For n greater than or equal to 4, the square of the volume of an n-simplexsatisfies a polynomial relation with coefficients depending on the squares ofthe areas of 2-faces of this simplex. First, we compute the minimal degree ofsuch polynomial relation. Second, we prove that the volume an n-simplexsatisfies a monic polynomial relation with coefficients depending on the areasof 2-faces of this simplex if and only if n is even and at least 6, and westudy the leading coefficients of polynomial relations satisfied by the volumefor other n.
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