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**Local densities and explicit bounds for representability by a quadratic form.**
*(English)*
Zbl 1090.11023

A classical theorem of V. A. Tartakovski (W. Tartakowsky) [Bull. Acad. Sci. Leningr. (7) (Izv. Akad. Nauk SSSR) 2, 111–122, 165–196 (1929; JFM 56.0882.04)] states that a positive definite integral quadratic form \(f\) in \(n\geq 5\) variables represents all sufficiently large integers that are represented by the genus of \(f\). This result extends to forms in 4 variables when the integers considered are restricted to those divisible by bounded powers of the primes at which \(f\) is anisotropic. Additional restrictions are needed in the ternary case, where there can exist infinite families of integers that are represented by a genus but not by all spinor genera within the genus. In this case, the results of W. Duke and R. Schulze-Pillot [Invent. Math. 99, No. 1, 49–57 (1990; Zbl 0692.10020)] show, in particular, that all sufficiently large integers primitively represented by the spinor genus of \(f\) are represented by \(f\) itself.

In the present paper, the author gives explicit lower bounds for an integer \(m\) (suitably restricted for the cases \(n=3,4\)) which ensure that \(m\) is represented by \(f\), provided that it is represented by the genus of \(f\). The main emphasis is on the cases \(n=3\) and \(n=4\) since reasonable bounds in these cases were not previously known (for \(n\geq5\) such bounds can be found in papers of G. L. Watson [Philos. Trans. R. Soc. Lond., Ser. A 253, 227–254 (1960; Zbl 0102.28102)] and J. S. Hsia and M. I. Icaza [Acta Arith. 89, 235–253 (1999; Zbl 0936.11021)]; previous results for \(n=4\) were obtained by R. Schulze-Pillot [Arch. Math. 77, 129–137 (2001; Zbl 0995.11030)]). The bounds obtained here also help to describe the general representation behavior when \(n=3\). In that case, an effective version of the result of Duke and Schulze-Pillot within any fixed square class is given, including an asymptotic formula for the number of representations by \(f\) of the integers within the square class. The author notes that the results of this paper can be extended to arbitrary totally real number fields, and all the local analysis in the paper is carried out in that generality.

The strength of the results obtained here is demonstrated by resolving the well known and long-standing conjecture that the only positive integers that fail to be represented by the form \(x^2+3y^2+5z^2+7w^2\) are 2 and 22.

In the present paper, the author gives explicit lower bounds for an integer \(m\) (suitably restricted for the cases \(n=3,4\)) which ensure that \(m\) is represented by \(f\), provided that it is represented by the genus of \(f\). The main emphasis is on the cases \(n=3\) and \(n=4\) since reasonable bounds in these cases were not previously known (for \(n\geq5\) such bounds can be found in papers of G. L. Watson [Philos. Trans. R. Soc. Lond., Ser. A 253, 227–254 (1960; Zbl 0102.28102)] and J. S. Hsia and M. I. Icaza [Acta Arith. 89, 235–253 (1999; Zbl 0936.11021)]; previous results for \(n=4\) were obtained by R. Schulze-Pillot [Arch. Math. 77, 129–137 (2001; Zbl 0995.11030)]). The bounds obtained here also help to describe the general representation behavior when \(n=3\). In that case, an effective version of the result of Duke and Schulze-Pillot within any fixed square class is given, including an asymptotic formula for the number of representations by \(f\) of the integers within the square class. The author notes that the results of this paper can be extended to arbitrary totally real number fields, and all the local analysis in the paper is carried out in that generality.

The strength of the results obtained here is demonstrated by resolving the well known and long-standing conjecture that the only positive integers that fail to be represented by the form \(x^2+3y^2+5z^2+7w^2\) are 2 and 22.

Reviewer: Andrew G. Earnest (Carbondale)

### MSC:

11E12 | Quadratic forms over global rings and fields |

11D09 | Quadratic and bilinear Diophantine equations |

11E25 | Sums of squares and representations by other particular quadratic forms |

11E20 | General ternary and quaternary quadratic forms; forms of more than two variables |

11Y50 | Computer solution of Diophantine equations |

### Citations:

Zbl 0692.10020; Zbl 0102.28102; Zbl 0936.11022; Zbl 0995.11030; JFM 56.0882.04; Zbl 0936.11021### Software:

QFLib
Full Text:
DOI

### References:

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