mathbb{F}_p[\mu]/(m(\mu))[...

**2**

votes

**1**answer

115 views

### Bases of the special form

Let $R = \mathrm{GF}(q)$, $S = \mathrm{GF}(q^n), \ n\geq 2$ be extension of $R$, $h$ be a primitive element of $S$. I want to count or estimate the number $N$ of bases of the following form.
Let $$\...

**3**

votes

**1**answer

131 views

### Non-zero coefficients of primitive polynomials

Let $R$ be the finite field with $q$ elements, and let $m,n\in \mathbb{N}$
be positive integers $\geq 2$. I want to prove that there exists a primitive
polynomial $$F(x) = x^{mn}-\sum\limits_{j=0}^{...

**2**

votes

**0**answers

65 views

### Carlitz factorials and Euler-like series

Let $q$ be a power of a prime $p$. For every $i\in\mathbb N$, one denotes $D_i=\prod_{\substack{h\in\mathbb F_q[T]\text{ monic}\\\deg h=i}}\limits h$. For $n\in\mathbb N$, write $$n=n_0+n_1q+\cdots+...

**2**

votes

**0**answers

59 views

### Property of a derivative in global field

Before posting I want to make it clear that I posted the same question in stack exchange awhile ago (http://math.stackexchange.com/questions/1533814/property-of-derivative-in-a-local-field) but didn't ...

**2**

votes

**0**answers

72 views

### the least point on a variety over a finite field

Let $p$ be a large prime parameter and $V\subseteq \mathbb{P}^n_{\mathbb{F}_p}$ a variety defined over the finite field $\mathbb{F}_p$ with bounded degree and dimension (w.r.t. $p$). Assume that $V$ ...

**2**

votes

**0**answers

93 views

### A reference about a problem of the number of the rational points on a projective scheme

Let $X\hookrightarrow\mathbb{P}^n_{\mathbb{F}_q}$ be a pure dimensional projective scheme of dimension $d$. So we know a trivial estimate of the number of $X(\mathbb F_q)$ is that $\#X(\mathbb F_q)\...

**36**

votes

**14**answers

5k views

### Fantastic properties of Z/2Z

Recently I gave a lecture to master's students about some nice properties of the group with two elements $\mathbb{Z}/2\mathbb{Z}$. Typically, I wanted to present simple, natural situations where the ...

**3**

votes

**0**answers

63 views

### How to construct large sets of $m$-dimensional vectors over a finite field such that any $m$ of them are independent?

Let $m\ge 2$ be an integer and $\mathbb{F}$ be a finite field of order $p^k$. I want to construct as many as possible $m$-dimensional vectors $v_1,v_2,\ldots,v_n$ in field $\mathbb{F}$ such that any $...

**6**

votes

**0**answers

98 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 ...

**14**

votes

**1**answer

1k views

### maximal order of elements in GL(n,p)

I am looking for a formula for the maximal order of an element in the group $\operatorname{GL}\left(n,p\right)$, where $ p$ is prime.
I recall seeing such a formula in a paper from the mid- or early ...

**5**

votes

**0**answers

84 views

### Gaussian Hypergeometric Functions and Legendre Character

I was hoping somebody might be able to point me to a good reference on Gaussian hypergeometric functions defined over a finite field. the reason I'm interested is that I've encountered sums of the ...

**3**

votes

**1**answer

93 views

### Characterization of Lagrangian planes in symplectic vector spaces over finite fields [closed]

EDIT: As L Spice pointed out, there is an error in the observation. The question is void therefore
Let $p$ be a prime and $q=p^r$. Let $V$ be a $\mathbb F_q$-vector space of dimension four, with a ...

**2**

votes

**0**answers

51 views

### Determining Inconsistency of (first-order) Non-linear System of Equations [closed]

Is there a way I can figure out what values of the coefficients of some system of non-linear equations makes the system inconsistent?
Take the following system of equations as an example. The ...

**0**

votes

**1**answer

105 views

### Functions of several variables over finite fields [closed]

For a finite field $F$ any function $f\colon F\to F$ is given by a polynomial. My question is what happens when we are given a function of two or more variables? Is this necessarily a polynomial ...

**10**

votes

**1**answer

260 views

### Counting isomorphism classes in open subsets of Bun_G

Let $G$ be a split semisimple algebraic group and let $C$ be a curve of genus $g$ over $\mathbb F_q$. Assume $g \geq 2$.
The number of $\mathbb F_q$-points of $\# \operatorname{Bun}_G(C)$, where each ...

**0**

votes

**1**answer

59 views

### Do tori in a symplectic group always have invariant maximal isotropic subspaces?

$\newcommand{\mbf}{\mathbf}$
Hi all,
I've been thinking about the following question for a while now, and got a little stuck trying to solve it. Hopefully, someone here might be able to help.
For ...

**5**

votes

**2**answers

331 views

### Splitting subspaces and finite fields

Hellow. I'm sure that the following is truth, but I can't prove it.
Let $R<S<K, R=\mathrm{GF}(q),\ S= \mathrm{GF}(q^n), \ K= \mathrm{GF}(q^{mn})$ be a chain of finite fields and
$A = \{\theta\...

**2**

votes

**2**answers

145 views

### A reference about Grassmannian over finite fields

Suppose $Gr_k(k,n)$ the Grassmannian which classifies all the dimension $k+1$ sub-spaces of a dimension $n+1$ linear space over the field $k$. For the case over a finite field $\mathbb F_{q}$, we can ...

**18**

votes

**1**answer

558 views

### Is hyperelliptic cryptography “practical”?

Previosly my impression on this subject was that hyperelliptic cryptography systems (as well as other possible cryptosystems based on abelian varieties of dimension $>1$) have no advantages over ...

**5**

votes

**1**answer

245 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.
...

**10**

votes

**2**answers

360 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,...

**3**

votes

**3**answers

267 views

### Minimal expression of 0 as a sum of kth powers in a finite field

Let $l=\min\{s\in \mathbb{N}|0\in s\cdot (\mathbb{F}_{p^n}^\times)^k\}$. Is any information known about this number already as a function of $k$? Any reference would be greatly appreciated!

**3**

votes

**1**answer

203 views

### Polynomial factoring over finite fields

What is known in general about the complexity of factoring polynomials over finite fields?
For instance given $\Bbb F_q$ where $q=p^n$ and total degree $d$ polynomial in $m$ variables what can we say ...

**4**

votes

**0**answers

137 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 $q(q+1)/2 + (q-1)/2$ ...