On the volume of a six-dimensional polytope

Details On the volume of a six-dimensional polytope On the volume of a six-dimensional polytope

2005-02-08

arxiv.org/abs/math/0502167v1

Mathematics

Metric Geometry

Scientific paper

This note is a comment to the paper by D.R.Heath-Brown and B.Z.Moroz (Math Proc. Camb. Phil. Soc. 125 (1999)). That paper concerns with the projective surface $S$ in $\mathbb{P}^{3}$ defined by the equation $x_{1}x_{2}x_{3}=x_{4}^{3}$. It is shown there that the evaluation of the leading term of the asymptotic formula for the number of rational points of bounded height in $S(\Q)$ is equivalent to the evaluation of the volume of some 6-dimensional polytope $\P$. The volume of $\P$ is known from several papers; we calculate this volume by elementary method using symmetry of the polytope $\P$. We also discuss a combinatorial structure of $\P$.

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