Positive-definite quadratic forms representing finite sets of integers. (English) Zbl 1454.11069
Let \(S\) be a set of positive integers. A positive definite integral quadratic form is said to be \(S\)-universal if it represents all the integers in \(S\). The goal of this paper is to study upper bounds for the discriminants of positive definite \(S\)-universal integral quadratic forms of minimal dimension, when \(S\) is a finite set. Since every positive integer can be written as a sum of four squares, this minimal dimension is at most 4 for any set \(S\).
Historical note: The upper bound of 861 for the discriminant of a universal positive definite quaternary integral quadratic form does not appear to be mentioned in the paper of A. E. Ross [Am. J. Math. 68, 29–46 (1946; Zbl 0060.11001)] cited here. To the reviewer’s knowledge, this bound is first mentioned in a paper of M. F. Willerding [Bull. Am. Math. Soc. 54, 334–337 (1947; Zbl 0032.26603)], who attributes it to Ross in an unpublished manuscript. This bound was subsequently improved by the reviewer and A. N. Khosravani [Mathematika 44, 342–347 (1997; Zbl 0895.11017)] to \(\frac{1073}{4}\), which was later shown to be best possible by H. Iwabuchi [Mathematika 49, 197–199 (2002; Zbl 1046.11021)], who proved that the form \(x^2+2y^2+xz+yz+5z^2+29w^2\) of that discriminant is universal.
Historical note: The upper bound of 861 for the discriminant of a universal positive definite quaternary integral quadratic form does not appear to be mentioned in the paper of A. E. Ross [Am. J. Math. 68, 29–46 (1946; Zbl 0060.11001)] cited here. To the reviewer’s knowledge, this bound is first mentioned in a paper of M. F. Willerding [Bull. Am. Math. Soc. 54, 334–337 (1947; Zbl 0032.26603)], who attributes it to Ross in an unpublished manuscript. This bound was subsequently improved by the reviewer and A. N. Khosravani [Mathematika 44, 342–347 (1997; Zbl 0895.11017)] to \(\frac{1073}{4}\), which was later shown to be best possible by H. Iwabuchi [Mathematika 49, 197–199 (2002; Zbl 1046.11021)], who proved that the form \(x^2+2y^2+xz+yz+5z^2+29w^2\) of that discriminant is universal.
Reviewer: Andrew G. Earnest (Carbondale)
MSC:
| 11E25 | Sums of squares and representations by other particular quadratic forms |
| 11E12 | Quadratic forms over global rings and fields |
| 11E20 | General ternary and quaternary quadratic forms; forms of more than two variables |
References:
| [1] | M. Bhargava, On the Conway-Schneeberge fifteen theorem, Contemp. Math 272 (2000), 27-37. · Zbl 0987.11027 |
| [2] | M. Bhargava and J. Hanke, Universal quadratic forms and the 290-Theorem, preprint. |
| [3] | B.M. Kim, M.-H. Kim and B.-K. Oh, A finiteness theorem for representability of quadratic forms by forms, J. Reine Angew. Math 581 (2005), 23-30. · Zbl 1143.11011 |
| [4] | O. T. O’Meara, Introduction to Quadratic Forms, Springer, New York, 1971. · Zbl 0207.05304 |
| [5] | A. E. Ross, On a problem of Ramanujan, Amer. J. Math 68 (1946), 29-46. · Zbl 0060.11001 |
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