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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.

The Characteristic Sequence and p -Orderings of the Set of d -th Powers of Integers