Abstract
AbstractLet
$\alpha $
be a totally positive algebraic integer of degree d, with conjugates
$\alpha _1=\alpha , \alpha _2, \ldots , \alpha _d$
. The absolute
$S_k$
-measure of
$\alpha $
is defined by
$s_k(\alpha )= d^{-1} \sum _{i=1}^{d}\alpha _i^k$
. We compute the lower bounds
$\upsilon _k$
of
$s_k(\alpha )$
for each integer in the range
$2\leq k \leq 15$
and give a conjecture on the results for integers
$k>15$
. Then we derive the lower bounds of
$s_k(\alpha )$
for all real numbers
$k>2$
. Our computation is based on an improvement in the application of the LLL algorithm and analysis of the polynomials in the explicit auxiliary functions.
Publisher
Cambridge University Press (CUP)