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CATEGORIES:Number Theory Seminar
SUMMARY:Integer valued polynomials and fast equdistributio
n in number fields - Mikolaj Fraczyk
DTSTART;TZID=Europe/London:20190205T143000
DTEND;TZID=Europe/London:20190205T153000
UID:TALK116077AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/116077
DESCRIPTION:In his early work on integer valued polynomials Bh
argava introduced the notion of a p-ordering. Let
A be Dedekind domain with the fraction field k and
let $\\frac p$ be a prime ideal. Roughly speaking
\, a $\\frac p$-ordering in A is a sequence that e
quidistributes modulo powers of $\\frac p$ as fast
as possible. Using $\\frac p$-orderings Bhargava
defined an analogue of the factorial function and
constructed generating sets of the modules of degr
ee n integer valued polynomials in $k[X]$.\nOf par
ticular importance are the sequences which are $\\
frac p$-orderings for all primes $\\frac p$ at the
same time. We call them simultaneous p-orderings.
Bhargava asked which Dedekind rings admit such se
quences. For a long time the answer was not even k
nown in the particular case of rings of integers o
f global fields. In a recent joint work with Anna
Szumowicz we prove that the only number field k wh
ose ring of integers O_k admits a simultaneous p-o
rdering is Q. The result follows from a stronger s
tatement that puts an obstacle on the simultaneous
equidistribution of finite subsets of O_k modulo
all primes at the same time. Our proof relies on e
ffective bounds on the number of solutions of cert
ain degree 2 norm inequalities that we establish
using Bakerâ€™s bounds theorem on linear forms in lo
garithms.
LOCATION:MR13
CONTACT:Beth Romano
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