Expand all Collapse all | Results 1 - 12 of 12 |
1. CMB 2014 (vol 57 pp. 538)
Infinite Families of $A_4$-Sextic Polynomials In this article we develop a test to determine whether a sextic
polynomial that is irreducible over $\mathbb{Q}$ has Galois group isomorphic
to the alternating group $A_4$. This test does not involve the
computation of resolvents, and we use this test to construct several
infinite families of such polynomials.
Keywords:Galois group, sextic polynomial, inverse Galois theory, irreducible polynomial Categories:12F10, 12F12, 11R32, 11R09 |
2. CMB 2012 (vol 56 pp. 759)
A Generalization of a Theorem of Boyd and Lawton The Mahler measure of a nonzero $n$-variable polynomial $P$ is the integral of
$\log|P|$ on the unit $n$-torus. A result of Boyd and Lawton says that
the Mahler measure of a multivariate polynomial is the limit of Mahler
measures of univariate polynomials. We prove the analogous
result for different extensions of Mahler measure such as generalized
Mahler measure (integrating the maximum of $\log|P|$ for possibly
different $P$'s),
multiple Mahler measure (involving products of $\log|P|$ for possibly
different $P$'s), and higher Mahler measure (involving $\log^k|P|$).
Keywords:Mahler measure, polynomial Categories:11R06, 11R09 |
3. CMB 2012 (vol 56 pp. 602)
Resultants of Chebyshev Polynomials: A Short Proof We give a simple proof of the value of the resultant of two Chebyshev polynomials
(of the first or the second kind),
values lately obtained by D. P. Jacobs, M. O. Rayes and V. Trevisan.
Keywords:resultant, Chebyshev polynomials, cyclotomic polynomials Categories:11R09, 11R04 |
4. 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 |
5. CMB 2011 (vol 54 pp. 739)
The Infimum in the Metric Mahler Measure Dubickas and Smyth defined the metric Mahler measure on the
multiplicative group of non-zero algebraic numbers.
The definition involves taking an infimum over representations
of an algebraic number $\alpha$ by other
algebraic numbers. We verify their conjecture that the
infimum in its definition is always achieved, and we establish its
analog for the ultrametric Mahler measure.
Keywords:Weil height, Mahler measure, metric Mahler measure, Lehmer's problem Categories:11R04, 11R09 |
6. 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 |
7. 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 |
8. CMB 2007 (vol 50 pp. 313)
On Cauchy--Liouville--Mirimanoff Polynomials Let $p$ be a prime greater than or equal to 17 and
congruent to
2 modulo 3. We use results of Beukers and Helou on
Cauchy--Liouville--Mirimanoff
polynomials to show that
the intersection of the Fermat curve of degree $p$ with the
line $X+Y=Z$ in the projective plane
contains no algebraic points of degree
$d$ with $3 \leq d \leq 11$.
We prove a result on
the roots of these polynomials and show that, experimentally,
they seem to satisfy
the conditions of a mild extension of
an irreducibility theorem of P\'{o}lya and Szeg\"{o}.
These conditions are \emph{conjecturally}
also necessary for irreducibility.
Categories:11G30, 11R09, 12D05, 12E10 |
9. CMB 2007 (vol 50 pp. 191)
Every Real Algebraic Integer Is a Difference of Two Mahler Measures We prove that every real
algebraic integer $\alpha$ is expressible by a
difference of two Mahler measures of integer polynomials.
Moreover, these polynomials can be chosen in such a way that they
both have the same degree as that of $\alpha$, say
$d$, one of these two polynomials is irreducible and
another has an irreducible factor of degree $d$, so
that $\alpha=M(P)-bM(Q)$ with irreducible polynomials
$P, Q\in \mathbb Z[X]$ of degree $d$ and a
positive integer $b$. Finally, if $d \leqslant 3$, then one can take $b=1$.
Keywords:Mahler measures, Pisot numbers, Pell equation, $abc$-conjecture Categories:11R04, 11R06, 11R09, 11R33, 11D09 |
10. CMB 2002 (vol 45 pp. 196)
Mahler Measures Close to an Integer We prove that the Mahler measure of an algebraic number cannot be too
close to an integer, unless we have equality. The examples of certain
Pisot numbers show that the respective inequality is sharp up to a
constant. All cases when the measure is equal to the integer are
described in terms of the minimal polynomials.
Keywords:Mahler measure, PV numbers, Salem numbers Categories:11R04, 11R06, 11R09, 11J68 |
11. CMB 1998 (vol 41 pp. 328)
Class number one and prime-producing quadratic polynomials revisited Over a decade ago, this author produced class number one criteria for
real quadratic fields in terms of prime-producing quadratic
polynomials. The purpose of this article is to revisit the problem
from a new perspective with new criteria. We look at the more general
situation involving arbitrary real quadratic orders rather than the
more restrictive field case, and use the interplay between the various
orders to provide not only more general results, but also simpler proofs.
Categories:11R11, 11R09, 11R29 |
12. CMB 1997 (vol 40 pp. 214)
Polynomials of quadratic type producing strings of primes The primary purpose of this paper is to provide necessary and
sufficient conditions for certain quadratic polynomials of negative
discriminant (which we call Euler-Rabinowitsch type), to produce
consecutive prime values for an initial range of input values less than
a Minkowski bound. This not only generalizes the classical work of
Frobenius, the later developments by Hendy, and the generalizations by
others, but also concludes the line of reasoning by providing a
complete list of all such prime-producing polynomials, under the
assumption of the generalized Riemann hypothesis ($\GRH$). We demonstrate
how this prime-production phenomenon is related to the exponent of the
class group of the underlying complex quadratic field. Numerous
examples, and a remaining conjecture, are also given.
Categories:11R11, 11R09, 11R29 |