The finite-fields tag has no wiki summary.

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votes

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533 views

### homogeneous polynomials over a finite field

Let $P(x_1, \ldots , x_n)$ be a homogeneous polynomials over a finite field with $q$ elements. Is there any way to count all the roots of $P$?

**2**

votes

**0**answers

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

**2**

votes

**1**answer

132 views

### Invariant subspaces of an $F_2$-representation of the affine linear group of dimension 1

Let $p$ be an odd prime (large if it matters) and let $G= Aff(\mathbb{F}_{p^2}) \cong \mathbb{F}_{p^2} \rtimes \mathbb{F}_{p^2}^*$ be the affine linear group acting on $\mathbb{F}_{p^2}$ by $x\mapsto ...

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votes

**0**answers

146 views

### Does the fundamental group identify group structure on subvarieties of products of curves?

Let $C_1,\dots, C_n$ be smooth curves over $\overline{\mathbb F}_p$, not necessarily proper. Let $X$ be a subvariety of $C_1 \times \dots \times C_n$. I'm interested in the natural map:
$$ ...

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votes

**2**answers

193 views

### Fixed space of the square of a symmetric matrix over $\mathbb{F}_2$

Let $M$ be an invertible symmetric $2n \times 2n$ matrix with entries in the finite field $\mathbb{F}_2$. Is $\mathrm{Ker}\ (M^2 - I_{2n})$ necessarily even dimensional?

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votes

**0**answers

53 views

### Matrices over a finite field with given Jordan normal form over the algebraic closure [migrated]

Can one describe the (conjugacy classes of) square matrices over a finite field such that over the algebraic closure of this finite field their Jordan normal form consists of one Jordan block? (Such ...

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vote

**0**answers

28 views

### Can we give efficiently the solution of a bilinear system of equations over a finite field?

Consider a finite field $F$ and suppose we have a system of equations
$$h_1(\alpha,\beta)=0,h_2(\alpha,\beta)=0,...,h_t(\alpha,\beta)=0$$
where $\alpha=(\alpha_1,...,\alpha_s)$ and ...

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votes

**3**answers

796 views

### Probability of coprime polynomials

Given positive integer $N$, we choose $m_1, m_2, n_1, n_2$ independently and with equal probabilities from $\{0,1,\ldots,N\}$, and let
$f_1 = x^{m_1} + (1+x)^{n_1}$ and $f_2 = x^{m_2} + (1+x)^{n_2}$ ...

**2**

votes

**1**answer

105 views

### Units in a finite semisimple group algebra

Let $G$ be a finite group and $k$ a finite field, with the characteristic of $k$ not dividing the order of $G$. Then $kG$ is a finite semisimple group algebra with the interesting property that an ...

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votes

**2**answers

198 views

### Picking codewords that are close

I posted this question in http://math.stackexchange.com/questions/1142698/picking-codewords-that-are-close a week back.
Let $[n,k,d]$ be a linear code over $\Bbb F_q$ with minimum distance $d$ and ...

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votes

**0**answers

55 views

### constant of functional equation of zeta function

Let $C$ be a smooth projective curve, of geometric genus $g$, over a finite field $\mathbb{F}_p$ and consider the zeta function $$
Z(C/\mathbb{F}_p, t)=\exp(\sum_{n=1}^{\infty} |C(\mathbb{F}_{q^n})| ...

**4**

votes

**1**answer

202 views

### Counting couples of square-free polynomials over finite fields

I have a curve defined by the following equations over the finite field $\mathbb{F}_q$ with $q=p^r$ with $p \geq 3$:
$$C_{h_1,h_2}:\begin{cases} y_1^2=h_1(t) & \\
y_2^2=h_2(t) ...

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vote

**0**answers

50 views

### Zero as a repeated permanental root for a matrix over a finite field

All,
Suppose $A \in Mat(n, \mathbb{F}_{q})$ for $q$ prime, $q \geq 5$, and $n \geq 2^{q-2}$ . Let $\pi_A(x)$ be the permanental polynomial for $A$. That is,
\begin{equation*}
\pi_{A}(x)=per(xI-A).
...

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votes

**0**answers

222 views

### Geometric vs combinatorial motives over Spec Z

Consider the category of reduced schemes of finite type over $\mathbb{Z}$. Take the Grothendieck group of this category, i.e. the free abelian group on isomorphism classes, modulo the usual "syzygy" ...

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votes

**1**answer

117 views

### cohomology of orthogonal (or general linear) group over finite fields

Let $\mathbb{Z}_2=\mathbb{Z}/2\mathbb{Z}$. Let
$$
O(\mathbb{Z}_2^{\oplus k})=\{A\mid A \text{ is a } k\times k \text{ - matrix with entries } 0,1, det(A)=\pm 1\}
$$
What is $$
...

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votes

**0**answers

134 views

### Counting points on Hessenberg varieties over a finite field

Let $G$ be an connected reductive group over finite field $k$. I will assume that $\text{char}(k)$ is very good for $G$ (or even larger, if preferred). Let $B\subset G$ be a Borel subgroup defined ...

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votes

**0**answers

247 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 singular ...

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vote

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171 views

### Polynomial existence over finite field

Denote $\mathcal{F_n}$ as collection of multiaffine polynomials $f\in\Bbb F_2[x_1,\dots,x_n]$.
Denote total degree of $f\in\mathcal{F_n}$ as $deg(f)$ (note $deg(f)\leq n$).
Denote ...

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votes

**0**answers

106 views

### Number of common solutions of polynomial system

Let $ \mathbb{F}_p$ be a finite field and $\{f_j\}_{j=1}^{j=r} \subseteq \mathbb{F}_p[X_1,...,X_n]$ be a set of polynomials.
Let consider the system of equations:
$f_j(x_1,...,x_n)=0$ for $j = ...

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votes

**0**answers

112 views

### Groupoid cardinality of DM stack and point counting on coarse moduli spaces

Let $X$ be a finite type DM stack over a finite field $k$ with a coarse moduli space $X_c$. (We only assume $X_c$ is an algebraic space and $X$ might have infinite inertia stack.)
Under which ...

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votes

**0**answers

259 views

### Enumerating certain types of permutation polynomials

Given a prime power $q$, I would like to enumerate (preferably up to isomorphism*) all the permutation polynomials $f(x)$ on $K = GF(q^3)$ satisfying the following conditions:
$f(ax) = af(x)$ for ...

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votes

**0**answers

124 views

### How many roots can $P(x):=\sum_s(x-s)^{(p-1)/2}$ have in ${\mathbb F}_p$?

For a prime $p$ and a set $S\subset{\mathbb F}_p$ of size $n:=|S|\approx \sqrt p$, what is the largest possible number of roots that the polynomial
$$ P(x) := \sum_{s\in S} ...

**7**

votes

**2**answers

405 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 ...

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votes

**1**answer

336 views

### Seeking conceptual explanation of these nice bijections on roots of unity

I proved the following facts by unenlightening calculations. Since the statements are quite clean, I think there should be a conceptual explanation for them, which my proof certainly is not.
Let $q$ ...

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votes

**1**answer

167 views

### Chebyshev polynomials factoring uniformly modulo all primes

Consider the Chebyshev polynomial of the first kind $T_n(x)$ and its factorization in $\mathbb F_p$ for a given prime $p$. Most often, this factorization is not uniform (meaning that the irreducible ...

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votes

**2**answers

280 views

### irreducible polynomials on the polynomial sequence

I suspect this problem is very famous and it must be studied very well. But I searched in Google and I did not find good reference. I will appreciate any answer and reference for any contribution ...

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votes

**0**answers

71 views

### Uniqueness of an embedding theorem for Real differential fields

I will follow a preliminary exposition for the problem in question, which will essentially follow the format on http://www4.ncsu.edu/~singer/papers/model_diff_fields.pdf [pg. 87]:
Let $K$ be a real ...

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votes

**1**answer

227 views

### Polynomiality of functions over residue rings

Suppose $\mathbb{Z}/m \mathbb{Z}$ is a residue ring for some $m \in \mathbb{N}$. If $m=p$ is a prime number then every function $f:\mathbb{Z}/p \mathbb{Z} \rightarrow \mathbb{Z}/p \mathbb{Z}$ is a ...

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votes

**1**answer

164 views

### Which criteria for “uniformly splitting” polynomials?

Let $P(x)$ be an irreducible monic polynomial of degree $\ge4$ with integer coefficients. We all know that over a finite field $\mathbb F_p$, $P$ will often split, and I am interested in polynomials ...

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votes

**1**answer

175 views

### Dihedral subgroups of $\mathrm{PSL}_2(\mathbb{F}_q)$

Let $\mathbb{F}_q$ be a finite field with $q=p^f$ elements. I need to know when $\mathrm{PSL}_2(\mathbb{F}_q)$ contains the group $D_{(q+1)/2}$, where by $D_n$ I mean the dihedral group of order $2n$. ...

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votes

**1**answer

579 views

### Roots of a polynomial in a finite field related to Fermat's Last Theorem

In my class, we proved the following condition: define the polynomial $P_l(x)$ as
$$P_l(x) = \sum_{r=1}^{l-1}{\frac{1}{r}x^{l-1-r}}$$
Then if for all $a \in \mathbb{Z}/l\mathbb{Z}-\{0,1\},$ ...

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votes

**1**answer

217 views

### Representation of GL(n, F_p) over F_p, for n small

The question is related to this post
Representation theory of the general linear group over a finite prime field
However, I am asking for more detailed references for n small, for example, for n=2, ...

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votes

**0**answers

37 views

### Comparing least degrees of certain polynomials

Let $\Bbb K$ be an infinite or a finite field with $\mathsf{char\mbox{ }}\Bbb K\neq 2$ and let $M\subsetneq\Bbb K[x_1,\dots,x_n]$ be the set of multilinear polynomials.
Fix $S\subsetneq\{0,1\}^n$ and ...

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votes

**0**answers

92 views

### Dimensions of two spaces

Let $R=\Bbb K[x_1,\dots,x_n]/I$ and $Z=\mathsf Z(I)$ where $\Bbb K$ is any field with $char(\Bbb K)\neq 2$.
Is there a way to describe space of $f_1,f_2\in R$ that satisfies $$f_1+f_2\neq 0,\mbox{ ...

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vote

**0**answers

191 views

### Eigenvalue of a linear map over finite field

Let $ F_q $ be a finite field with $ q $ elements.
Let $ g $ be a multiplicative generator of $ F_{q^2}^* $.
It implies that
$ <g^{q+1}> = F_q^* $.
Let $ l $ be a prime greater than $ q^2-1 ...

**29**

votes

**1**answer

1k views

### Algebraic dependency over $\mathbb{F}_{2}$

Let $f_{1},f_{2},\ldots,f_{n}$ be $n$ polynomials in $\mathbb{F}_{2}[x_{1},x_{2},\ldots,x_{n}]$
such that $\forall a=(a_1,a_2,\ldots,a_n)\in\mathbb{F}_{2}^{n}$ we have $\forall ...

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votes

**1**answer

186 views

### Distribution of the permanent modulo $p$

We know that the order of $SL_n({\mathbb F}_p)$ is
$$p^{n(n-1)/2}(p^n-1)(p^{n-1}-1)\cdots(p^2-1).$$
Dividing by $p^{n^2}$, we deduce the probability that $\det$ takes the value $1$ over $M_n({\mathbb ...

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votes

**3**answers

389 views

### An infinite product associated with random matrices

Motivation
Let ${\mathbb F}_q$ be the field with $q$ (a power of some prime number) elements. Then the order of $GL_n({\mathbb F}_q)$ is
$$(q^n-1)(q^n-q)\cdots(q^n-q^{n-1}).$$
The fact that this ...

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votes

**4**answers

914 views

### Books on advanced galois theory

I have been studying galois theory on my own and find it very fascinating. I have gone through Ian Stewarts book: http://www.amazon.co.uk/Galois-Theory-Third-Chapman-Mathematics/dp/1584883936. I am ...

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votes

**14**answers

3k 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 ...

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votes

**2**answers

294 views

### Is there a proof of Warning's Second Theorem using p-adic cohomology?

Let $\mathbb{F}_q$ be a finite field, $n \in \mathbb{Z}^+$, and $f_1,\ldots,f_r \in \mathbb{F}_q[t_1,\ldots,t_n]$ with $\operatorname{deg}(f_i) = d_i$. Put $d = \sum_{i=1}^n d_i$ and suppose $d< ...

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votes

**1**answer

126 views

### Sylow-subgroups of the group of units of a finite field [closed]

Let $K$ be a finite field with $p^n$ elements. Then the group of unit is cyclic of order $p^n-1$. What are the primes which are dividing $p^n-1$? What are the Sylow-subgroups of the group of units of ...

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votes

**3**answers

1k views

### Classification of rings satisfying $a^4=a$

We have the famous classification of rings satisfying $a^2=a$ (for each element $a$) in terms of Stone spaces, via $X \mapsto C(X,\mathbb{F}_2)$. Similarly, rings satisfying $a^3=a$ are classified by ...

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votes

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145 views

### Partial differential Equation over characteristic p

I want some references on partial differential equations over characteristic $p$. If we have a first order partial differential equation, how can we check whether there exists polynomials or rational ...

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vote

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79 views

### Points on the intersection of an affine quadric and cubic over a finite field

Are there absolute constants $N$ and $B$ such that the following is true?
Let $p>B$ be a prime. Let $q(x_0,\dotsc,x_n)$ and $c(x_0,\dotsc,x_n)$ be homogeneous of degree $2$ and $3$ with ...

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votes

**1**answer

183 views

### orthogonal group in characteristic 2

Let $O(2,\mathbb{Z}_2)$ be the orthogonal group of order two matrices. On $\mathbb{Z}_2$ there should exist just one odd quadratic form, hence the stabilizer subgroup $O^-$ of an odd quadratic for ...

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votes

**1**answer

261 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 ...

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votes

**0**answers

100 views

### Existence of roots of high order polynomial over finite fields

I want to solve the following question:
Consider a polynomial $f(x)=a_0+a_1*x^{e_1}+a_2*x^{e_2}+\cdots+x^{e_m}\in F_p[x]$ where $p$ is a prime such that $\log(p)\sim m$ and $e_m\sim 2^m$, I want to ...

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vote

**0**answers

47 views

### Generate Finite Field power of g [closed]

consider the field F_2^4, defined by using polynomial representation with the irreducible polynomial f(x) = x^4 + x + 1.
Given element g = (0010) as a generator for the field. How are the powers of g ...

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votes

**1**answer

240 views

### Over a finite field, does a torsor under the component group of G lift to a torsor under G?

Let $k$ be a finite field and $G$ a finite type smooth $k$-group scheme. Let $G^0$ and $\Gamma$ be the connected component of identity and the component group of $G$, so there is an exact sequence $1 ...