On the GKZ discriminant locus

Let $A$ be an integral matrix and let $P$ be the convex hull of its columns. By a result of Gelfand, Kaparanov 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.