THE PROBABILITY THAT A RANDOM MONIC p-ADIC POLYNOMIAL SPLITS INTO LINEAR FACTORS

The Probability That a Random Monic p-adic Polynomial Splits

Galois Groups of p-Adic Polynomials of Fixed Degree

Given an integer n and a group G we estimate the probability that a random, monic, degree n, p-adic polynomial has Galois group isomorphic to G. A comparison is made with classic results concerning Galois groups of random, monic, degree n polynomials with integer coefficients. We conclude by analyzing the probability that a random, monic, degree n polynomial in Z[x] has these properties: its Ga...

Resultants and subresultants of p-adic polynomials

We address the problem of the stability of the computations of resultants and subresultants of polynomials defined over complete discrete valuation rings (e.g. Zp or k[[t]] where k is a field). We prove that Euclide-like algorithms are highly unstable on average and we explain, in many cases, how one can stabilize them without sacrifying the complexity. On the way, we completely determine the d...

Hausdorff dimension in a family of self-similar groups

For each prime p and a monic polynomial f , invertible over p, we define a group Gp,f of p-adic automorphisms of the p-ary rooted tree. The groups are modeled after the first Grigorchuk group, which in this setting is the group G2,x2+x+1. We show that the constructed groups are self-similar, regular branch groups. This enables us to calculate the Hausdorff dimension of their closures, providing...

Algorithms for solving linear systems over cyclotomic fields

We consider the problem of solving a linear system Ax = b over a cyclotomic field. What makes cyclotomic fields of special interest is that we can easily find a prime p that splits the minimal polynomial m(z) for the field into linear factors. This makes it possible to develop very fast modular algorithms. We give two output sensitive modular algorithms, one using multiple primes and Chinese re...