  Let $A$ be an integral matrix and let $P$ be the convex hull of its columns.
By a result of Gelfand, Kapranov and Zelevinski, the so-called principal
$A$-determinant locus is equal to the union of the closures of the discriminant
loci of the Laurent polynomials associated to the faces of $P$ that are
hypersurfaces. In this short note we show that it is also the straightforward
union of all the discriminant loci, i.e. we may include those of higher
codimension, and there is no need to take closures. This answers a question by
Kite and Segal.
