The finite-fields tag has no wiki summary.

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**1**answer

42 views

### Roots of $X^{l-1}+1$ in a quadratic extension $F_q, q= l^2$ [on hold]

Consider a finite field $F_q$ where $q=l^2$ ($l$ can be of the form $p^m$). Does $F_q$ has a root of $X^{l-1}+1$?
As $X^l+X = X(X^{l-1}+1)$ we can show that $X^{l-1}+1$ splits if it has a root. This ...

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**0**answers

390 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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**0**answers

44 views

### 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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**0**answers

47 views

### Must a multivariate polynomial be zero subject to constraint on the values of all pairs of distinct variables at solutions?

I suppose this is both easy and false.
Let $f \in \mathbb{F}_2[x_1, \ldots x_n]$.
Suppose in all solutions of $f(x_1, \ldots x_n)=0$
all pairs of variables $(x_i,x_j),i \ne j$ can take all
possible ...

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votes

**6**answers

3k 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 ...

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votes

**2**answers

581 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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votes

**2**answers

226 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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votes

**1**answer

128 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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**2**answers

88 views

### 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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votes

**1**answer

288 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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votes

**1**answer

173 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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votes

**1**answer

93 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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votes

**3**answers

583 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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**1**answer

101 views

### 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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**0**answers

68 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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**0**answers

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

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**1**answer

142 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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156 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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**2**answers

205 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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38 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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**3**answers

810 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 $f_2 = x^{m_2} + (1+x)^{n_2}$ ...

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**1**answer

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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**2**answers

204 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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**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})| ...

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**1**answer

210 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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53 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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234 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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**1**answer

126 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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137 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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262 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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177 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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**0**answers

109 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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**0**answers

117 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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**0**answers

260 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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**0**answers

126 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} ...

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**2**answers

417 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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**1**answer

361 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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**1**answer

174 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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**2**answers

285 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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72 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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**1**answer

229 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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**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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**1**answer

177 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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**1**answer

596 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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**1**answer

222 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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**0**answers

42 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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**0**answers

95 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

197 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 ...

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**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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**1**answer

191 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 ...