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### Binary `neighbors` perfect polynomials, instead of `consecutive`

Inspired by Andrei's nice solution of 52609, (namely consecutive perfect polynomials):
Denote by $A$ the full ring of polynomials in one variable $t$ over the ...

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### Can two Consecutive Polynomials both be perfect ?

Denote by $A$ the full ring of polynomials in one variable $t$ over the finite field with $q$ elements.
For any monic polynomial $P \in A$ define
$$
\sigma(P) = \sum_{d \mid P, d\, \text{monic}} d.
...

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### J. T. B. Beard's algorithm to catch `perfect polynomials`

For any monic polynomial $P$ with coefficients in a finite field of order $q$, we put
$$
\sigma(P) = \sum_{d \mid P, d \text{ monic}} d.
$$
Observe that $P$ and $\sigma(P)$ have the same degree.
...

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### All polynomials over a finite field are sums of $2$ square-free polynomials

I quote Proposition 2.3, page 14 lines -3 and -4 of Michael Rosen's book
Number Theory in Function Fields:
Let $b_n$ be the number of square-free monics in $A= \mathbb{F}_q[t]$ of degree $n.$
Then ...

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### Quadratic forms without common zeroes

A question which, I suppose, is as easy as abc for an expert. For a given finite field $F$ of odd characteristic (if needed, the characteristic is $3$ and the size of $F$ is large), do there exist two ...

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### elementary question on ECDLP

If $\mathbb{F}_q$ is a finite field and the elliptic curve E is defined over finite field $\mathbb{F}_p$ such that ECDLP is hard in E($\mathbb{F}_p$), where $q, p$ are prime and $q \ll p$. Let T $\in ...

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### Finch's sequence over $\mathbb{F}_3$

In http://algo.inria.fr/csolve/seqmod3.pdf -- "Periodicity in sequences mod 3" Steven Finch (also cited in Sloane's OEIS A112683) defines the following sequences in $\mathbb{F}_3$:
For each positive ...

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### Does the etale fundamental group of the projective line minus a finite number of points over a finite field depend on the points?

Clearly the etale fundamental group of $\mathbb{P}^1_{\mathbb{C}} \setminus \{a_1,...,a_r\}$ doesn't depend on the $a_i$'s, because it is the profinite completion of the topological fundamental group. ...

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### Finite fields: Is multiplicative order of $x^p - x - 1$ equal to $\frac{p^p-1}{p-1}$ over GF(p)? [duplicate]

Possible Duplicate:
Multiplicative order of zeros of the Artin-Schreier Polynomial
I will be grateful for any reference to some literature on the following question (to the best of my ...

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### Noether-Deuring for injections and surjections?

Noether-Deuring theorem (not in the strongest form, but in the one I usually need):
Let $L\diagup K$ be a field extension. Let $A$ be a $K$-algebra which is finite-dimensional as a vector space over ...

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### Multiplicative order of zeros of the Artin-Schreier Polynomial

This question was asked on NMBRTHRY by Kurt Foster:
If $p$ is a prime number and $\mathbb{F}_p$ the field of $p$ elements, the zeroes of the Artin-Schreier polynomial
$x^p - x - 1 \in ...

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### A silly question: is the number of points on a Jacobian (of a curve, over a finite field) known?

In a cryptography book I read that people does not known how to compute the number of points on a Jacobian of a hyperelliptic curve $C$ over a finite field $F_q$? Is this true? It seems easy to ...

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### Gelfand-Tsetlin bases for Lie groups over finite fields

There is a method of constructing representations of classical Lie algebras via Gelfand-Tsetlin bases. It has also been applied to Symmetric groups by Vershik and Okounkov. Does anybody know of any ...

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### A family of hypersurfaces with many points

This question is a sequel to an earlier question, which asked about the zeta function of a certain affine variety over a finite field $k$. The unusual thing about this variety is that it had the ...

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### 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 ...

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### Polynomials over Z evaluated with finite field arguments

A) Given a non-constant polynomial $q\in\mathbb{Z}[\alpha_1,\alpha_2,\ldots,\alpha_n],$ if we pick random $\omega_i\in\mathbb{F}$ (a finite field) uniformly and independently across $1\leq i\leq n,$ ...

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### 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 ...

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### Existence question on rational points on a curve

I am puzzled about the following question:
Let C be a smooth, projective, absolutely irreducible curve defined over GF(q) and let g denote the genus of C. O is a rational point on C, and the divisor ...

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### Number of irreducible polynomials with some coefficients fixed over a finite field

I am interested in the following problem: I have a finite field $F_q$, two positive integers
$n>m$ and elements $a_1,...,a_m\in F_q$. How many of the polynomials
...

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455 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 $x,y\in F^n$, ...

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### how to generate the n-torsion group

Hello,
I was curious about the following sentence: "then the $n$-torsion on $E(\overline{K})$ has known structure, as a Cartesian product of two cyclic groups of order $n$" (found at ...

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### 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 ...

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### Order of finite unitary group

This may be an easy exercise but I am not getting it. Let $\mathbf F_q$ be a finite field with $q$ elements and $\mathbf F_{q^2}$ be its degree two extension. Define an automorphism $\sigma$ of ...

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### Number of subset sums

Let $\mathbb{F}_q$ be a finite field with characteristic $p$ and $p < q$ (i.e. not a prime field). Let $D\subseteq \mathbb{F}_q$ be a some set with $|D|=n$. Find a non-empty subset ...

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### Elements of trace zero in a field extension

Let $K=F_q$ and $F=F_{q^3}$, define the set A={$x \in F$ : $Tr_{F/K} (x)=0$}.
Is it true that for every $x \in A$ there are $y,z \in A$ such that $x=yz$?

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### Elliptic curves over proper variety over $\mathbf{F}_q$ isotrivial

Why is every elliptic curve over a proper (edit: smooth and geometrically connected) base over $\mathbf{F}_q$ isotrivial, i.e. is constant after base changing with $\bar{\mathbf{F}}_q$? If the moduli ...

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### Sub-representations of the affine group

Let $F=\mathrm{GF}\left(p^k\right)$ be any finite field. Let $G$ be the group of all affine permutations on $F$ (i.e. permutations of form $x\mapsto ax+b$). Then the set of all functions from $F$ to ...

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### Non-split extensions of $GL_n(F_q)$ by $F_q^n$ ?

A very naive question :
I just learned that there is a non-split extension of $GL_3(F_2)$ by $F_2^3$ (with standard action). It can be realized as the subgroup of the automorphism group $G_2$ of ...

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### Number of solutions of a linear equation in a small subset.

Let $p$ be a prime. Let $F_p$ be the finite field of
$p$ elements. Let $A$ be a subset of $F_p$ of size $s$.
Assume that $s > 2$ is polylogarithmic in $p$.
Suppose that we want to count ...

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### Roots of unity in different completions of a number field

For any field $k$, let $\mu(k)$ denote the roots of unity in $k$. Now let $k$ be a number field and let $v, w$ be non-archimedean primes of $k$ with distinct residual characteristics. Does there ...

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### Hartshorne Exercise III.4.7 (cohomology of closed subschemes in $\mathbb{P}^2$)

I have some questions about the following exercise in Hartshorne (III.4.7):
Let $f \in k[x_0,x_1,x_2]$ be a homogeneous polynomial of degree $d \geq 1$ and $f \neq 0$ and let $X$ be the closed ...

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### 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" ...

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### Cyclic order relation in Zn

The ring Zn:={0,1,..,n-1} under addition and multiplication modulo n.
Suppose a,b,c,x $\in$ Zn are nonzero and the cyclic order R(a,b,c) holds, then under what conditions does R(ax,bx,cx) hold ?

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### How many groups of size at most n are there? What is the asymptotic growth rate? And what of rings, fields, graphs, partial orders, etc.?

Question. How many (isomorphism types of) finite groups of size at most n are there? What is the asymptotic growth rate? And the same question for rings,
fields, graphs, partial orders, etc.
...

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### additive structure in a small multiplicative group of a finite field?

Let $p$ be a prime. Given a positive integer $n$, does there exist a
$\beta$ in an extension of $F_p$ such that
1) If $F_p[\beta] = F_{p^N}$, then $N > n^n$; ( $\beta$ lies in a high extension)
...

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### Inverse for a permutation over GF(2)

Given a permutation $f: \{0,1\}^n \rightarrow \{0,1\}^n$ as $n$ polynomials over $GF(2)$ how to get formulas for the inverse permutation $f^{-1}$?
I am interested in the answer to the previous ...

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### Restriction theorems over finite fields

A short while ago, Dvir proved the Kakeya conjecture over finite fields. Does this have any implications for restriction theorems over finite fields? I am aware only of implications going in the ...

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### How to factorize X^n - 1 in Z/pZ?

How do I factorize a polynomial $X^n - 1$ over $\mathbb{F}_p$? In particular I need to find factors of the polynomial $X^{3^3 - 1} - 1 = X^{26} - 1$ over $\mathbb{F}_3$.

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### étale cohomology with G_m coefficients

Most calculations of étale cohomology in Milne's book deal with constructible or torsion sheaves. Are there references where the cohomology of varieties with $\mathbf{G}_m$ coefficients are ...

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### Complex powers in finite fields

Is it possible to compute complex powers in finite fields? Given a $\in \mathbb{F}_p$ ($p$ prime), how can one compute $a^i$ per example?

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### 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 ...

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### algorithm for calculating the Chow groups of a variety over a finite field

Is there an algorithm for calculating the Chow groups of a variety over a finite field?
It is know that $H^{2i,i}_\mathrm{mot}(X,\mathbf{Z}) = CH^i(X)$. In how many cases does this help us?

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### Computer program to solve a system of polynomial equations over a finite field

I have a set of polynomial equations for which I want to know the solutions (actually really the number of solutions). It would be great if I could get a computer to do it, but I'm not sure exactly ...

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### 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 ...

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### Elementary theory of finite fields

I read on Ax's article that the elementary theory of finite fields is decidable if one assumes the continuum hypothesis to be true. What about if one assumes the hypothesis to be false?

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### A hypersurface with many points

Ok, it's time for me to ask my first question on MO.
Consider the affine curve $Y+Y^q=X^{q+1}$ over the finite field $\mathbf{F}_q$. It's interesting because it has the largest number of points over ...

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### Weil Conjectures for Grassmannians

To establish the Weil conjectures for $n$-dimensional projective space over a finite field is elementary. Does there exist a simple direct proof of the conjectures for finite field Grassmannians?

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### Counting solutions to x^{p+1}=y^4 in a finite field

I need to compute the number of solutions to the equation $x^{p+1} = y^4$ in the field with $p^2$ elements (for p sufficiently large). The form of the equation suggests to me that the solution would ...

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### Richardson varieties over finite fields

Let me start with some background to set the notation before I ask my question.
Let G be a semisimple algebraic group over some algebraically closed field K, and suppose we have fixed a Borel ...

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### Induction from split and non-split tori for GL_2 over a finite field

Let k be a finite field, G the k-points of GL_2, T1, T2 the k-points of the split and non-split tori of G.
Then the G-representations C[G/T1] and C[G/T2] are almost the same.
More precisely, they ...