Positive definite \(n\)-regular quadratic forms. (English) Zbl 1132.11018
For a positive integer \(n\), a positive definite integral quadratic form \(f\) is said to be \(n\)-regular if it represents every quadratic form of rank \(n\) that is represented by its genus, and to be (even) \(n\)-universal if it represents every (even) positive definite integral quadratic form of rank \(n\). So every \(n\)-universal form is \(n\)-regular. However, for small values of \(n\) there exist \(n\)-regular forms that are not \(n\)-universal. For example, the ternary form \(x_1^2+x_2^2+x_3^2\) is \(1\)-regular, but not \(1\)-universal. The main result of the present paper is that if \(f\) is an \(n\)-regular (even) form for any \(n\geq 27\), then \(f\) is (even) \(n\)-universal. As an application of this result, it is shown that the minimal rank of an \(n\)-regular form is bounded from below by a function that grows exponentially in \(n\). The author comments that every known example of an \(n\)-regular form with \(n\geq 10\) is either an \(n\)-universal form or an even \(n\)-universal form.
Reviewer: Andrew G. Earnest (Carbondale)
MSC:
| 11E12 | Quadratic forms over global rings and fields |
| 11E08 | Quadratic forms over local rings and fields |
| 11E25 | Sums of squares and representations by other particular quadratic forms |
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