The regular octagon and the signature of nonsingular integral quadratic forms. With an appendice of Géraldine Gahide. (Q1395357)

Let \(L\) be a \({\mathbb Z}\)-lattice of rank \(n\) in an \(n\)-dimensional \({\mathbb Q}\)-vector space \(V\) equipped with a nondegenerate quadratic form \(q:V\to {\mathbb Q}\) such that \(q(L)\subset {\mathbb Z}\), and let \(L^\#\) be the dual lattice in \(V\). There is a well known Gaussian sum relation stating that \[ \sum_{x\in L^\#/L}\exp(2\pi iq(x))=\sqrt{[L^\#:L]}\exp(2\pi i\sigma (q)/8) \] where \(\sigma(q)\) denotes the signature of \(q\) [see, e.g. \textit{W. Scharlau}\ 's book: Quadratic and Hermitian forms. Grundlehren der Mathematischen Wissenschaften, 270. Berlin etc.: Springer-Verlag (1985; Zbl 0584.10010), Ch. 5 \S 8]. In the present paper, the authors give a new elementary albeit somewhat lengthy and quite technical proof using as one of the ingredients the following geometric fact: Let \(L\) be the length of a side of a regular octagon circumscribed around a circle of radius \(R\), and let \(l\) be the length of a side of a regular octagon inscribed in that same circle. Then \(L^2<Rl\).