Formes quadratiques contiguës à \(D_ 7\). (Quadratic forms neighbouring \(D_ 7)\). (French) Zbl 0693.10025

Jaquet, David-Olivier; Sigrist, François

C. R. Acad. Sci., Paris, Sér. I 309, No. 10, 641-644 (1989).

Let A be (the symmetric coefficient matrix of) an n-ary perfect positive quadratic form. The convex cone in \({\mathbb{R}}^{n(n+1)/2}\) generated by all \(n\times n\)-matrices \(vv^{to}\) where \(v\in {\mathbb{Z}}^ d\setminus \{0\}\) is a minimum vector of A is a convex polyhedral cone of dimension \(n(n+1)/2\). For a normal vector (matrix) P of a facet of this cone let \(\rho >0\) be minimal such that the positive form \(A+\rho P\) has a set of minimum vectors different from that of A. Then \(A+\rho P\) is also a perfect form, called a contiguous or neighbouring form of A. The authors determine the 14 inequivalent contiguous forms of \[ D_ 7=(x_ 1-x_ 2)^ 2+x^ 2_ 3+...+x^ 2_ 7+(x_ 1+...+x_ 7)^ 2. \] (See also J. H. Conway and N. J. A. Sloane [Proc. R. Soc. Lond., Ser. A 418, 43-80 (1988; Zbl 0655.10022)], P. M. Gruber and C. G. Lekkerkerker [Geometry of numbers (North-Holland, 1987; Zbl 0611.10017)]).

Reviewer: P.Gruber

MSC:

11H55

Quadratic forms (reduction theory, extreme forms, etc.)

Keywords:

Voronoi algorithm; quadratic form; perfect form; neighbouring form; contiguous forms

Citations:

Zbl 0198.380; Zbl 0655.10022; Zbl 0611.10017