Event Type:

Number Theory Seminar

Date/Time:

Tuesday, April 11, 2017 -

16:00 to 16:45

Location:

Glikey 115

Guest Speaker:

Institution:

U. of Oregon

Abstract:

We consider the distribution of N p-adic particles with interaction energy proportional to the log of the p-adic distance between two particles. When the particles are constrained to the ring of integers of a local field, the distribution of particles is proportional to a power of the p-adic absolute value of the Vandermonde determinant. This leads to a first question: What is the normalization constant necessary to make this a probability measure? This sounds like a triviality, but this normalization constant as a function of extrinsic variables (like number of particles, or temperature) holds much information about the statistics of the particles. Viewed another way, this normalization constant is a p-adic analog of the now famous Selberg integral. While a closed form for this seems out of reach, I will derive a remarkable identity that may hold the key to unlocking more nuanced information about p-adic electrostatics. Along the way we’ll find an identity for the generating function of probabilities that a degree N polynomial with p-adic integer coefficients split completely. Joint work with Jeff Vaaler.