A q-analog of a number N is a polynomial in q that simplifies to N when we set q = 1 and also encodes some “additional structure”.
In this talk we will explore many different examples of what may be meant by “additional structure”, focusing on the classical q-analog of the factorial and moving to more recent work on a q-analog of a “Fubini number”, which gives the number of surjections from an n-element set to a k-element set.
February 10, 2020
Joe Webster, University of Oregon
p-adic volume estimates via coin changing problems
The Vandermonde polynomial V(x1,x2,…,xN) is defined as the product of (xi - xj) over all i < j.
If the xi are p-adic integers, then so is the Vandermonde polynomial, and hence its p-adic absolute value is either zero or |V(x1,x2,…,xN)|p = p-n for some nonnegative integer n.
If (x1,x2,…,xN) is chosen uniformly randomly from (ℤp)N, what is the probability that |V(x1,x2,…,xN)|p = p-n, and how
does it vary with N, p, and n?
In the first half of the talk we will answer this question by developing a combinatorial formula for the probability.
In the second half we will use the formula to compute the probabilities in the N = 2 and N = 3 cases (and maybe N = 4 if we have enough time), then discuss how estimates for more general N, p, and n can be made by solving coin changing problems.