The Characteristic Sequence and P -orderings of the Set of D-th Powers of Integers
نویسندگان
چکیده
If E is a subset of Z then the n-th characteristic ideal of the algebra of rational polynomials taking integer values on E, Int(E, Z), is the fractional ideal consisting of 0 and the leading coefficients of elements of Int(E, Z) of degree no more than n. For p a prime the characteristic sequence of Int(E, Z) is the sequence of negatives of the p-adic values of these ideals. We give recursive formulas for these sequences for the sets {nd : n = 0, 1, 2, . . . } 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.
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