CYCLOTOMIC FACTORS OF BORWEIN POLYNOMIALS

A cyclotomic polynomial $\unicode[STIX]{x1D6F7}_{k}(x)$ is an essential cyclotomic factor of $f(x)\in \mathbb{Z}[x]$ if $\unicode[STIX]{x1D6F7}_{k}(x)\mid f(x)$ and every prime divisor of $k$ is less than or equal to the number of terms of $f.$ We show that if a monic polynomial with coefficients from $\{-1,0,1\}$ has a cyclotomic factor, then it has an essential cyclotomic factor. We use this result to prove a conjecture posed by Mercer [‘Newman polynomials, reducibility, and roots on the unit circle’, Integers12(4) (2012), 503–519].

A Basis of Finite and Infinite Sets with Small Representation Function

Let $A$ be a subset of the set of nonnegative integers $\mathbb{N}\cup\{0\}$, and let $r_A(n)$ be the number of representations of $n\geq 0$ by the sum $a+b$ with $a,b \in A$. Then $\big(\sum_{a \in A}x^a\big)^2=\sum_{n=0}^{\infty} r_A(n)x^n$. We show that an old result of Erdős asserting that there is a basis $A$ of $\mathbb{N}\cup \{0\}$, i.e., $r_A(n) \geq 1$ for $n \geq 0$, whose representation function $r_A(n)$ satisfies $r_A(n) < (2e+\epsilon)\log n$ for each sufficiently large integer $n$. Towards a polynomial version of the Erdős-Turán conjecture we prove that for each $\epsilon>0$ and each sufficiently large integer $n$ there is a set $A \subseteq \{0,1,\dots,n\}$ such that the square of the corresponding Newman polynomial $f(x):=\sum_{a \in A} x^a$ of degree $n$ has all of its $2n+1$ coefficients in the interval $[1, (1+\epsilon)(4/\pi)(\log n)^2]$. Finally, it is shown that the correct order of growth for $H(f^2)$ of those reciprocal Newman polynomials $f$ of degree $n$ whose squares $f^2$ have all their $2n+1$ coefficients positive is $\sqrt{n}$. More precisely, if the Newman polynomial $f(x)=\sum_{a \in A} x^a$ of degree $n$ is reciprocal, i.e., $A=n-A$, then $A+A=\{0,1,\dots,2n\}$ implies that the coefficient for $x^n$ in $f(x)^2$ is at least $2\sqrt{n}-3$. In the opposite direction, we explicitly construct a reciprocal Newman polynomial $f(x)$ of degree $n$ such that the coefficients of its square $f(x)^2$ all belong to the interval $[1, 2\sqrt{2n}+4]$.

Newman Polynomials, Reducibility, and Roots on the Unit Circle

A We also show that certain plausible conjectures imply that the proportion of length 5 Newman polynomials with roots on the unit circle is

Pisot Numbers from {0, 1}-Polynomials

A Pisot number is a real algebraic integer greater than 1, all of whose conjugates lie strictly inside the open unit disk; a Salem number is a real algebraic integer greater than 1, all of whose conjugate roots are inside the closed unit disk, with at least one of them of modulus exactly 1. Pisot numbers have been studied extensively, and an algorithm to generate them is well known. Our main result characterises all Pisot numbers whose minimal polynomial is derived from a Newman polynomial — one with {0, 1}-coefficients — and shows that they form a strictly increasing sequence with limit (1 + √5)/2. It has long been known that every Pisot number is a limit point, from both sides, of sequences of Salem numbers. We show that this remains true, from at least one side, for the restricted sets of Pisot and Salem numbers that are generated by Newman polynomials.