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1. CMB 2011 (vol 56 pp. 510)
Linear Forms in Monic Integer Polynomials We prove a necessary and sufficient condition on the list of
nonzero integers $u_1,\dots,u_k$, $k \geq 2$, under which a monic
polynomial $f \in \mathbb{Z}[x]$ is expressible by a linear form
$u_1f_1+\dots+u_kf_k$ in monic polynomials $f_1,\dots,f_k \in
\mathbb{Z}[x]$. This condition is independent of $f$. We also show that if
this condition holds, then the monic polynomials $f_1,\dots,f_k$
can be chosen to be irreducible in $\mathbb{Z}[x]$.
Keywords:irreducible polynomial, height, linear form in polynomials, Eisenstein's criterion Categories:11R09, 11C08, 11B83 |
2. CMB 2009 (vol 53 pp. 140)
Pisot Numbers from $\{ 0, 1 \}$-Polynomials A \emph{Pisot number} is a real algebraic integer greater than 1, all of whose conjugates lie strictly inside the open unit disk; a \emph{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 Â- one with $\{0,1\}$-coefficients Â- and shows that they form a strictly increasing sequence with limit $(1+\sqrt{5}) / 2$. 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.
Categories:11R06, 11R09, 11C08 |
3. CMB 2009 (vol 52 pp. 511)
The Irreducibility of Polynomials That Have One Large Coefficient and Take a Prime Value We use some classical estimates for polynomial roots to provide
several irreducibility criteria for polynomials with integer
coefficients that have one sufficiently large coefficient and take a
prime value.
Keywords:Estimates for polynomial roots, irreducible polynomials Categories:11C08, 11R09 |
4. CMB 2008 (vol 51 pp. 497)
Expected Norms of Zero-One Polynomials Let $\cA_n = \big\{ a_0 + a_1 z + \cdots + a_{n-1}z^{n-1} : a_j \in \{0, 1 \
} \big\}$, whose elements are called \emf{zero-one polynomials}
and correspond naturally to the $2^n$ subsets of $[n] := \{ 0, 1,
\ldots, n-1 \}$. We also let $\cA_{n,m} = \{ \alf(z) \in \cA_n :
\alf(1) = m \}$, whose elements correspond to the ${n \choose m}$
subsets of~$[n]$ of size~$m$, and let $\cB_n = \cA_{n+1} \setminus
\cA_n$, whose elements are the zero-one polynomials of degree
exactly~$n$.
Many researchers have studied norms of polynomials with restricted
coefficients. Using $\norm{\alf}_p$ to denote the usual $L_p$ norm
of~$\alf$ on the unit circle, one easily sees that $\alf(z) = a_0 +
a_1 z + \cdots + a_N z^N \in \bR[z]$ satisfies $\norm{\alf}_2^2 = c_0$
and $\norm{\alf}_4^4 = c_0^2 + 2(c_1^2 + \cdots + c_N^2)$, where $c_k
:= \sum_{j=0}^{N-k} a_j a_{j+k}$ for $0 \le k \le N$.
If $\alf(z) \in \cA_{n,m}$, say $\alf(z) = z^{\beta_1} + \cdots +
z^{\beta_m}$ where $\beta_1 < \cdots < \beta_m$, then $c_k$ is the
number of times $k$ appears as a difference $\beta_i - \beta_j$. The
condition that $\alf \in \cA_{n,m}$ satisfies $c_k \in \{0,1\}$ for $1
\le k \le n-1$ is thus equivalent to the condition that $\{ \beta_1,
\ldots, \beta_m \}$ is a \emf{Sidon set} (meaning all differences of
pairs of elements are distinct).
In this paper, we find the average of~$\|\alf\|_4^4$ over $\alf \in
\cA_n$, $\alf \in \cB_n$, and $\alf \in \cA_{n,m}$. We further show
that our expression for the average of~$\|\alf\|_4^4$ over~$\cA_{n,m}$
yields a new proof of the known result: if $m = o(n^{1/4})$ and
$B(n,m)$ denotes the number of Sidon sets of size~$m$ in~$[n]$, then
almost all subsets of~$[n]$ of size~$m$ are Sidon, in the sense that
$\lim_{n \to \infty} B(n,m)/\binom{n}{m} = 1$.
Categories:11B83, 11C08, 30C10 |