Let $f(x) in mathbb{Z}[x]$; for each integer $alpha$ it is interesting to consider the number of iterates $n_{alpha}$, if possible, needed to satisfy $f^{n_{alpha}}(alpha) = alpha$. The sets ${alpha, f(alpha), ldots, f^{n_{alpha} - 1}(alpha), alpha}$
generated by the iterates of $f$ are called cycles. For $mathbb{Z}[x]$ it is known that cycles of length 1 and 2 occur, and no others. While much is known for extensions to number fields, we concentrate on extending $mathbb{Z}$ by adjoining reciprocals of primes. Let $mathbb{Z}[1/p_1, ldots, 1/p_n]$ denote $mathbb{Z}$ extended by adding in the reciprocals of the $n$ primes $p_1, ldots, p_n$ and all their products and powers with each other and the elements of $mathbb{Z}$. Interestingly, cycles of length 4, called 4-cycles, emerge for polynomials in $mathbb{Z}left[1/p_1, ldots, 1/p_nright][x]$ under the appropriate conditions. The problem of finding criteria under which 4-cycles emerge is equivalent to determining how often a sum of four terms is zero, where the terms are $pm 1$ times a product of elements from the list of $n$ primes. We investigate conditions on sets of primes under which 4-cycles emerge. We characterize when 4-cycles emerge if the set has one or two primes, and (assuming a generalization of the ABC conjecture) find conditions on sets of primes guaranteed not to cause 4-cycles to emerge.
We estimate several probability distributions arising from the study of random, monic polynomials of degree $n$ with coefficients in the integers of a general $p$-adic field $K_{mathfrak{p}}$ having residue field with $q= p^f$ elements. We estimate t
he distribution of the degrees of irreducible factors of the polynomials, with tight error bounds valid when $q> n^2+n$. We also estimate the distribution of Galois groups of such polynomials, showing that for fixed $n$, almost all Galois groups are cyclic in the limit $q to infty$. In particular, we show that the Galois groups are cyclic with probability at least $1 - frac{1}{q}$. We obtain exact formulas in the case of $K_{mathfrak{p}}$ for all $p > n$ when $n=2$ and $n=3$.
We analyze the probability that, for a fixed finite set of primes S, a random, monic, degree n polynomial f(x) with integer coefficients in a box of side B around 0 satisfies: (i) f(x) is irreducible over the rationals, with splitting field over the
rationals having Galois group $S_n$; (ii) the polynomial discriminant Disc(f) is relatively prime to all primes in S; (iii) f(x) has a prescribed splitting type at each prime p in S. The limit probabilities as $B to infty$ are described in terms of values of a one-parameter family of measures on $S_n$, called splitting measures, with parameter $z$ evaluated at the primes p in S. We study properties of these measures. We deduce that there exist degree n extensions of the rationals with Galois closure having Galois group $S_n$ with a given finite set of primes S having given Artin symbols, with some restrictions on allowed Artin symbols for p<n. We compare the distributions of these measures with distributions formulated by Bhargava for splitting probabilities for a fixed prime $p$ in such degree $n$ extensions ordered by size of discriminant, conditioned to be relatively prime to $p$.
A binary renewal process is a stochastic process ${X_n}$ taking values in ${0,1}$ where the lengths of the runs of 1s between successive zeros are independent. After observing ${X_0,X_1,...,X_n}$ one would like to predict the future behavior, and the
problem of universal estimators is to do so without any prior knowledge of the distribution. We prove a variety of results of this type, including universal estimates for the expected time to renewal as well as estimates for the conditional distribution of the time to renewal. Some of our results require a moment condition on the time to renewal and we show by an explicit construction how some moment condition is necessary.
