The Difference Kapranov Theorem

This paper creates a link between $\mathit{\text{Tropical}}\mathit{\text{Geometry}}$ and $\mathit{\text{Difference}}\mathit{\text{Algebra}}$. The main result is a difference version of $\mathit{\text{Kapranov’s}}\mathit{\text{Theorem}}$. In this theorem, we extend Kapranov’s Theorem to the case of a Laurent difference polynomial with coefficients from a multiplicative valued difference field, where the residue field is an algebraically closed field with a generic automorphism (ACFA). A result of this paper that plays a critical role in the proof of the Difference Kapranov Theorem is a difference version of $\mathit{\text{Newton’s}}\mathit{\text{Lemma}}$.