Irreducible polynomials over F2r 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 an equidistribution result of Katz.     

Sequences of binary irreducible polynomials

In this paper we construct an infinite sequence of binary irreducible polynomials starting from any irreducible polynomial $f_0\in F_2\left[x\right]$. If $f_0$ is of degree $n=2^l?m$, where $m$ is odd and $l$ is a nonnegative integer, after an initial finite sequence of polynomials $f_0,f_1,\dots ,f_s$, with $s\le l+3$, the degree of $f_{i+1}$ is twice the degree of $f_i$ for any $i\ge s$.     

On the enumeration of polynomials with prescribed factorization pattern

We use generating functions over group rings to count polynomials over finite fields with the first few coefficients and a factorization pattern prescribed. In particular, we obtain different exact formulas for the number of monic n-smooth polynomials of degree m over a finite field, as well as the number of monic n-smooth polynomials of degree m with the prescribed trace coefficient.     

Haglund's conjecture for multi-t Macdonald polynomials

We provide new approaches to prove identities for the modified Macdonald polynomials via their LLT expansions. As an application, we prove a conjecture of Haglund concerning the multi-t-Macdonald polynomials of two rows.     

The Bateman–Horn conjecture: Heuristic,history, and applications

The Bateman–Horn conjecture is a far-reaching statement about the distribution of the prime numbers. It implies many known results, such as the prime number theorem and the Green–Tao theorem, along with many famous conjectures, such the twin prime conjecture and Landau’s conjecture. We discuss the Bateman–Horn conjecture, its applications, and its origins.     

Irreducibility of Hypersurfaces

Given a polynomial P in several variables over an algebraically closed field, we show that except in some special cases that we fully describe, if one coefficient is allowed to vary, then the polynomial is irreducible for all but at most deg(P)² ? 1 values of the coefficient. We more generally handle the situation where several specified coefficients vary.     

Groups of prime degree and the Bateman–Horn Conjecture

As a consequence of the classification of finite simple groups, the classification of permutation groups of prime degree is complete, apart from the question of when the natural degree $(q^n-1)/(q-1)$ of ${\mathrm{PSL}}_n\left(q\right)$ is prime. We present heuristic arguments and computational evidence based on the Bateman–Horn Conjecture to support a conjecture that for each prime $n\ge 3$ there are infinitely many primes of this form, even if one restricts to prime values of $q$. Similar arguments and results apply to the parameters of the simple groups ${\mathrm{PSL}}_n\left(q\right)$, ${\mathrm{PSU}}_n\left(q\right)$ and ${\mathrm{PSp}}_{2n}\left(q\right)$ which arise in the work of Dixon and Zalesskii on linear groups of prime degree.     

A refined Gallai-Edmonds structure theorem for weighted matching polynomials

In this work, we prove a refinement of the Gallai-Edmonds structure theorem for weighted matching polynomials by Ku and Wong. Our proof uses a connection between matching polynomials and branched continued fractions. We also show how this is related to a modification by Sylvester of the classical Sturm's theorem on the number of zeros of a real polynomial in an interval. In addition, we obtain some other results about zeros of matching polynomials.     

PSC Galois Extensions of Hilbertian Fields

We prove the following result: Theorem. Let K be a countable Hilbertian field, S a finite set of local primes of K, and e ≥ 0 an integer. Then, for almost all σ ∈ G (K)^e, the field K_s [ σ ] ∩ K_tot,S is PSC. Here a local prime is an equivalent class 𝔭 of absolute values of K whose completion is a local field, _𝔭. Then K_𝔭 = K_s ∩ _𝔭 and K_tot,S = ∩_{𝔭 ∈ S}∩_{σ ∈ G(K)} K^σ_𝔭. G(K) stands for the absolute Galois group of K. For each σ = (σ₁, …, σ_e ) ∈ G(K)^e we denote the fixed field of σ₁, …, σ_e in K_s by K_s( σ ). The maximal Galois extension of K in K_s( σ ) is K_s[ σ ]. Finally “almost all” means “for all but a set of Haar measure zero”.     

Fibre products of supersingular curves and the enumeration of irreducible polynomials with prescribed coefficients

For any positive integers $n\ge 3$, $r\ge 1$ we present formulae for the number of irreducible polynomials of degree n over the finite field ${\mathbb{F}}_{2^r}$ where the coefficients of $x^{n-1}$, $x^{n-2}$ and $x^{n-3}$ are zero. Our proofs involve counting the number of points on certain algebraic curves over finite fields, a technique which arose from Fourier-analysing the known formulae for the ${\mathbb{F}}_2$ base field cases, reverse-engineering an economical new proof and then extending it. This approach gives rise to fibre products of supersingular curves and makes explicit why the formulae have period 24 in n.     

Values of coefficients of cyclotomic polynomials

Let a(k,n) be the k-th coefficient of the n-th cyclotomic polynomials. In 1987, J. Suzuki proved that . In this paper, we improve this result and prove that for any prime p and any integer l≥1, we have

{a(k,p^ln)∣n,k∈N}=Z.     

Values of coefficients of cyclotomic polynomials II

Let a(n,k) be the kth coefficient of the nth cyclotomic polynomial. In part I it was proved that , in the case when m is a prime power. In this paper we show that the result also holds true in the case when m is an arbitrary positive integer.