# Tagged Questions

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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### 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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### 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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### When does composing polynomials reduce the degree?

Let $\mathbb{F}$ be the field of size 2. For a function $f : \mathbb{F}^n \to \mathbb{F}$, let $d(f)$ be the smallest integer such that there exists a degree-$d(f)$, $n$-variate, multilinear ...
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### What is the function like when its Mobius inversion is $\sum_{w|r, (w,t)=1}\mu(w)q^{r/w}$?

Everyone, I am now reading a paper named The Irreducible Factors of $(cx+d)x^{q^m}-(ax+b)$ over $GF(q)$, http://qjmath.oxfordjournals.org/content/14/1/61.extract. And I’m confused with one of its ...
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### 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 $C=\{x\}$...
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### 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 $t < p$?
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### 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 ...
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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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### 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 n=n_0+n_1q+\cdots+...
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### 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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### 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 $V$ ...
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 $\#X(\mathbb F_q)\... 0answers 124 views ### A question on vector space over finite field Let$\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$be a ... 0answers 39 views ### Computations of some character sums/zeta function I'm currently trying to "compute" the zeta function of some hyperelliptic curves over a finite field (of odd characteristic). Precisely, let$b\in\mathbb{F}_q^\times$be a non zero element (I also ... 0answers 81 views ### A question about polynomial over finite field For the positive integer$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$denote the set ... 0answers 84 views ### The significance of the Parvaresh-Vardy curve Even though this seems like a computer science question, it is purely mathematical and concerns polynomials and curves over finite fields. Consider the Parvaresh-Vardy list decoder. As I understand ... 0answers 146 views ### Number of common solutions of polynomial system Let$ \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 = 1,......
I want to solve the following question: Consider a polynomial $f(x)=a_0+a_1*x^{e_1}+a_2*x^{e_2}+\cdots+x^{e_m}\in F_p[x]$ where $p$ is a prime such that $\log(p)\sim m$ and $e_m\sim 2^m$, I want to ...