Schwach positive ganze quadratische Formen, die eine aufrichtige, positive Wurzel mit einem Koeffizienten 6 besitzen. (Weakly positive integral quadratic forms, having a positive sincere root with coefficient 6)
An integral quadratic form is a polynomial \[ f(X_ 1,...,X_ n)=\sum^{n}_{j=1}X^ 2_ i+\sum_{i<j}f_{ij}X_ iX_ j\in {\mathbb{Z}}[X_ 1,...,X_ n]. \] An element \(z=(z_ 1,...,z_ n)\in {\mathbb{Z}}^ n\) is called a root of f if \(f(z)=1\), it is called positive if \(z_ i\geq 0\) for all i and \(z_ i>0\) for at least one i, and it is called sincere if \(z_ i\neq 0\) for all i. f is called weakly positive, if \(f(z)>0\) for all positive \(z\in {\mathbb{Z}}^ n\). \textit{S. A. Ovsienko} [Integral weakly positive forms, in ``Schur Matrix Problems and Quadratic Forms'', Kiev 1978, 3-17 (1978)] showed that a positive root z of a weakly positive integral quadratic form satisfies \(z_ i\leq 6\) for all i, and that \(z_ i=6\) for some i can only occur in case \(n\geq 8\). For instance the form \[ g_{(8)}:=\sum^{8}_{i=1}X^ 2_ i-X_ 1X_ 3-X_ 2X_ 4-X_ 3X_ 4-X_ 4X_ 5-X_ 5X_ 6-X_ 6X_ 7-X_ 7X_ 8 \] which belongs to the Dynkin graph \(E_ 8\) has the root \(z=(2,3,4,6,5,4,3,2).\) In the present interesting article the picture is completed as follows. Suppose that a weakly positive integral quadratic form f has a sincere positive root \(z\in {\mathbb{Z}}^ n\) with \(z_ i=6\) for some i. Then the number of variables satisfies \(n\leq 24\), f is positive semidefinite and can be computed in a well-described way from a certain weakly positive integral quadratic form \(g_{(24)}\in {\mathbb{Z}}[X_ 1,...,X_{24}]\) which is derived from \(g_{(8)}\) by a process called spreading by the authors. Furthermore the authors obtain an estimation for the number of positive roots of f and show that f is a radical extension of \(g_{(8)}\) and \(g_{(24)}\) a radical extension of f. (The radical rad f is defined as the subgroup of all \(x\in {\mathbb{Z}}^ n\) with \(f(z+x)=f(z)\) for all \(z\in {\mathbb{Z}}^ n\); if there exist subgroups \(U\subset {\mathbb{Z}}^ n\) and \(V\subset rad f\) such that \(U\oplus V={\mathbb{Z}}^ n\), then f is called a radical extension of \(f|_ U.)\) Finally an application of these results in the representation theory of finite dimensional algebras is mentioned.
representation theory of finite dimensional algebras
scientific article; zbMATH DE number 4137866