A finite field is a field with a finite number of elements. For each prime power $q^k$, there is a unique (up to isomorphism) finite field with $q^k$ elements. Up to isomorphism, these are the only finite fields.

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Gcd of polynomials over a finite field [on hold]

Let $\mathbb{F}_q[X]$ be the polynomial ring over the finite field with $q$ elements. Let $f$ be a polynomial of the form $x^n-a$ and let $g$ be a polynomial of the form $x^m-b$. Is it known whether ...
4
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1answer
230 views

Vector with many non-zero coordinates

Given finite field $\mathbb{F}_q$, positive integers $n$ and $k<n$. Given $k$-dimensional subspace $X$ of $\mathbb{F}_q^n$, for which $m=m(q,k,n)$ may we find for sure a vector in $X$ with at least ...
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39 views

Number of Skew Symmetric Matrices of fixed rank

The number of symmetric matrices of order $n$ and rank $r$ over finite fields has been counted e.g. http://www.math.clemson.edu/~kevja/REU/2004/SymmetricRankRMatrices.pdf Is the number of ...
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votes
3answers
644 views

Hypersurface missing just one point

Let $\mathbb F_q$ be a finite field and $n$ an integer. What is the minimal degree $d = d(q,n)$ of a polynomial $f \in \mathbb F_q[X_1,\dots,X_n]$ such that the set $Z(f)$ of zeros of $f$ in the ...
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63 views

Calculation of cardinality of Jacobians

The problem of calculation the number of rational points on curves over finite fields is $\#P$-complete - "Counting curves and their projections". Is it true for calculation of number of rational ...
4
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1answer
148 views

Schwartz-Zippel lemma for an algebraic variety

Let $X $ be a smooth affine subvariety of $(\overline{\mathbb{F}_q})^n$ defined by a prime ideal $I$. Let $f$ $\in \mathbb{F}_q[x_1,\ldots,x_n]$ be a polynomial such that $f \notin I$. Let $r_1, ...
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2answers
293 views

Irreducible algebraic sets via irreducible polynomials

There are many results about irreducible polynomials over finite fields: we know a cardinality of all irreducible polynomials with given degree, we know explicit examples of irreducible polynomials, ...
2
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1answer
114 views

Bases of the special form

Let $R = \mathrm{GF}(q)$, $S = \mathrm{GF}(q^n), \ n\geq 2$ be extension of $R$, $h$ be a primitive element of $S$. I want to count or estimate the number $N$ of bases of the following form. Let ...
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59 views

Carlitz factorials and Euler-like series

Let $q$ be a power of a prime $p$. For every $i\in\mathbb N$, one denotes $D_i=\prod_{\substack{h\in\mathbb F_q[T]\text{ monic}\\\deg h=i}}\limits h$. For $n\in\mathbb N$, write ...
3
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1answer
129 views

Non-zero coefficients of primitive polynomials

Let $R$ be the finite field with $q$ elements, and let $m,n\in \mathbb{N}$ be positive integers $\geq 2$. I want to prove that there exists a primitive polynomial $$F(x) = ...
2
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0answers
59 views

Property of a derivative in global field

Before posting I want to make it clear that I posted the same question in stack exchange awhile ago (http://math.stackexchange.com/questions/1533814/property-of-derivative-in-a-local-field) but didn't ...
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69 views

the least point on a variety over a finite field

Let $p$ be a large prime parameter and $V\subseteq \mathbb{P}^n_{\mathbb{F}_p}$ a variety defined over the finite field $\mathbb{F}_p$ with bounded degree and dimension (w.r.t. $p$). Assume that $V$ ...
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0answers
92 views

A reference about a problem of the number of the rational points on a projective scheme

Let $X\hookrightarrow\mathbb{P}^n_{\mathbb{F}_q}$ be a pure dimensional projective scheme of dimension $d$. So we know a trivial estimate of the number of $X(\mathbb F_q)$ is that $\#X(\mathbb ...
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0answers
60 views

How to construct large sets of $m$-dimensional vectors over a finite field such that any $m$ of them are independent?

Let $m\ge 2$ be an integer and $\mathbb{F}$ be a finite field of order $p^k$. I want to construct as many as possible $m$-dimensional vectors $v_1,v_2,\ldots,v_n$ in field $\mathbb{F}$ such that any ...
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92 views

Counting nonzero hyperdeterminants over $\mathbb{F}_q$

The hyperdeterminant $D(A)$ is a multidimensional generalization of the determinant. It is a polynomial in the entries of a $(k_1+1)\times (k_2+1)\times\cdots \times (k_n+1)$ array $A$. The ...
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84 views

Gaussian Hypergeometric Functions and Legendre Character

I was hoping somebody might be able to point me to a good reference on Gaussian hypergeometric functions defined over a finite field. the reason I'm interested is that I've encountered sums of the ...
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51 views

Determining Inconsistency of (first-order) Non-linear System of Equations [closed]

Is there a way I can figure out what values of the coefficients of some system of non-linear equations makes the system inconsistent? Take the following system of equations as an example. The ...
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1answer
96 views

Functions of several variables over finite fields [closed]

For a finite field $F$ any function $f\colon F\to F$ is given by a polynomial. My question is what happens when we are given a function of two or more variables? Is this necessarily a polynomial ...
3
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1answer
90 views

Characterization of Lagrangian planes in symplectic vector spaces over finite fields [closed]

EDIT: As L Spice pointed out, there is an error in the observation. The question is void therefore Let $p$ be a prime and $q=p^r$. Let $V$ be a $\mathbb F_q$-vector space of dimension four, with a ...
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2answers
285 views

Splitting subspaces and finite fields

Hellow. I'm sure that the following is truth, but I can't prove it. Let $R<S<K, R=\mathrm{GF}(q),\ S= \mathrm{GF}(q^n), \ K= \mathrm{GF}(q^{mn})$ be a chain of finite fields and $A = ...
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1answer
253 views

Counting isomorphism classes in open subsets of Bun_G

Let $G$ be a split semisimple algebraic group and let $C$ be a curve of genus $g$ over $\mathbb F_q$. Assume $g \geq 2$. The number of $\mathbb F_q$-points of $\# \operatorname{Bun}_G(C)$, where each ...
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2answers
142 views

A reference about Grassmannian over finite fields

Suppose $Gr_k(k,n)$ the Grassmannian which classifies all the dimension $k+1$ sub-spaces of a dimension $n+1$ linear space over the field $k$. For the case over a finite field $\mathbb F_{q}$, we can ...
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1answer
544 views

Is hyperelliptic cryptography “practical”?

Previosly my impression on this subject was that hyperelliptic cryptography systems (as well as other possible cryptosystems based on abelian varieties of dimension $>1$) have no advantages over ...
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1answer
58 views

Do tori in a symplectic group always have invariant maximal isotropic subspaces?

$\newcommand{\mbf}{\mathbf}$ Hi all, I've been thinking about the following question for a while now, and got a little stuck trying to solve it. Hopefully, someone here might be able to help. For ...
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2answers
348 views

Blocking sets in three dimensional finite affine spaces

What is the smallest possible size of a set of points in $\mathbb{F}_q^3$ which intersects (blocks) every line? Clearly the union of three affine hyperplanes that intersect in a singleton, say $x = ...
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1answer
198 views

Polynomial factoring over finite fields

What is known in general about the complexity of factoring polynomials over finite fields? For instance given $\Bbb F_q$ where $q=p^n$ and total degree $d$ polynomial in $m$ variables what can we say ...
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75 views

How order of divisor with support at infinity is changed at reduction?

Jing Yu in his paper "On Arithmetic Of Hyperelliptic Curves" on page 5 asserts the following The most interesting case is certainly the case $k = \mathbb Q$ and $D \in \mathbb Z[t]$. To ...
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159 views

Polynomials representing locally constant functions

Let $K$ be a finite field with $p$ elements. (a) Let $f\in K\lbrack x\rbrack$ be such that (i) $\deg(f)<p$ and (ii) $f(2x) = f(x)$ for $\geq (1-\epsilon) p$ values of $x$ in $K$. What can we say ...
2
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1answer
75 views

Distribution of values of quadratic polynomials over a finite field

Let $S$ denote the set of squares in the finite field ${\mathbb F}_p$. Now let $f(x) \in {\mathbb F}_p[x]$ denote a degree $2$ polynomial, which is not a square of a linear polynomial. If we consider ...
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1answer
211 views

Solving Non-Linear Equations over a Finite Field of a Large Prime Order

I want to know is there is an efficient way to figure out whether or not a ( underdetermined) system of non-linear equations have a solution over a finite field of large prime order. The equations ...
3
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1answer
112 views

embedding of $O_4^-(q)$ in $U_4(q)$

For $q$ an odd prime power, one can construct $H:=O_4^-(q)<U_4(q)$ by treating the $H$-invariant bilinear form as Hermitean. On the other hand $U_4(q)$ is isomorphic to $O_6^-(q)$; I need to ...
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103 views

When is a Hermitian matrix of the form $g^*g$ for some matrix $g$

I tried asking this question on Math StackExchange and didn't get any replies. I was hoping that maybe someone here could help, sorry for duplicating. I'm trying to figure out some properties of ...
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1answer
383 views

Algebraic dynamics in finite fields

What is known about combinatorial structure of the rational maps of degree 2 over finite fields? From some general reasons I think it was studied. For being more specific, consider the field ...
5
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1answer
157 views

Finite field analogue of Chebotaryov theorem on roots of unity?

Chebotarev's theorem on roots of unity says that all the minors of a prime-length DFT matrix over the complex numbers are nonzero. I was wondering if there was an analogue for finite fields. More ...
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0answers
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Fields whose algebraic closure is a finite extension [duplicate]

It is well-known that the complex numbers $\mathbb{C}$ is a degree two extension of $\mathbb{R}$, where one possible minimal polynomial is $x^2 + 1$. Further, $\mathbb{C}$ is algebraically closed. ...
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1answer
307 views

What are some open problems regarding elliptic curves in finite fields?

I accept that my question seems so vague and broad, and I already looked into some similar questions in MO. But I would like to learn specifically about some open problems and conjectures regarding ...
4
votes
1answer
142 views

How to calculate log or exp of a value in GF(2^n) using log/exp table of GF((2^k)^m) where n=k*m?

Consider Galois fields $\mathbb{F}_{2^n}$ and $\mathbb{F}_{2^k}$, where $n=km$, and $\mathbb{F}_{2^k}$ is a ground field of $\mathbb{F}_{2^n}$. I’d appreciate pointers to papers or suggestions on: ...
5
votes
1answer
242 views

$(n-2)$-blocking sets in $AG(n,2)$

Let's define $k$-blocking set in affine space $AG(n,q)$ a set that meets every coset (translate of subspace) of dimension $k$. I have seen a lot work related to minimal $(n-1)$-blockings set. ...
2
votes
3answers
118 views

How to evaluate this function in F_p efficiently?

For the positive prime integer $p$, Let $\mathbb{F}_p=\{0,1,\cdots, p-1\}$ be the finite field of order $p$. For $x\in \mathbb{F}_p$, define $f_p(x)$ to be the maximum element in the set $\{ ...
7
votes
1answer
279 views

Reinterpreting Galois descent over finite fields

This question is indirectly related to my previous question Is an elliptic curve that is isomorphic to its Frobenius conjugate defined over $\mathbb{F}_p$? Let $\mathbb{F}_{q^n}/\mathbb{F}_q$ be an ...
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0answers
119 views

A $k \times n$ matrix with a lot of invertible $k \times k$ submatrices over $\mathbb{F}_2$

In the appendix of the paper by Tolhuizen ( http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=841182) there is a very fast and easy probabilistic proof that for $k=cn$ where $c \in [0,1]$ is fixed, ...
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A question on vector space over finite field

Let $\mathbb{F}_{2^\sigma}$ be a finite field of $\sigma$-bit elements, and use $\mathbb{F}_{2^\sigma}^{\ell}$ to denote an $\ell$-dimensional vector space over $\mathbb{F}_{2^\sigma}$. Let $V$ be a ...
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Order of zeros for sparse polynomials mod $p$

It is a fairly well known fact that sparse polynomials $f(x)$ cannot have large order zeros other than at $x=0$. If $f(x)=a_1x^{r_1}+\cdots+a_kx^{r_k}$ then at $c \neq 0$, $f$ has a zero of order at ...
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0answers
38 views

Computations of some character sums/zeta function

I'm currently trying to "compute" the zeta function of some hyperelliptic curves over a finite field (of odd characteristic). Precisely, let $b\in\mathbb{F}_q^\times$ be a non zero element (I also ...
3
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3answers
264 views

Minimal expression of 0 as a sum of kth powers in a finite field

Let $l=\min\{s\in \mathbb{N}|0\in s\cdot (\mathbb{F}_{p^n}^\times)^k\}$. Is any information known about this number already as a function of $k$? Any reference would be greatly appreciated!
9
votes
1answer
252 views

Are $E_n$-operads not formal in characteristic not equal to zero?

This is a short question: Is it just unproven folklore (yet), or is it definitively known that $E_n$-operads are not formal, if the characteristic of the underlying field is not equal to zero?
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A question about polynomial over finite field

For the positive integer $m$, let $n=\varphi(m)$ denote the totient of $m$. Given a rational prime $p$ such that $p\equiv 1 \bmod m$, let $A=\{a_1,\cdots, a_n\}\subseteq \mathbb{F}_p$ denote the set ...
5
votes
1answer
170 views

When is Chevalley Warning's bound best possible?

Chevalley Warning's theorem (a form of) states that any homogeneous form over a finite field of degree $d$ in more than $d$ variables has a nontrivial zero in the field. However, for diagonal forms, ...
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276 views

Counting abelian varieties over finite fields in a given isogeny class

Let $f(x) \in \mathbb Z[x]$ be a monic polynomial of degree $g$ with all roots having absolute value $\sqrt{q}$. How many principally polarized abelian varieties over $\mathbb F_q$ have $f(x)$ as the ...
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About the rank of a Pell equation-related matrix

I have a question about the solution of Pell-equation over a prime field. I want prove that the matrix $M$ is of rank $\frac{p-1}{2}$, with $M=(m_{i,j})\in\left(\mathbb{Z}/(p^p-1)\mathbb{Z} ...