Positive Values of Non-homogeneous Indefinite Quadratic Forms of Type (2, 5)

✍ Scribed by R. Sehmi; V.C. Dumir

• Publisher: Elsevier Science
• Year: 1994
• Tongue: English
• Weight: 827 KB
• Volume: 48
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.1994.1049

✦ Synopsis

The minimum (\Gamma_{r, n-r}) of positive values of non-homogeneous indefinite quadratic forms of type ((r, n-r)) is defined as the infimum of all constants (\Gamma>0) such that for any indefinite quadratic form (Q) of type ( (r, n-r) ) and determinant (D \neq 0) and any real numbers (c_{1}, \ldots, c_{n}) there exist integers (x_{1}, \ldots, x_{n}) such that

[
0<Q\left(x_{1}+c_{1}, \ldots, x_{n}+c_{n}\right)<(\Gamma|D|)^{1 / n}
]

In this paper it is proved that (\Gamma_{2.5}=32), thereby confirming the conjecture of Bambah, Dumir, and Hans-Gill. Also, all the critical forms for which equality is needed are determined. O 1994 Academic Press, Inc.