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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.