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We give explicit formulas for the dimensions and the degrees of$A$-discriminant varieties introduced by Gelfand-Kapranov-Zelevinsky. Ourformulas can be applied also to the case where the $A$-discriminant varietiesare higher-codimensional and their degrees are described by the geometry of theconfigurations $A$. Moreover combinatorial formulas for the Euler obstructionsof general (not necessarily normal) toric varieties will be also given.

A geometric degree formula for $A$-discriminants and Euler obstructions of toric varieties