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SUMMARY:Integer valued polynomials and fast equdistribution in number fiel
 ds - Mikolaj Fraczyk
DTSTART:20190205T143000Z
DTEND:20190205T153000Z
UID:TALK116077@talks.cam.ac.uk
CONTACT:Beth Romano
DESCRIPTION:In his early work on integer valued polynomials Bhargava intro
 duced the notion of a p-ordering. Let A be Dedekind domain with the fracti
 on field k and let $\\frac p$ be a prime ideal. Roughly speaking\, a $\\fr
 ac p$-ordering in A is a sequence that equidistributes 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 t
 he modules of degree n integer valued polynomials in $k[X]$.\nOf particula
 r importance are the sequences which are $\\frac p$-orderings for all prim
 es $\\frac p$ at the same time. We call them simultaneous p-orderings. Bha
 rgava asked which Dedekind rings admit such sequences. For a long time the
  answer was not even known in the particular case of rings of integers of 
 global fields. In a recent joint work with Anna Szumowicz we prove that th
 e only number field k whose ring of integers O_k admits a simultaneous p-o
 rdering is Q. The result follows from a stronger statement that puts an ob
 stacle on the simultaneous equidistribution of finite subsets of O_k modul
 o all primes at the same time. Our proof relies on effective bounds on the
  number of solutions of certain  degree 2 norm inequalities that we establ
 ish using Baker’s bounds theorem on linear forms in logarithms. 
LOCATION:MR13
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