Two consequences of Minkowski's 2n theorem

✍ Scribed by Xueqing Tang; Adi Ben-Israel

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 136 KB
• Volume: 169
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/s0012-365x(96)00119-7

✦ Synopsis

Consider the inequalities

(a) IAxl <~ b, A ~ ~×~, r < n, b positive vector (here lYl denotes the vector of absolute values of components of the vector y) and (b) xTAx <~ c~, A positive semi-definite e R7 ×~, r < n, ~ > 0. Both inequalities are guaranteed a nonzero integer solution x for every positive right-hand side (b, ~ respectively). Such solutions will generally have a nonzero orthogonal projection XN/AI on the null space of A. We prove that a nonzero integer solution x exists with II XNtAill bounded, for (a): /~/' volA ~1/¢,,-,) II xNrA/II ~< ~/n --r ~) for (b):

(2"x/7~ ~ la"-'' II xN{,,,/II ~< t <x "i2 K,, ,) '

where vol k = x/~det 2 AIj summing over all r x r submatrices klx, and x, is the volume of the Euclidean unit ball in R".