If A is a subset of Z, then the n-th characteristic ideal of A is the fractional ideal of Z consisting of 0 and the leading coecients of polynomials in Q[x] of degree no more than n which are integer-valued on A. The valuative capacity with respect to a prime p of A is a measure of the rate of growth of the p-adic part of these characteristic ideals of A and is defined, for a given p, to be the value of the limit limn!1 ,p(n) n , where ,p(n) is the p-adic valuation of the inverse of the n-th characteristic ideal of A. In this paper we compute this valuative capacity when A is the set of those integers which are expressible as the sum of two and of three squares.
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