Dequantization Transform and Generalized Newton Polytopes

For functions defined on Cn or Rn+ we construct a dequantization transform f 7→ f̂ ; this transform is closely related to the Maslov dequantization. If f is a polynomial, then the subdifferential ∂f̂ of f̂ coincides with the Newton polytope of f . For the semiring of polynomials with nonnegative coefficients, the dequantization transform is a homomorphism of this semiring to the idempotent semiring of convex polytopes with respect to the well-known Minkowski operations. Using the dequantization transform we generalize these results to a wide class of functions and convex sets.