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In this short note, we introduce our recent results on the explicit description of the degrees of A-discriminant varieties introduced by Gelfand-Kapranov-Zelevinsky [10]. Our formulas can be applied also to the case where the A-discriminant varieties are higher-codimensional and their degrees are described by the geometry of the congurations A. For the detail, see [23]. x 1. Degree formulas for A-discriminants In this section, we rst introduce the formula for the degrees of A-discriminants obtained by Gelfand-Kapranov-Zelevinsky [10] and announce our generalization in [23]. Let M ' Zn be a Z-lattice (free Z-module) of rank n and MR: = R Z M the real vector space associated with M. Let A M be a nite subset of M and denote by P its convex hull in MR. In this note, such a polytope P will be called an integral polytope in MR. If A = f(1); (2); : : : ; (N + 1)g, we can dene a morphism 'A: T ! PN from an algebraic torus T: = (C)n to a complex projective space PN (N: =]A 1) by (1.1) x = (x1; x2; : : : ; xn) 7! [x (1) : x(2) : : x(N+1)]; where for each (i) 2 A M ' Zn we set x(i) = x (i)1 1