24 research outputs found
Can polylogarithms at algebraic points be linearly independent?
Let be positive integers. Let be a rational number. Let
be the -th Lerch function
with . When
, this is the polylogarithmic function. Let be
pairwise distinct algebraic numbers with . In
this article, we state a linear independence criterion over algebraic number
fields of all the numbers
and . This is the first result that gives a sufficient
condition for the linear independence of values of the Lerch functions
at distinct algebraic points
without any assumption for and , even for the case , the
polylogarithms. We give an outline of our proof and explain basic idea
-unit equation in two variables and Pad\'{e} approximations
In this article, we use Pad\'{e} approximations constructed for binomial
functions, to give a new upper bound for the number of the solutions of the
-unit equation. Combining explicit formulae of these Pad\'{e} approximants
with a simple argument relying on Mahler measure and on the local height, we
refine the bound due to J.-H. Evertse.Comment: 13 page
ON THE NAGELL-LJUNGGREN EQUATION (Analytic Number Theory : Distribution and Approximation of Arithmetic Objects)
We show that there exists an effective upper bound for the solutions to the Nagell-Ljunggren equation of the form --=y^{q} in 4 unknowns in integers x>1, y>1, m>2, q>1, when x is a cube of an integer. Our method relies on a refined estimate of linear forms in logarithms