It turns out that this collection of derivative elements has more
structure than had previously been recognized. In this talk I
will discuss joint work with Barry Mazur, in which we attempt to
understand more fully what these derivative elements are, and what
they are good for.
Joe Buhler
Title: The probability that a p-adic polynomial splits
Abstract:
This talk will begin by calculating the probability that a uniformly
chosen polynomial, with degree n and coefficients in the p-adic
integers, factors completely into a product of linear polynomials.
(Warm-up question for you: find the answer for the case n = 2, i.e.,
quadratic polynomials.) This was motivated by ideas on generalizing
Serre's ``mass formula'' to not necessarily irreducible polynomials;
further results in this direction will be discussed. This is joint
work with Asher Auel.
Stephen Choi
Title: Small Prime Solutions for Quadratic Equations with Five Variables
Abstract: Initiated by a diophantine problem considered by A. Baker, the solubility
and small solutions over primes of the following equation were studied by
M.C. Liu and K. M. Tsang:
b=a1p1 + a2p2 + a3p3
When a1 = a2 = a3, their results recover the celebrated Vinogradov's three prime theorem. In this talk, we will discuss their results for other diophantine equations especially the quadratic equations
b=a1(p1)2 + a2(p2)2 +
a3(p3)2 +
a4(p4)2 . +
a5(p5)2