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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:
 1.1e+13 Quadratic forms over global rings and fields 1.1e+09 Quadratic forms over local rings and fields
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