Youssef Fares

The characteristic sequence and p-orderings of the set of d-th powers of integers

Abstract. If E is a subset of , then the n-th characteristic ideal of the algebra of rational polynomials taking integer values on E, , is the fractional ideal consisting of 0 and the leading coefficients of elements of of degree no more than n. For p a prime the characteristic sequence of is the sequence of negatives of the p-adic values of these ideals. We give recursive formulas for these sequences for the sets by describing how to recursively p-order them in the sense of Bhargava. We describe the asymptotic behavior of these sequences and identify primes, p, and exponents, d, for which there is a formula in closed form for the terms.

Subsets of ℤ with Simultaneous Orderings

Abstract We are interested in subsets S of ℤ which admit simultaneous orderings, that is, sequences {an } n≥0 such that divides for every x ∈ S (analogously to n! which divides (m + n)!/m! for every m). In particular, we characterize the polynomials ƒ of degree 2 such that ƒ(ℕ) admits a simultaneous ordering.

Poonens conjecture and Ramsey numbers

For cQ, let c:QQ denote the quadratic map c(X)=X2+c. How large can the period of a rational periodic point of c be? Poonen conjectured that it cannot exceed 3. Here, we tackle this conjecture by graph-theoretical means with the Ramsey numbers Rk(3). We show that, for any cQ whose denominator admits at most k distinct prime factors, the map c admits at most 2Rk(3)2 periodic points. As an application, we prove that Poonens conjecture holds for all cQ whose denominator is a power of 2.

Preservation of the residual classes numbers by polynomials

Let K be a global field and let O K,S be the ring of S-integers of K for some finite set S of primes of K. We prove that whatever the infinite subset E C O K,S and the polynomial f(X) ∈ K[X], the subsets E and f(E) have the same number of residual classes modulo m for almost all maximal ideals m of O K,S if and only if deg(f) = 1 when the characteristic of K is 0 and f(X) = g(X pk ) for some integer k and some polynomial g with deg(g) = 1 when the characteristic of K is p > 0.