Abstract

In his early work on integer valued polynomials Bhargava 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 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 the modules of degree n integer valued polynomials in $k[X]$. Of particular 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 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 the only number field k whose ring of integers O_k admits a simultaneous p-ordering is Q. The result follows from a stronger statement 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 effective bounds on the number of solutions of certain degree 2 norm inequalities that we establish using Baker’s bounds theorem on linear forms in logarithms.