Almost-universal quadratic forms: an effective solution of a problem of Ramanujan
type: Item
instance of: scholarly article
title: Almost-universal quadratic forms: an effective solution of a problem of Ramanujan
zbMATH Open document ID: 1242.11028
DOI: 10.1215/00127094-2009-008
published in: Duke Mathematical Journal
publication date: 2009-03-25T00:00:00Z
review text: From the introduction: The object of this paper is to prove several results giving an effective method for deciding whether a positive definite integral quaternary quadratic form is almost universal, that is, whether it represents all large positive integers. In this way we obtain an effective and definitive solution to a problem first addressed and investigated by \textit{S. Ramanujan} 90 years ago [Proc. Camb. Philos. Soc. 19, 11--21 (1917; JFM 46.0240.01)]. The following set \(\Sigma=\{1,2,3,5,6,7,10, 14\}\) plays an important role in our investigations. Observe that the 8 elements of \(\Sigma\) are the smallest possible positive integers representing all 8 different square classes of \(Q\). Theorem 1.1. For any positive definite integral quaternary quadratic form \(f\), the following conditions are equivalent: (i) \(f\) is almost universal. (ii) \(f\) is locally universal and, either (a) \(f\) is \(p\)-isotropic for every prime number \(p\), or (b) \(f\) is 2-anisotropic and represents every element of the set \[ 16\Sigma = \{16,32,48,80,96,112, 160,224\}, \] or (c) \(f\) is equivalent to one of the following four forms \[ x^2 + 2y^2 + 5z^2 + 10t^2,\quad x^2 + 2y^2 + 3z^2 + 5t^2 + 2yz,\quad x^2 + y^2 + 3z^2 + 3t^2,\quad x^2 + 2y^2 + 4z^2 + 7t^2 + 2yz. \] The set \(16\Sigma\) in Theorem 1.1 is optimal in the sense that removing any of its elements would lead to a false statement. Unless stated otherwise, here by an ``integral form'' is always meant a positive definite quadratic form having integer matrix. An integral form \(f\) is called locally universal if \(f\) is \(p\)-universal for every prime \(p\), that is, if \(f\) represents every \(p\)-adic integer over the ring \(\mathbb Z_p\) of \(p\)-adic integers. If \(f\) represents zero nontrivially over \(\mathbb Z_p\), we say that \(f\) is \(p\)-isotropic, otherwise \(f\) is called \(p\)-anisotropic. Effective criteria for \(p\)-universality are known and easy to verify [cf. \textit{G. Pall}, Am. J. Math. 68, 47…
reviewed by: Olaf Ninnemann
Mathematics Subject Classification ID: 11E20
Mathematics Subject Classification ID: 11E12
zbMATH DE Number: 5532604
author: Jacek Bochnak
author: Byeong-Kweon Oh
MaRDI profile type: Publication
OpenAlex ID: W2065984539
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description: scientific article; zbMATH DE number 5532604