The finite-fields tag has no wiki summary.

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### What is known about order of torsion of jacobian of hyperelliptic curve over finite field? [closed]

Suppose $J$ is jacobian of hyperelliptic curve $C$ over $F_p$ of genus $g$. Suppose $T$ is torsion of $J(F_p)$.
What is known about order of $T$?
Are there some bounds on order of $T$? Can one say ...

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92 views

### References for modular curves over finite fields [closed]

I'm looking for a detailed reference for modular curves over finite fields, such as $X(N)$, $X_1(N)$, and $X_0(N)$. There seems to be a lot of literature dealing with them over $\mathbb{C}$, but I'm ...

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62 views

### The significance of the Parvaresh-Vardy curve

Even though this seems like a computer science question, it is purely mathematical and concerns polynomials and curves over finite fields.
Consider the Parvaresh-Vardy list decoder.
As I understand ...

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192 views

### Covering all, but $k$ points with affine subspaces

For non-negative integer $d\le n$ and $k\le 2^n$, how many affine subspaces of co-dimension $d$ are needed to cover all, but exactly $k$ elements of the vector space ${\mathbb F}_2^n$, and what are ...

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### When is $f(x^d)$ irreducible?

Let $f(x)$ be an irreducible polynomial of degree $n$ over a finite field $\mathbb F_p$. What can we say about $f(x^d)$? When is it irreducible ?

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130 views

### Obstruction to get a galois invariant cycle

Let $X$ be a smooth projective variety over a finite field $k$, $G=Gal(\bar{k}/k)$ and $\Gamma\in CH^i(\bar{X})$ such that:
$cl(\Gamma) \in H_{et}^{2i}(\bar{X},\mathbb{Z}_l(i))^G$ and
$\exists$ ...

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89 views

### Polynomial generated with primitive element modulo p

This question is equivalent to the question "Normal basis in cyclotomic number fields" that I asked recently. I am posing this question because maybe in this format somebody can have an answer:
Let ...

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507 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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394 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 ...

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### non-intersecting families of subspaces

Given $V$, a vector space over a finite field $F$ of size $k$, if $\dim(V)=m$, and $r$ divides $m$, there exists a family of $r$-dimensional subspaces, whose size is equal to $(k^m-1)/(k^r-1)$ and ...

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### 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 ...

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601 views

### Quotient of $Z[x_1,…,x_n]$ by a maximal ideal is a finite field [duplicate]

I am seeing the proof of the Ax-Groethendieck theorem from commutative algebra and I have a problem. How can I prove that if $x_1,...,x_n$ are complex numbers and $I$ is a maximal ideal of ...

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240 views

### On MDS code property

Is there a code that is Maximum Distance Separable and not isomorphic to Reed Solomon Codes? When is a MDS code isomorphic to Reed Solomon Code?
Is there an easy test? If so, could someone provide ...

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139 views

### An upper bound on the number of sets of parallel lines covering points in a finite plane?

Let $\mathbb{F}$ be a finite field of characteristic $2$. Let $L_m$ denote the set of lines in $\mathbb{F}^2$ with slope $m\in\mathbb{F}$, that is, all parallel lines of the form $y=mx+b$. Consider a ...

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### How does one calculate/estimate/guarantee the girth of a non-Abelian Cayley graph?

This question is in reference to this other question,
Can someone point out references (or explain!) which give techniques of being able to prove for any Cayley graph this property of having a girth ...

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295 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( ...

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**1**answer

181 views

### Uncountable cardinals and Prufer $p$-groups

Let $A$ be an elementary Abelian uncountable $p$-group. Is it known if there is an action of a Prufer $q$-group (here $q$ is a prime not necessarily distinct from $p$) $C_{q^{\infty}}$ onto $A$ such ...

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**1**answer

94 views

### Binary algebra, is it possible to partition the elements in GF(2^12) into 65 subgroups closed under addition?

The set of all binary vectors with 12 components forms a field with 2^12 elements containing 000000000000 and another 65*63 elements. Is it possible to partition these elements into 65 subgroups of 63 ...

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585 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 $P$?

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### What is the best known lower bound for: $\max_{2\leq i\leq p-1}(ord_n(i)) $?

Given an in integer $n$, and let $p$ be its smallest prime divisor (you can assume that $p$ is very large ). Let $ord_n(i)$ denotes the order of $i$ as an element of $\Bbb Z_n^*$ the multiplicative ...

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

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### 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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159 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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208 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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### 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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### 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 $f_2 = x^{m_2} + (1+x)^{n_2}$ ...

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111 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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209 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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### 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})| ...

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213 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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### 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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240 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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131 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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144 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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269 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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181 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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115 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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118 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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263 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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### 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} ...

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### 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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180 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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287 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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### 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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232 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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166 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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180 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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### 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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232 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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96 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{ ...