Non-Beiter ternary cyclotomic polynomials with an optimally large set of coefficients

Constrained ternary integers

An integer $n$ is said to be ternary if it is composed of three distinct odd primes. In this paper, we asymptotically count the number of ternary integers $n \leq x$ with the constituent primes satisfying various constraints. We apply our results to the study of the simplest class of (inverse) cyclotomic polynomials that can have coefficients that are greater than 1 in absolute value, namely to the $n^{th}$ (inverse) cyclotomic polynomials with ternary $n$. We show, for example, that the corrected Sister Beiter conjecture is true for a fraction $\ge 0.925$ of ternary integers.

Neighboring ternary cyclotomic coefficients differ by at most one

A cyclotomic polynomial Phi_n(x) is said to be ternary if n=pqr with p,q and r distinct odd prime factors. Ternary cyclotomic polynomials are the simplest ones for which the behaviour of the coefficients is not completely understood. Eli Leher showed in 2007 that neighboring ternary cyclotomic coefficients differ by at most four. We show that, in fact, they differ by at most one. Consequently, the set of coefficients occurring in a ternary cyclotomic polynomial consists of consecutive integers. As an application we reprove in a simpler way a result of Bachman from 2004 on ternary cyclotomic po…

Bounds on ternary cyclotomic coefficients

We present a new bound on $A = \max_n |a_{pqr}(n)|$, where $a_{pqr}(n)$ are the coefficients of a ternary cyclotomic polynomial. We also prove that two consecutive coefficients of such a polynomial differ by at most one.

Jumps of ternary cyclotomic coefficients

It is known that two consecutive coefficients of a ternary cyclotomic polynomial $Φ_{pqr}(x)=\sum_k a_{pqr}(k)x^k$ differ by at most one. In this paper we give a criterion on $k$ to satisfy $|a_{pqr}(k)-a_{pqr}(k-1)|=1$. We use this to prove that the number of nonzero coefficients of the $n$th ternary cyclotomic polynomial is greater than $n^{1/3}$.

On a certain family of inverse ternary cyclotomic polynomials

We study a family of inverse ternary cyclotomic polynomials $Ψ_{pqr}$ in which $r\leφ(pq)$ is a positive linear combination of $p$ and $q$. We derive a formula for the height of such polynomial and characterize all flat polynomials in this family.

The family of ternary cyclotomic polynomials with one free prime

A cyclotomic polynomial Φ_n(x) is said to be ternary if n=pqr with p,q and r distinct odd primes. Ternary cyclotomic polynomials are the simplest ones for which the behaviour of the coefficients is not completely understood. Here we establish some results and formulate some conjectures regarding the coefficients appearing in the polynomial family Φ_{pqr}(x) with p<q<r, p and q fixed and r a free prime.

Explicit expression for a family of ternary cyclotomic polynomials

In this paper, we give an explicit expression for a certain family of ternary cyclotomic polynomials: specifically $Φ_{p_{1}p_{2}p_{3}}$, where $p_{1}<p_{2}<p_{3}$ are odd primes such that $p_{2} \equiv1 \mod p_{1}$ and $p_{3} \equiv1 \mod {p_{1}p_{2}}$. As an application of the explicit expressions, we give an exact formula for the number of nonzero terms in the polynomials in the family, which in turn immediately shows that the density (number of non-zeros terms / degree) is roughly inversely proportional to $p_{2}$, when $p_{1}$ is sufficiently large.

关于三次循环多项式和容斥多项式的高度和直径

On Heights and Diameters of Ternary Cyclotomic and Inclusion-Exclusion Polynomials

本文证明了对于任意奇素数p和整数h，存在足够大的素数q和r，使得对应的循环多项式Φ_{pqr}的高度为h，并揭示了其直径的可能取值。

On the number of terms of some families of the ternary cyclotomic polynomials $Φ_{3p_2p_3}$

We study the number of non-zero terms in two specific families of ternary cyclotomic polynomial, we find formulas for the number of terms by writing the cyclotomic polynomial as a sum of smaller sub-polynomials and study the properties of these polynomial.

On Ternary Inclusion-Exclusion Polynomials

Taking a combinatorial point of view on cyclotomic polynomials leads to a larger class of polynomials we shall call the inclusion-exclusion polynomials. This gives a more appropriate setting for certain types of questions about the coefficients of these polynomials. After establishing some basic properties of inclusion-exclusion polynomials we turn to a detailed study of the structure of ternary inclusion-exclusion polynomials. The latter subclass is exemplified by cyclotomic polynomials $Φ_{pqr}$, where $p<q<r$ are odd primes. Our main result is that the set of coefficients of $Φ_{pqr}$ is si…

实分圆域的Kitaoka猜想

Kitaoka's conjecture for real cyclotomic fields

该研究证明了所有实分圆域上的Kitaoka猜想，明确这类域存在通用经典三元二次型当且仅当它是实二次域。

On a Class of Ternary Inclusion-Exclusion Polynomials

A ternary inclusion-exclusion polynomial is a polynomial of the form \[ Q_{p,q,r}=\frac{(z^{pqr}-1)(z^p-1)(z^q-1)(z^r-1)} {(z^{pq}-1)(z^{qr}-1)(z^{rp}-1)(z-1)}, \] where $p$, $q$, and $r$ are integers $\ge3$ and relatively prime in pairs. This class of polynomials contains, as its principle subclass, the ternary cyclotomic polynomials corresponding to restricting $p$, $q$, and $r$ to be distinct odd prime numbers. Our object here is to continue the investigation of the relationship between the coefficients of $Q_{p,q,r}$ and $Q_{p,q,s}$, with $r\equiv s\pmod{pq}$. More specifically, we conside…

二元分圆多项式：通过字表示与算法

Binary Cyclotomic Polynomials: Representation via Words and Algorithms

本文研究二元分圆多项式，将其系数向量视为三元字母表上的字，设计了一种线性时间的最优算法以计算其紧凑表示，并推广到二维数值半群相关多项式。