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3
votes
1answer
544 views

Does there exist a polar decomposition of matrices over finite fields?

There exists a polar decomposition for matrices over the reals. What I would like to know is if an analog has been shown for groups of matrices over finite fields. If not, it would be great to get ...
7
votes
2answers
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 ...
9
votes
0answers
240 views

Noether-Lefschetz over finite fields

The classical Noether-Lefschetz theorem asserts the following: Over the complex numbers, a very general surface $S\subset \mathbb{P}^3$ has Picard number 1 (that is, $Pic(S)\simeq \mathbb Z$), ...
8
votes
4answers
964 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 ...
3
votes
1answer
520 views

Trivializing principal bundles on a curve over a finite field

This is related to my question Adelic description of moduli of $G$-bundles on a curve. Let $X$ be a smooth, projective, and geometrically connected curve over a finite field $\mathbb{F}_q$ and $G$ a ...
5
votes
1answer
381 views

More expanders?

Having received several exhausting answers to my recent question about the expansion properties of a certain graph, I now wonder whether anything is known on the following graphs of a similar nature: ...
7
votes
4answers
952 views

The “interplay” between additive and multiplicative structure in a field

A field is an ordered triple $(F, +,\cdot)$ of a set $F$ and binary operations $+,\times$ on $F$ such that $(F,+)$ and $(F\backslash 0,\times)$ are abelian groups satisfying the distributive laws ...
11
votes
2answers
435 views

An expander (?) graph

For a prime $p$, consider the graph on the vertex set ${\mathbb F}_p$, in which every vertex $z$ is adjacent to $z\pm 1$ and also to $z^{-1}$ (unless $z=0$). I was told that this graph is known to be ...
2
votes
0answers
110 views

Family with a fixed special fiber over finite fields

Let $X$ be a smooth projective variety over a finite field $\mathbb{F}_p$. What are the conditions for the existence of a projective variety $X'$ over $\mathbb{Q}_p$ such that $X$ is a special fiber ...
4
votes
2answers
1k views

A formula for a generator of the multiplicative group of $\mathbb{F}_p$ ?

Let $p$ be a prime. It is a common statement that the multiplicative group $(\mathbb{F}_p)^*$ of the prime field has no canonical generator. It is however no so easy to say exactly what this means, in ...
5
votes
2answers
273 views

Polynomials over $\mathbb F_2$ without zeros in $\mathbb F_2$ having an inverse series with support of large density.

Does there exist a sequence $A_n=A_n(x)\in\mathbb F_2[x]$ over the field $\mathbb F_2$ of two elements (represented by $0$ and $1$) such that $A_n(0)=A_n(1)=1$ and the inverse series ...
3
votes
1answer
835 views

Order of an element in a finite field

Let $\mathbb F_p$ be the finite field of a prime order $p$, $f(x)\in \mathbb F_p[x]$ an irreducible polynomial, $E = \mathbb F_p[x]/\langle f(x)\rangle$ a finite extension of $\mathbb F_p$, ...
6
votes
0answers
107 views

Sum of densities of support of $A$ and $A^{-1}$ for $A=1+\dots\in \mathbb F_2[[x]]$

Let $A=1+\dots\in\mathbb F[[x]]$ be a (multiplicatively) invertible series over the field $\mathbb F_2$ of two elements. Writing $A=\sum_{n\geq 0}\alpha_n x^n$ and $\frac{1}{A}=\sum_{n\geq 0} ...
2
votes
2answers
158 views

linear independence of orbits via a set of transformations in char p

Let $T_1, \ldots, T_n \in GL(n,\mathbb{F}_p)$. Suppose for all $\vec{v} \in \mathbb{F}_p^n$ we have $\det (T_1 \vec{v}, T_2 \vec{v}, \ldots, T_n \vec{v}) = 0$. Now, let $k$ be a finite extension of ...
1
vote
1answer
246 views

Distribution of the powers of a primitive element of a finite field

What are known results regarding the distribution of the powers of a primitive element (generator of the multiplicative group) of a finite field? Specifically, compare the ordered list of ascending ...
1
vote
2answers
279 views

When is PSU(2,q^2) = PSL(2,q) ?

The context for this question is from page 284 - 287 of Berger's paper: ...
4
votes
2answers
392 views

The Lang isogeny

Let $G$ be a connected commutative algebraic group over $\mathbb{F}_q$. If $\text{Fr}_q : G \to G$ denotes the $q$-Frobenius morphism, we define the Lang isogeny $L_q$ to be the endomorphism of $G$ ...
0
votes
1answer
96 views

Probability of summing products of irreducible polynomials in a finite field to zero

Let $f(x), g(x), h(x)$ be randomly chosen irreducible polynomials over the finite field $GF(2^n)$. What would the probability be for $\sum_{(i,j,k:i,j,k\in\mathbb{N},i+j+k=C)} f^i(x)g^j(x)h^k(x)=0$, ...
4
votes
0answers
254 views

Tangent space of the moduli stack of Drinfeld modules

I am going through the proof of Thm 1.5.1 of Laumon, Cohomology of Drinfeld modular varieties, which says that a certain map of stacks is smooth. To prove this, Laumon considers the tangent space of a ...
4
votes
0answers
304 views

Does the Riemann hypothesis for liftable varieties over a finite field imply the Riemann hypothesis for all varieties over a finite field

The Riemann hypothesis for varieties over a finite field has been proven by Deligne. Still I would like to ask the following question. A variety $X$ over a finite field $k$ is liftable if there ...
3
votes
0answers
147 views

Number of Inverse Pairs Modulo Prime $p$

Is there a result which gives a lower bound on the number of inverse pairs $(a, a^{-1})$ modulo prime $p$ lying in the interval $[1,t]$, where $t < p$?
2
votes
1answer
301 views

Finding a subspace disjoint from a union of subspaces

Let $k$ be a finite field (I care about $\mathbb F_p$, especially $\mathbb F_2$) and let $V_1,...,V_N\subset k^n$ be subspaces. I want to find a subspace $S\subset k^n$ such that $S\cap V_i=0$ for ...
3
votes
1answer
149 views

Does the Hilbert polynomial determine the weight of the Galois representation associated to a variety

Let $X$ be a curve or an abelian variety (over a finite field). Then the Galois representation associated to $X$ via the etale cohomology of $X$ (in degree $1$) is integral of weight $1$ and its ...
0
votes
0answers
219 views

Vanishing of motivic cohomology with finite coefficients in negative degrees

I wonder whether the "finite coefficient version" of Beilinson-Soule conjecture i.e. the following statement holds or not. STATEMENT: Let $X$ be a smooth and projective scheme over a finite field ...
1
vote
0answers
136 views

Special values of zeta functions and extensions of base fields.

Let $X$ be a scheme of finite type over a finite field $k=\mathbb{F}_{q}$ of $q$ elements. Then, one can define the zeta function $Z_{X/k}(T)$ of $X$ ovet $k$ as $\prod_{x\in ...
1
vote
1answer
1k views

Primitive $k$th root of unity in a finite field $\mathbb{F}_p$

I am given a prime $p$ and another number $k$ ($k$ is likely a power of $2$). I want an efficient algorithm to find the $k$th root of unity in the field $\mathbb{F}_p$. Can someone tell me how to do ...
3
votes
1answer
306 views

Abelianized fundamental group of a curve over a finite field

Let $X$ be a smooth, projective, and geometrically connected curve over a finite field $\mathbb{F}_q$ and fix a geometric point $\overline{x} : \text{Spec } \overline{\mathbb{F}_q} \to X$. Then there ...
4
votes
3answers
1k views

Number of solutions of an equation over finite fields

Does anyone know of any result that deals with the following problem of counting the number of solutions of a certain algebraic equation over a finite field? Let $p$ be an odd prime and ...
6
votes
2answers
1k views

Algorithms to find irreducible polynomials of a given degree

I need to know what are the efficient algorithms to find all the irreducible polynomials of a given degree, say $d$ over a given finite field, say $\mathbb{F}_{p^n}.$ One way is to factorize the ...
2
votes
0answers
209 views

Small geometric progression modulo N

An problem related to integer factorization using the General Number Field Sieve is the following: Let $N$ be a composite. Must there exist a 5-term geometric progression $\lbrace ...
5
votes
4answers
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 ...
0
votes
1answer
146 views

trace cotrace Matrix

Hello I want to know whats mean (trace) cotrace matrix. In the context, mapping a matrix (t x n) $\in$ GF($2^m$) to a cotrace matrix (tm x n) $\in$ GF($2$)?
3
votes
0answers
369 views

Finite Field Varieties and the de Rham Complex of Kähler Differentials

In an answer to a previous question I asked, where the Kahler differentials of a variety over a finite field were discussed, it was stated that: You can certainly define de Rham cohomology using ...
17
votes
4answers
922 views

A mixing property for finite fields of characteristic $2$

In connection with this MO post, here is a question somewhat implicitly contained in a joint paper of S. Kopparty, S. Saraf, M. Sudan, and myself. Let ${\mathbb F}$ be a finite field, and suppose ...
8
votes
2answers
392 views

A mixing property of linear map over finite fields

Let $F$ be a finite field of odd size $q$, and $\phi_0 : F \mapsto F$ be any map from $F$ to itself. For each $a \in F$, set $\phi_a : x \in F \mapsto \phi_0 (x) + ax $. When $\phi_0 : x \mapsto x^2 ...
1
vote
1answer
200 views

Given g1(x), g2(x) minimize over p(x) Hamming weight of [p(x)g1; p(x)g2(x) ] ? (Or how to find minimal distance of convolutional code?)

Fix polynoms g1(x), g2(x) over F_2[x]. Question: How to find minimum over polynoms p(x) of the: HammingWeight(p(x) g1(x) ) + HammingWeight(p(x) g2(x) ) ? By HammingWeight of polynom I mean number ...
7
votes
4answers
1k views

Orthogonal Groups over finite fields

Hello Let $\mathbb{F}_p$ be the finite field with $p$ elements. One can show that over finite fields, there are just two non-degenerate quadratic forms. So here I want to pick any non-degenerate ...
3
votes
1answer
252 views

Special polynomials over finite fields

My field of research is coding theory and I am working on cyclic codes. During my research, I tackled an algebraic problem. After some simple definitions, I asked my question. I will appreciate any ...
0
votes
0answers
259 views

Combinatorial Interpretation of an Extension of Gaussian Polynomials

It is well-known that the Gaussian polynomial (or Gaussian coefficient, q-binomial coefficient) $\binom{n}{k}_q$ counts the number of $k$-dimensional subspaces of an $n$-dimensional vector space over ...
1
vote
1answer
190 views

How many k-nomials of deg N divisible by X^16+x^12+x^5 +1 ? (Spectrum of CRC-16-CCITT erroc-correcting code ?)

Let us consider polynoms over $F_2$. Consider the linear SUBSPACE of polynoms divisible by $x^{16}+x^{12}+x^5 +1$ and of degree less or equal $N$ (e.g. 40). Question: How many k-nomials belong to ...
1
vote
2answers
134 views

Number of solutions to $mx^2+ny^2 \equiv k\pmod{p}$

I need a reference for the result which gives the number of solutions to the congruence $mx^2+ny^2 \equiv k\pmod{p}$. This result seems to be something that would be discussed in Gauss' ...
7
votes
3answers
922 views

Quadratic reciprocity and Weil reciprocity theorem

I was told that Weil reciprocity theorem (one has two meromorphic function $f,g$ on a complex curve $C$, so $\prod\limits_{x\in C} g(x)^{ord_xf}=\prod\limits_{x\in C}f(x)^{ord_xg} \ $ where $ord_xf$ ...
0
votes
2answers
164 views

Matrices whose range is equal to the column set [closed]

Is there such a thing? I suppose there are two cases in which they may exist: over a finite field or infinite matrices (but let's stick to countable matrices then).
4
votes
0answers
297 views

Matching a binary matrix

Given a MxN 0-1 matrix D, with the property that both M and N are odd numbers its row sums and column sums in the $\mathbb{Z}_2$ field are all equal to the same number (0 or 1). How do we find M ...
5
votes
3answers
355 views

Source for embedding multiplicative group of an algebraic closure of a finite field?

It's easy to embed the (cyclic) multiplicative group of a finite field into the multiplicative group of $\mathbb{C}$ (or other algebraically closed field of characteristic 0): assign to a generator of ...
19
votes
2answers
1k views

Sums of powers mod p

For prime $p > 7$ with $p-1=rs$, $r>1$, $s>1$, let $A=\{x^r|x \in \mathbb{Z}_p\}$ and $B = \{x^s|x \in \mathbb{Z}_p\}$. If $g$ is a primitive root mod $p$ then $A = \{0\} \cup \{g^{ir}|0 \leq ...
4
votes
1answer
446 views

Finite Field Grassmannians as Homogeneous Spaces

For the real Grassmannian Gr$(N,k)$ we have the well-known isomorphism $$ \text{Gr}(N,k) = O(N)/(O(k) \times O(N-k)) $$ For the complex case, we have $$ \text{Gr}(N,k) = U(N)/(U(k) \times U(N-k)) $$ ...
3
votes
3answers
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 $P$?
4
votes
2answers
871 views

Vandermonde matrices and general position

I was wondering if it is known whether a Vandermonde matrix over a sufficiently large finite field is in general position with respect to intersections of subspaces spanned by subsets of columns, i.e. ...
5
votes
1answer
391 views

Probability of a set of random vectors over finite field being a spanning set

Suppose I have a set of random vectors $f(a_1, \ldots, a_\ell) := (v_1, \ldots, v_m) \subset F_p^n$, $m \ge n$, given by a matrix valued polynomial function $f$, where the $a_i$'s are independent, ...