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Effective version of Tartakowsky’s theorem. (English) Zbl 0936.11021
The authors prove an effective version of a theorem of Tartakowsky on the representation of integers by positive definite integral quadratic forms. In order to describe this result in detail, let \(f\) be a positive definite integral quadratic form of rank \(m\geq 5\), and let \(N\) be a natural number which is represented \(p\)-adically by \(f\) for all primes \(p\). W. Tartakowsky [Izv. Akad. Nauk SSSR 7, 111-122, 165-196 (1929; JFM 56.0882.04)] proved that there exists a constant \(C\) such that if \(N\geq C\), then \(N\) is represented by \(f\) over \(\mathbb{Z}\). However, the work of Tartakowsky does not yield an estimate for the size of the constant \(C\). The question of obtaining an upper bound for \(C\) was first addressed by G. L. Watson [Philos. Trans. R. Soc. Lond., Ser. A 253, 227-254 (1960; Zbl 0102.28102)], who proved that if \(f\) and \(N\) satisfy the conditions in Tartakowsky’s theorem, but \(N\) is not represented by \(f\), then there is an exponent \(l\) (where \(l\) depends on \(m\) when \(5\leq m\leq 9\), and \(l= 1\) when \(m\geq 10\)) such that \(N\ll D^l\), where \(D\) is the determinant of \(f\). The implied constants in these asymptotic bounds were not explicitly obtained by Watson.
In the present paper, the authors refine the algebraic techniques of M. Kneser [Quadratische Formen, Göttingen Lecture Notes (1973/74)] and the first author, Y. Kitaoka and M. Kneser [J. Reine Angew. Math. 301, 132-141 (1978; Zbl 0374.10013)] in order to improve on these previous results in several ways. First, an explicit bound for the size of the constant in Tartakowsky’s theorem is given. Second, the power of \(D\) appearing in the authors’ bound improves the exponent obtained by Watson for small values of \(m\) (for example, when \(m=5\), Watson’s exponent of 5.2 is improved here to 3.4). Third, the analogous result is obtained for positive definite integral quadratic lattices over the ring of integers of a totally real algebraic number field in which 2 does not ramify (for example, for a totally real number field whose absolute discriminant is an odd integer).

MSC:
11E12 Quadratic forms over global rings and fields
11E08 Quadratic forms over local rings and fields
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