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On conjectures of Minkowski and Woods for \(n=7\). (English) Zbl 1175.11033

One of the long-standing open question of the geometry of numbers is a conjecture by Minkowski about the minimum of the product of non-homogeneous linear forms. Consider \(n\) linear forms \(L_i = a_{i,1} x_1 + \cdots + a_{i,n} x_n\), the conjecture of Minkowski states that for any real numbers \((c_i)\), there exist integers \((x_i)\) such that \( | (L_1 + c_1) \cdots (L_n + c_n) | \leq \det (a_{i,j}) /2^n\). This result was only known up to dimension 6 and is extended here to dimension 7.
The proof relies on two arguments: (i) for any lattice \(L\), there is a positive unimodular diagonal matrix \(D\) such that \(D\cdot L\) is well-rounded, i.e. its minimal vectors span the full space, (ii) the value of the covering radius of well-rounded lattices is less than \(\sqrt{n}/2\). Item (i) was shown by C. T. McMullen [J. Am. Math. Soc. 18, No. 3, 711–734 (2005; Zbl 1132.11034)] for any dimension provided the homogeneous minimum of the lattice is non-zeros; this is enough for proving Minkowski’s conjecture according to B. J. Birch and H. P. F. Swinnerton-Dyer [Mathematika 3, 25–39 (1956; Zbl 0074.03702)]. The authors show a restricted version of (ii) in dimension 7, which implies (ii) in dimension 7 according to A. C. Woods [J. Number Theory 4, 157–180 (1972; Zbl 0232.10020)]. Their arguments rely on a case by case inquiry and a handy manipulation of classical inequalities.

MSC:

11H46 Products of linear forms
11H31 Lattice packing and covering (number-theoretic aspects)
11J20 Inhomogeneous linear forms
52C17 Packing and covering in \(n\) dimensions (aspects of discrete geometry)
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