arXiv subjects

Lenny Jones

Publications and source records attributed to Lenny Jones.

At least 37 records · Page 2Linked to original sources

arXiv2024

Monogenic Even Octic Polynomials and Their Galois Groups

A monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree $N$ is called monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,θ,θ^2,\ldots ,θ^{N-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(θ)$, where $f(θ)=0$. In a series of recent articles, complete classifications of the Galois groups were given for irreducible polynomials \[{\mathcal F}(x):=x^8+ax^4+b\in {\mathbb Z}[x]\] and \[{\mathcal G}(x):=x^8+ax^6+bx^4+ax^2+1\in {\mathbb Z}[x], \quad a\ne 0.\] In this article, for each Galois group $G$ arising in these classifications, we either construct an infinite family of monogenic octic polynomials ${\mathcal F}(x)$ or ${\mathcal G}(x)$ having Galois group $G$, or we prove that at most a finite such family exists. In the finite family situations, we determine all such polynomials. Here, a ``family" means that no two polynomials in the family generate isomorphic octic fields.

math.NT↗

arXiv2024

Monogenic Cyclic Quartic Trinomials

A monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree $N$ is called monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,θ,θ^2,\ldots ,θ^{N-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(θ)$, where $f(θ)=0$. In this brief note, we prove that there exist exactly three distinct monogenic trinomials of the form $x^4+bx^2+d$ whose Galois group is the cyclic group of order 4.

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arXiv2024

Monogenic Quartic Polynomials and Their Galois Groups

A monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree $N$ is called monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,θ,θ^2,\ldots ,θ^{N-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(θ)$, where $f(θ)=0$. In this article, we use the classification of the Galois groups of quartic polynomials, due to Kappe and Warren, to investigate the existence of infinite collections of monogenic quartic polynomials having a prescribed Galois group, such that each member of the collection generates a distinct quartic field. With the exception of the cyclic case, we provide such an infinite single-parameter collection for each possible Galois group. We believe these examples are new, and we provide evidence to support this belief by showing that they are distinct from other infinite collections in the current literature. Finally, we devote a separate section to a discussion concerning, what we believe to be, the still-unresolved cyclic case.

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arXiv2023

A new condition for $k$-Wall-Sun-Sun primes

Let $k\ge 1$ be an integer, and let $(U_n)$ be the Lucas sequence of the first kind defined by \begin{equation*}\label{Eq:Lucas} U_0=0,\quad U_1=1\quad \mbox{and} \quad U_n=kU_{n-1}+U_{n-2} \quad \mbox{ for $n\ge 2$}. \end{equation*} It is well known that $(U_n)$ is periodic modulo any integer $m\ge 2$, and we let $π(m)$ denote the length of this period. A prime $p$ is called a $k$-Wall-Sun-Sun prime if $π(p^2)=π(p)$. Let $f(x)\in {\mathbb Z}[x]$ be a monic polynomial of degree $N$ that is irreducible over ${\mathbb Q}$. We say $f(x)$ is monogenic if $Θ=\{1,θ,θ^2,\ldots ,θ^{N-1}\}$ is a basis for the ring of integers ${\mathbb Z}_K$ of ${\mathbb Q}(θ)$, where $f(θ)=0$. If $Θ$ is not a basis for ${\mathbb Z}_K$, we say that $f(x)$ is non-monogenic. Define ${\mathcal D}:=k^2+4$ if $k\equiv 1 \pmod{2}$, and ${\mathcal D}:=(k/2)^2+1$ if $k\equiv 0 \pmod{2}$. Suppose that $k\not \equiv 0 \pmod{4}$ and that ${\mathcal D}$ is squarefree. In this article, we prove that $p$ is a $k$-Wall-Sun-Sun prime if and only if ${\mathcal F}_p(x)=x^{2p}-kx^p-1$ is non-monogenic. This result, combined with previous work, shows that ${\mathcal F}_p(x)$ is monogenic if $p$ is a prime divisor of $k^2+4$.

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arXiv2023

On the monogenicity of power-compositional Shanks polynomials

Let $f(x)\in {\mathbb Z}[x]$ be a monic polynomial of degree $N$ that is irreducible over ${\mathbb Q}$. We say $f(x)$ is \emph{monogenic} if $Θ=\{1,θ,θ^2,\ldots ,θ^{N-1}\}$ is a basis for the ring of integers ${\mathbb Z}_K$ of $K={\mathbb Q}(θ)$, where $f(θ)=0$. If $Θ$ is not a basis for ${\mathbb Z}_K$, we say that $f(x)$ is \emph{non-monogenic}. Let $k\ge 1$ be an integer, and let $(U_n)$ be the sequence defined by \[U_0=U_1=0,\quad U_2=1 \quad \mbox{and}\quad U_n=kU_{n-1}+(k+3)U_{n-2}+U_{n-3} \quad \mbox{for $n\ge 3$}.\] It is well known that $(U_n)$ is periodic modulo any integer $m\ge 2$, and we let $π(m)$ denote the length of this period. We define a \emph{$k$-Shanks prime} to be a prime $p$ such that $π(p^2)=π(p)$. Let ${\mathcal S}_k(x)=x^{3}-kx^{2}-(k+3)x-1$. Let ${\mathcal D}=(k/3)^2+k/3+1$ if $k\equiv 0 \pmod{3}$, and ${\mathcal D}=k^2+3k+9$ otherwise. Suppose that $k\not \equiv 3 \pmod{9}$ and that ${\mathcal D}$ is squarefree. In this article, we prove that $p$ is a $k$-Shanks prime if and only if ${\mathcal S}_k(x^p)$ is non-monogenic, for any prime $p$ such that ${\mathcal S}_k(x)$ is irreducible in ${\mathbb F}_p[x]$. Furthermore, we show that ${\mathcal S}_k(x^p)$ is monogenic for any prime divisor $p$ of $k^2+3k+9$. These results extend previous work of the author on $k$-Wall-Sun-Sun primes.

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arXiv2023

Generalized Wall-Sun-Sun primes and monogenic power compositional trinomials

For positive integers $a$ and $b$, we let $[U_n]$ be the Lucas sequence of the first kind defined by \[U_0=0,\quad U_1=1\quad \mbox{and} \quad U_n=aU_{n-1}+bU_{n-2} \quad \mbox{ for $n\ge 2$},\] and let $π(m):=π_{(a,b)}(m)$ be the period length of $[U_n]$ modulo the integer $m\ge 2$, where $\gcd(b,m)=1$. We define an \emph{$(a,b)$-Wall-Sun-Sun prime} to be a prime $p$ such that $π(p^2)=π(p)$. When $(a,b)=(1,1)$, such a prime $p$ is referred to simply as a \emph{Wall-Sun-Sun prime}. We say that a monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree $N$ is \emph{monogenic} if $f(x)$ is irreducible over ${\mathbb Q}$ and \[\{1,θ,θ^2,\ldots, θ^{N-1}\}\] is a basis for the ring of integers of ${\mathbb Q}(θ)$, where $f(θ)=0$. Let $f(x)=x^2-ax-b$, and let $s$ be a positive integer. Then, with certain restrictions on $a$, $b$ and $s$, we prove that the monogenicity of \[f(x^{s^n})=x^{2s^n}-ax^{s^n}-b\] is independent of the positive integer $n$ and is determined solely by whether $s$ has a prime divisor that is an $(a,b)$-Wall-Sun-Sun prime. This result improves and extends previous work of the author in the special case $b=1$.

math.NT↗

arXiv2022

A Connection Between the Monogenicity of Certain Power-Compositional Trinomials and $k$-Wall-Sun-Sun Primes

We say that a monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree $N$ is monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and \[\{1,θ,θ^2,\ldots, θ^{N-1}\}\] is a basis for the ring of integers of ${\mathbb Q}(θ)$, where $f(θ)=0$. Let $k$ be a positive integer, and let $U_n:=U_n(k,-1)$ be the Lucas sequence $\{U_n\}_{n\ge 0}$ of the first kind defined by \[U_0=0,\quad U_1=1\quad \mbox{and} \quad U_n=kU_{n-1}+U_{n-2} \quad \mbox{ for $n\ge 2$}.\] A $k$-Wall-Sun-Sun prime is a prime $p$ such that \[U_{π_k(p)}\equiv 0 \pmod{p^2},\] where $π_k(p)$ is the length of the period of $\{U_n\}_{n\ge 0}$ modulo $p$. Let ${\mathcal D}=k^2+4$ if $k\equiv 1 \pmod{2}$, and ${\mathcal D}=(k/2)^2+1$ if $k\equiv 0 \pmod{2}$. Suppose that $k\not \equiv 0 \pmod{4}$ and ${\mathcal D}$ is squarefree, and let $h$ denote the class number of ${\mathbb Q}(\sqrt{\mathcal D})$. Let $s\ge 1$ be an integer such that, for every odd prime divisor $p$ of $s$, ${\mathcal D}$ is not a square modulo $p$ and $\gcd(p,h{\mathcal D})=1$. In this article, we prove that $x^{2s^n}-kx^{s^n}-1$ is monogenic for all integers $n\ge 1$ if and only if no prime divisor of $s$ is a $k$-Wall-Sun-Sun prime.

math.NT↗

arXiv2022

The Irreducibility and Monogenicity of Power-Compositional Trinomials

A polynomial $f(x)\in {\mathbb Z}[x]$ of degree $N$ is called \emph{monogenic} if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,θ,θ^2,\ldots ,θ^{N-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(θ)$, where $f(θ)=0$. Define ${\mathcal F}(x):=x^m+Ax^{m-1}+B$. In this article, we determine sets of conditions on $m$, $A$, and $B$, such that the power-compositional trinomial ${\mathcal F}(x^{p^n})$ is monogenic for all integers $n\ge 0$ and a given prime $p$. Furthermore, we prove the actual existence of infinite families of such trinomials ${\mathcal F}(x)$.

math.NT↗

arXiv2021

Monogenic trinomials with non-squarefree discriminant

For each integer $n\ge 2$, we identify new infinite families of monogenic trinomials $f(x)=x^n+Ax^m+B$ with non-squarefree discriminant, many of which have small Galois group. These families are thus different from many previous examinations of specific trinomial forms in the literature. Moreover, in certain situations when $A=B\ge 2$ with fixed $n$ and $m$, we produce asymptotics on the number of such trinomials with $A\le X$.

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arXiv2020

Characteristic Polynomials of Simple Ordinary Abelian Varieties over Finite Fields

We provide an easy method for the construction of characteristic polynomials of simple ordinary abelian varieties ${\mathcal A}$ of dimension $g$ over a finite field ${\mathbb F}_q$, when $q\ge 4$ and $2g=ρ^{b-1}(ρ-1)$ for some prime $ρ\ge 5$, with $b\ge 1$. Moreover, we show that ${\mathcal A}$ is absolutely simple if $b=1$ and $ρ=5$, but ${\mathcal A}$ is not absolutely simple for arbitrary $ρ$ with $b>1$.

math.NT↗

arXiv2019

Monogenic cyclotomic compositions

Let $m$ and $n$ be positive integers, and let $p$ be a prime. Let $T(x)=Φ_{p^m}\left(Φ_{2^n}(x)\right)$, where $Φ_k(x)$ is the cyclotomic polynomial of index $k$. In this article, we prove that $T(x)$ is irreducible over $\mathbb Q$ and that \[\left\{1,θ,θ^2,\ldots,θ^{2^{n-1}p^{m-1}(p-1)-1}\right\}\] is a basis for the ring of integers of $\mathbb Q(θ)$, where $T(θ)=0$.

math.NT↗

arXiv2014

On Primitive Covering Numbers

In 2007, Zhi-Wei Sun defined a \emph{covering number} to be a positive integer $L$ such that there exists a covering system of the integers where the moduli are distinct divisors of $L$ greater than 1. A covering number $L$ is called \emph{primitive} if no proper divisor of $L$ is a covering number. Sun constructed an infinite set $\mathcal L$ of primitive covering numbers, and he conjectured that every primitive covering number must satisfy a certain condition. In this paper, for a given $L\in \mathcal L$, we derive a formula that gives the exact number of coverings that have $L$ as the least common multiple of the set $M$ of moduli, under certain restrictions on $M$. Additionally, we disprove Sun's conjecture by constructing an infinite set of primitive covering numbers that do not satisfy his primitive covering number condition.

math.NT↗

arXiv2014

Representing Integers as the Sum of Two Squares in the Ring $\Z_n$

A classical theorem in number theory due to Euler states that a positive integer $z$ can be written as the sum of two squares if and only if all prime factors $q$ of $z$, with $q\equiv 3 \pmod{4}$, have even exponent in the prime factorization of $z$. One can consider a minor variation of this theorem by not allowing the use of zero as a summand in the representation of $z$ as the sum of two squares. Viewing each of these questions in $\Z_n$, the ring of integers modulo $n$, we give a characterization of all integers $n\ge 2$ such that every $z\in \Z_n$ can be written as the sum of two squares in $\Z_n$.

math.NT↗

arXiv2012

Maximal Denumerant of a Numerical Semigroup With Embedding Dimension Less Than Four

Given a numerical semigroup $S = < a_1, a_2,..., a_t>$ and $s\in S$, we consider the factorization $s = c_1 a_1 + c_2 a_2 +... + c_t a_t$ where $c_i\ge0$. Such a factorization is {\em maximal} if $c_1+c_2+...+c_t$ is a maximum over all such factorizations of $s$. We show that the number of maximal factorizations, varying over the elements in $S$, is always bounded. Thus, we define $\dx(S)$ to be the maximum number of maximal factorizations of elements in $S$. We study maximal factorizations in depth when $S$ has embedding dimension less than four, and establish formulas for $\dx(S)$ in this case.

math.AC↗

arXiv2012

Multiplicative Bases for the Centres of the Group Algebra and Iwahori-Hecke Algebra of the Symmetric Group

Let $\H_n$ be the Iwahori-Hecke algebra of the symmetric group $S_n$, and let $Z(\H_n)$ denote its centre. Let $B={b_1,b_2,...,b_t}$ be a basis for $Z(\H_n)$ over $R=\Z[q,q^{-1}]$. Then $B$ is called \emph{multiplicative} if, for every $i$ and $j$, there exists $k$ such that $b_ib_j= b_k$. In this article we prove that there are no multiplicative bases for $Z(\Z S_n)$ and $Z(\H_n)$ when $n\ge 3$. In addition, we prove that there exist exactly two multiplicative bases for $Z(\Z S_2)$ and none for $Z(\H_2)$.

math.RT↗

arXiv2012

Enumerating Invariant Subspaces of ${\mathbb R}^n$

In this article, we develop an algorithm to calculate the set of all integers $m$ for which there exists a linear operator $T$ on ${\mathbb R}^n$ such that ${\mathbb R}^n$ has exactly $m$ $T$-invariant subspaces. A brief discussion is included as how these methods might be extended to vector spaces over arbitrary fields.

math.RA↗

arXiv2011

Using Lucas Sequences to Generalize a Theorem of Sierpiński

In 1960, Sierpiński proved that there exist infinitely many odd positive integers $k$ such that $k\cdot 2^n+1$ is composite for all positive integers $n$. In this paper, we prove some generalizations of Sierpiński's theorem with $2^n$ replaced by expressions involving certain Lucas sequences $U_n(α,β)$. In particular, we show the existence of infinitely many Lucas pairs $(α,β)$, for which there exist infinitely many positive integers $k$, such that $k (U_n(α,β)+(α-β)^2)+1$ is composite for all integers $n\ge 1$. Sierpiński's theorem is the special case of $α=2$ and $β=1$. Finally, we establish a nonlinear version of this result by showing that there exist infinitely many rational integers $α>1$, for which there exist infinitely many positive integers $k$, such that $k^2 (U_n(α,1)+(α-1)^2)+1$ is composite for all integers $n\ge 1$.

math.NT↗

arXiv2011

Polynomial Cunningham Chains

Let $ε\in \{-1,1\}$. A sequence of prime numbers $p_1, p_2, p_3, ...$, such that $p_i=2p_{i-1}+ε$ for all $i$, is called a {\it Cunningham chain} of the first or second kind, depending on whether $ε=1$ or -1 respectively. If $k$ is the smallest positive integer such that $2p_k+ε$ is composite, then we say the chain has length $k$. Although such chains are necessarily finite, it is conjectured that for every positive integer $k$, there are infinitely many Cunningham chains of length $k$. A sequence of polynomials $f_1(x), f_2(x), ...$, such that $f_i(x)\in \Z[x]$, $f_1(x)$ has positive leading coefficient, $f_i(x)$ is irreducible in $\Q[x]$, and $f_i(x)=xf_{i-1}(x)+ε$ for all $i$, is defined to be a {\it polynomial Cunningham chain} of the first or second kind, depending on whether $ε=1$ or -1 respectively. If $k$ is the least positive integer such that $f_{k+1}(x)$ is reducible over $\Q$, then we say the chain has length $k$. In this article, for chains of each kind, we explicitly give infinitely many polynomials $f_1(x)$, such that $f_{k+1}(x)$ is the only term in the sequence $\{f_i(x)\}_{i=1}^{\infty}$ that is reducible. As a first corollary, we deduce that there exist infinitely many polynomial Cunningham chains of length $k$ of both kinds, and as a second corollary, we have that, unlike the situation in the integers, there exist infinitely many polynomial Cunningham chains of infinite length of both kinds.

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