Speaker:
Igor Shparlinski
Date:
Thu, 07/04/2016 - 14:00 - 14:50
Motivated by work of Shanks (1969), we study the distribution
of the fields
${\mathbb Q}\big(\sqrt{f(g^n)}\big)$ for a polynomial $f \in {\mathbb Z}[X]$
and an integer $g >1$.
Using a variety of known results and techniques such as
the abundance of shifted primes with a large prime divisor
(R. Baker and G. Harman) and the square sieve (R. Heath-Brown) together
with new bounds on character
sums, we improve an upper bound of Luca and Shparlinski (2009)
on the number of $n \in \{M+1,\ldots,M+N\}$ with
${\mathbb Q}\bigl(\sqrt{f(g^n)}\bigr) ={\mathbb Q}\bigl(\sqrt{s}\bigr)$ for a
given squarefree integer $s$, individually and on average over $s \le S$.
(Joint work with Bill Banks.)
