Expand all Collapse all | Results 1 - 7 of 7 |
1. 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 |
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 2008 (vol 51 pp. 57)
A Note on Integer Symmetric Matrices and Mahler's Measure We find a lower bound on the absolute value of the discriminant of
the minimal polynomial of an integral symmetric matrix and apply
this result to find a lower bound on Mahler's measure of related
polynomials and to disprove a conjecture of D. Estes and R. Guralnick.
Keywords:integer matrices, Lehmer's problem, Mahler's measure Categories:11C20, 11R06 |
4. 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 |
5. CMB 2006 (vol 49 pp. 108)
A Dynamical Proof of Pisot's Theorem We give a geometric proof of classical results that characterize
Pisot numbers as algebraic $\lambda>1$ for which
there is $x\neq0$ with $\lambda^nx \to 0 \mod$ and identify such
$x$ as members of $\Z[\lambda^{-1}] \cdot \Z[\lambda]^*$ where $\Z[\lambda]^*$ is the dual module of $\Z[\lambda]$.
Category:11R06 |
6. 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 |
7. CMB 1998 (vol 41 pp. 125)
Uniform approximation to Mahler's measure in several variables If $f(x_1,\dots,x_k)$ is a polynomial with complex coefficients, the Mahler measure
of $f$, $M(f)$ is defined to be the geometric mean of $|f|$ over the $k$-torus
$\Bbb T^k$. We construct a sequence of approximations $M_n(f)$ which satisfy
$-d2^{-n}\log 2 + \log M_n(f) \le \log M(f) \le \log M_n(f)$. We use these to prove
that $M(f)$ is a continuous function of the coefficients of $f$ for polynomials
of fixed total degree $d$. Since $M_n(f)$ can be computed in a finite number
of arithmetic operations from the coefficients of $f$ this also demonstrates
an effective (but impractical) method for computing $M(f)$ to arbitrary
accuracy.
Categories:11R06, 11K16, 11Y99 |