Elliptic curves and explicit enumeration of irreducible polynomials with two coefficients prescribed

Irreducible polynomials with several prescribed coefficients

We study the number of irreducible polynomials over $\mathbf{F}_{q}$ with some coefficients prescribed. Using the technique developed by Bourgain, we show that there is an irreducible polynomial of degree $n$ with $r$ coefficients prescribed in any location when $r \leq \left[\left(1/4 - ε\right)n \right]$ for any $ε>0$ and $q$ is large; and when $r\leqδn$ for some $δ>0$ and for any $q$. The result is improved from the earlier work of Pollack that the similar result holds for $r\leq\left[(1-ε)\sqrt{n}\right]$.

Enumeration of a special class of irreducible polynomials in characteristic 2

A-polynomials were introduced by Meyn and play an important role in the iterative construction of high degree self-reciprocal irreducible polynomials over the field F_2, since they constitute the starting point of the iteration. The exact number of A-polynomials of each degree was given by Niederreiter. Kyuregyan extended the construction of Meyn to arbitrary even finite fields. We relate the A-polynomials in this more general setting to inert places in a certain extension of elliptic function fields and obtain an explicit counting formula for their number. In particular, we are able to show t…

Irreducible polynomials with prescribed trace and restricted norm

Let GF(q), q=p^r, be a finite field with a primitive element g. In this paper we use exponential sums and Jacobi sums to compute the number of the irreducible polynomials of degree m over GF(q) with trace fixed and norm restricted to a coset of a subgroup <g^s>, s|(q-1). We give the number explicitly for s=2, 3, 4 when q=p, and for s|(p^e+1) when r=2en. Finally, we give explicit formulae for the number when both trace and norm are fixed, p=2 and m =< 30.

Irreducible polynomials over $\mathbb{F}_{2^r}$ with three prescribed coefficients

For any positive integers $n \ge 3$ and $r \ge 1$, we prove that the number of monic irreducible polynomials of degree $n$ over $\mathbb{F}_{2^r}$ in which the coefficients of $T^{n-1}$, $T^{n-2}$ and $T^{n-3}$ are prescribed has period $24$ as a function of $n$, after a suitable normalization. A similar result holds over $\mathbb{F}_{5^r}$, with the period being $60$. We also show that this is a phenomena unique to characteristics $2$ and $5$. The result is strongly related to the supersingularity of certain curves associated with cyclotomic function fields, and in particular it complements a…

Sequences of irreducible polynomials without prescribed coefficients over finite fields of even characteristic

In this paper we deal with the construction of sequences of irreducible polynomials with coefficients in finite fields of even characteristic. We rely upon a transformation used by Kyuregyan in 2002, which generalizes the $Q$-transform employed previously by Varshamov and Meyn for the synthesis of irreducible polynomials. While in the iterative procedure described by Kyuregyan the coefficients of the initial polynomial of the sequence have to satisfy certain hypotheses, in the present paper we construct infinite sequences of irreducible polynomials of non-decreasing degree starting from any ir…

有限域中间扩张中具有给定迹的范数与原始范数元素

Normal and primitive normal elements with prescribed traces in intermediate extensions of finite fields

本文研究有限域中间扩张中具有给定迹的范数及原始范数元素，完整刻画了范数情形下的存在条件并给出显式计数，对原始范数情形提供了渐近结果。

Sequences of irreducible polynomials without prescribed coefficients over odd prime fields

In this paper we construct infinite sequences of monic irreducible polynomials with coefficients in odd prime fields by means of a transformation introduced by Cohen in 1992. We make no assumptions on the coefficients of the first polynomial $f_0$ of the sequence, which belongs to $\F_p [x]$, for some odd prime $p$, and has positive degree $n$. If $p^{2n}-1 = 2^{e_1} \cdot m$ for some odd integer $m$ and non-negative integer $e_1$, then, after an initial segment $f_0, ..., f_s$ with $s \leq e_1$, the degree of the polynomial $f_{i+1}$ is twice the degree of $f_i$ for any $i \geq s$.

On the Enumeration of Irreducible Polynomials over $\text{GF}(q)$ with Prescribed Coefficients

We present an efficient deterministic algorithm which outputs exact expressions in terms of $n$ for the number of monic degree $n$ irreducible polynomials over $\mathbb{F}_{q}$ of characteristic $p$ for which the first $l < p$ coefficients are prescribed, provided that $n$ is coprime to $p$. Each of these counts is $\frac{1}{n}(q^{n-l} + \mathcal{O}(q^{n/2}))$. The main idea behind the algorithm is to associate to an equivalent problem a set of Artin-Schreier curves defined over $\mathbb{F}_q$ whose number of $\mathbb{F}_{q^n}$-rational affine points must be combined. This is accomplished by c…

Irreducible polynomials with prescribed sums of coefficients

Let $q$ be a power of a prime, let $\mathbb{F}_q$ be the finite field with $q$ elements and let $n \geq 2$. For a polynomial $h(x) \in \mathbb{F}_q[x]$ of degree $n \in \mathbb{N}$ and a subset $W \subseteq [0,n] := \{0, 1, \ldots, n\}$, we define the sum-of-digits function $$S_W(h) = \sum_{w \in W}[x^{w}] h(x)$$ to be the sum of all the coefficients of $x^w$ in $h(x)$ with $w \in W$. In the case when $q = 2$, we prove, except for a few genuine exceptions, that for any $c \in \mathbb{F}_2$ and any $W \subseteq [0,n]$ there exists an irreducible polynomial $P(x)$ of degree $n$ over $\mathbb{F}_…

二维表示的规定提升

Prescribed lifts of 2-dimensional representations

在标准Taylor-Wiles假设下，证明全实域F的绝对Galois群的不可约二维全奇模p表示可以提升到局部形变环的任意指定分支上，允许具有任意正则Hodge-Tate权的潜在半稳定条件。

关于Reed-Muller码的距离分布

On the Distance Distribution of Reed-Muller Codes

本文通过特征和方法，首次系统研究了Reed-Muller码在大有限域上的余集权分布问题，并扩展了Reed-Solomon码的距离分布研究。

Existence of Primitive Pairs with Prescribed Traces over Finite Fields

Let $F=\mathbb{F}_{q^m}$, $m>6$, $n$ a positive integer, and $f=p/q$ with $p$, $q$ co-prime irreducible polynomials in $F[x]$ and deg$(p)$ $+$ deg$(q)= n$. A sufficient condition has been obtained for the existence of primitive pairs $(α, f(α))$ in $F$ such that for any prescribed $a, b$ in $E=\mathbb{F}_q$, Tr$F/E (α) = a$ and Tr$F/E (α^{-1}) = b$. Further, for every positive integer $n$, such a pair definitely exists for large enough $(q,m)$. The case $n = 2$ is dealt separately and proved that such a pair exists for all $(q,m)$ apart from at most $64$ choices.

Character sums over products of prime polynomials

We study sums of Dirichlet characters over polynomials in $\mathbb{F}_q[t]$ with a prescribed number of irreducible factors. Our main results are explicit formulae for these sums in terms of zeros of Dirichlet L-functions. We also exhibit new phenomena concerning Chebyshev-type biases of such sums when the number of irreducible factors is very large.

具有多个指定系数的光滑多项式

Smooth polynomials with several prescribed coefficients

本文研究了具有预设系数的m-光滑多项式分布问题，采用特征和估计、Ha对多项式应用的Bourgain论证以及光滑多项式上的双重特征和来分析。

Abelian varieties over $\mathbb{F}_2$ of prescribed order

We prove that for every positive integer $m$, there exist infinitely many simple abelian varieties over $\mathbb{F}_2$ of order $m$. The method is constructive, building on the work of Madan--Pal in the case $m=1$ to produce an explicit sequence of Weil polynomials giving rise to abelian varieties over $\mathbb{F}_2$ of order $m$. This sequence itself depends on the choice of a suitable generalized binary representation of $m$; by making careful choices of this representation, we can ensure that the the resulting sequence of polynomials have 2-adic Newton polygons which guarantee the existence…

Existence of primitive pairs with two prescribed traces over finite fields

Given $F= \mathbb{F}_{p^{t}}$, a field with $p^t$ elements, where $p $ is a prime power, $t\geq 7$, $n$ are positive integers and $f=f_1/f_2$ is a rational function, where $f_1, f_2$ are relatively prime, irreducible polynomials with $deg(f_1) + deg(f_2) = n $ in $F[x]$. We construct a sufficient condition on $(p,t)$ which guarantees primitive pairing $(ε, f(ε))$ exists in $F$ such that $Tr_{\mathbb{F}_{p^t}/\mathbb{F}_p}(ε) = a$ and $Tr_{\mathbb{F}_{p^t}/\mathbb{F}_p}(f(ε)) = b$ for any prescribed $a,b \in \mathbb{F}_{p}$. Further, we demonstrate for any positive integer $n$, such a pair defi…

计数具有系数来自薄子群的不可约多项式

Counting Irreducible polynomials with coefficients from thin subgroups

本文提出了一种新的方法，用于计数具有系数来自有限域F_q小子群的不可约多项式，并扩展了该方法以考虑判别式的预设条件。

基 $b(T)$ 下数字受限的不可约多项式

Irreducible polynomials with restricted digits in base $b(T)$

本文研究有限域上多项式环中数字受限的不可约多项式计数，利用傅里叶分析得到渐近公式，并推广圆法至任意多项式基。

Single-factor lifting and factorization of polynomials over local fields

Let $f(x)$ be a separable polynomial over a local field. Montes algorithm computes certain approximations to the different irreducible factors of $f(x)$, with strong arithmetic properties. In this paper we develop an algorithm to improve any one of these approximations, till a prescribed precision is attained. The most natural application of this "single-factor lifting" routine is to combine it with Montes algorithm to provide a fast polynomial factorization algorithm. Moreover, the single-factor lifting algorithm may be applied as well to accelerate the computational resolution of several glo…

A new proof of the Hansen-Mullen irreducibility conjecture

We give a new proof of the Hansen-Mullen irreducibility conjecture. The proof relies on an application of a (seemingly new) sufficient condition for the existence of elements of degree $n$ in the support of functions on finite fields. This connection to irreducible polynomials is made via the least period of the discrete Fourier transform (DFT) of functions with values in finite fields. We exploit this relation and prove, in an elementary fashion, that a relevant function related to the DFT of characteristic elementary symmetric functions (which produce the coefficients of characteristic polyn…

具有给定范数和迹的元素个数

On the number of elements with prescribed norm and trace

本文研究有限域扩张中范数和迹均为给定值的元素个数，通过Artin-Schreier曲线上的有理点计数改进估计，并应用于环面Calabi-Yau簇及不可约多项式计数。

序的唯一分解

Unique decomposition of orders

建立序的基本定理，将数域中的任意序唯一表示为不可约序的交集，并利用该分解推广Furtwängler准则。

具有指定分解长度的全局域离散赋值环上的整值多项式

Integer-valued polynomials on discrete valuation rings of global fields with prescribed lengths of factorizations

该研究针对全局域赋值环上整值多项式的分解问题，证明对指定的分解长度序列，存在对应整值多项式，并将结果推广到满足条件的离散赋值环，解决了相关公开问题。