Expand all Collapse all | Results 1 - 6 of 6 |
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 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 |
3. CMB 2009 (vol 53 pp. 58)
Ranks in Families of Jacobian Varieties of Twisted Fermat Curves In this paper, we prove that the unboundedness of ranks in families of Jacobian varieties of twisted Fermat curves is equivalent to the divergence of certain infinite series.
Keywords:Fermat curve, Jacobian variety, elliptic curve, canonical height Categories:11G10, 11G05, 11G50, 14G05, 11G30, 14H45, 14K15 |
4. CMB 2009 (vol 53 pp. 87)
Elliptic Curves over the Perfect Closure of a Function Field We prove that the group of rational points of a non-isotrivial elliptic curve defined over the perfect closure of a function field in positive characteristic is finitely generated.
Keywords:elliptic curves, heights Categories:11G50, 11G05 |
5. CMB 2009 (vol 52 pp. 237)
Points of Small Height on Varieties Defined over a Function Field We obtain a Bogomolov type of result for the affine space defined
over the algebraic closure of a function field of transcendence
degree $1$ over a finite field.
Keywords:heights, Bogomolov conjecture Categories:11G50, 11G25, 11G10 |
6. CMB 2004 (vol 47 pp. 398)
A Reduction of the Batyrev-Manin Conjecture for Kummer Surfaces Let $V$ be a $K3$ surface defined over a number field $k$. The
Batyrev-Manin conjecture for $V$ states that for every nonempty open
subset $U$ of $V$, there exists a finite set $Z_U$ of accumulating
rational curves such that the density of rational points on $U-Z_U$ is
strictly less than the density of rational points on $Z_U$. Thus,
the set of rational points of $V$ conjecturally admits a stratification
corresponding to the sets $Z_U$ for successively smaller sets $U$.
In this paper, in the case that $V$ is a Kummer surface, we prove that
the Batyrev-Manin conjecture for $V$ can be reduced to the
Batyrev-Manin conjecture for $V$ modulo the endomorphisms of $V$
induced by multiplication by $m$ on the associated abelian surface
$A$. As an application, we use this to show that given some restrictions
on $A$, the set of rational points of $V$ which lie on rational curves
whose preimages have geometric genus 2 admits a stratification of
Keywords:rational points, Batyrev-Manin conjecture, Kummer, surface, rational curve, abelian surface, height Categories:11G35, 14G05 |