The tag has no usage guidance.

learn more… | top users | synonyms

10
votes
2answers
522 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 $q$ ...
5
votes
1answer
233 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 ...
0
votes
1answer
244 views

Number of Minimal left ideals in the full matrix ring over a finite commutative local ring

Inspired with another QUESTION I would like to know the number of minimal left ideals of $M_n(R)$ in terms of $n$ and $R$ where $R$ is a finite local commutative ring with identity ?
8
votes
1answer
306 views

A Balog-Szemeredi-Gowers-type question

A nice Lemma due to Konyagin asserts that for any subset $B \subset \mathbb{F}_p$ holds $$ |B.B - B.B + B.B - B.B + B.B - B.B| \geq \frac{1}{2}\min\{p, |B|^2 \}, $$ where the standard notation for the ...
1
vote
2answers
403 views

Equations of elliptic curves

First part of question I have asked on mathoverflow already: http://math.stackexchange.com/questions/467088/explict-form-of-the-equation-of-elliptic-curve 1) Let $E(\mathbb{F}_{q^2})$ is elliptic ...
16
votes
1answer
237 views

Do mutually dual finite vector spaces have the same orbit cardinalities under a linear group action?

Let $G$ be a finite group acting linearly on a finite dimensional vector space $V$ over a finite field. By Burnside's lemma, $$ |V/G| = \frac 1{|G|} \sum_{g\in G} q^{\dim(ker(g - I))}. $$ Since $g-I$ ...
4
votes
2answers
992 views

Calculus over finite fields

$P(x,y,z)$ is a polynomial function on an algebraic surface $S$ in $F_{q}^{3}$. Suppose that the derivative of $P$ along any tangent vector of $S$ is zero. Can we say that $P$ is constant on $S$? ...
3
votes
0answers
105 views

Closest sumset to a set

Suppose $A$ is a subset of the finite field with $p$ elements. What is the best approximation of $A$ by a sumset $B+C$ in the sense that $|A\Delta (B+C)|$ is minimal? Of course if $B=A-x$ and ...
5
votes
1answer
430 views

Are there Carlitz analogues of quadratic residues and reciprocity?

Let $q$ be a prime power. I will use the notations of Keith Conrad's Carlitz extensions paper (but I'll work over $\mathbb{F}_q$ rather than $\mathbb{F}_p$). The most general question I'm asking here ...
1
vote
1answer
247 views

probability of having linearly independent sparse vectors over finite fields

Suppose that there are $i< N$ linearly independent $N$-dimensional vectors $\mathbf{v}_1,\mathbf{v}_2,\ldots,\mathbf{v}_i$ over a finite field $\mathbb{F}_q$, whose elements are denoted as ...
5
votes
0answers
222 views

Transferring addition and multiplication over finite fields to $\mathbb{Z}$

It seems to me that the most basic wisdom on why many number-theoretic conjectures are hard is because the interplay between addition and multiplication is subtle and delicate (much of the lay chatter ...
0
votes
1answer
271 views

Existence of nontrivial roots of a homogeneous polynomial over a finite field in extension fields

Consider a homogeneous polynomial, f, of total degree n in n variables, with coefficients in a prime order finite field, GF(p). Are there any general rules regarding the existence of nontrivial roots ...
3
votes
1answer
545 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
436 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 ...
10
votes
0answers
252 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, $Pic(S)\simeq \mathbb Z$), ...
9
votes
4answers
1k 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 ...
4
votes
1answer
533 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
384 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
969 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
472 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
112 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
275 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
873 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
108 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
162 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
252 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
288 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
408 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
99 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
261 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
305 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 $t < p$?
4
votes
1answer
199 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 ...
2
votes
1answer
315 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
150 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
225 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
137 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
318 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
219 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
148 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
374 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
925 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
397 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
2k 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 ...