In constrained polynomial optimization (CPOP) we minimize real, multivariate polynomials under polynomial constraints. These problems are closely connected to nonnegativity of real polynomials, which has been investigated in real algebraic geometry since the 19th century. As nonnegativity problems are notoriously hard to solve (e.g., various NP-complete problems admit a CPOP formulation), one uses certificates of nonnegativity like - prominently - sums of squares (SOS) or - more recently - sums of nonnegative circuit polynomials (SONC), which I introduced joint with Iliman in 2014.

In fact, our original motivation for developing SONCs was not to create a certificate, but to investigate the relation of nonnegativity of a polynomial $f$ to its amoeba. Amoebas are the log-absolute image of the (complex) hypersurface corresponding to $f$ - an object that not only plays a significant role in the context of tropical geometry, but also one that Frank made several contributions about.

In this talk I will discuss said connection, recalling initial developments, particularly during my time at Texas A&M 2014-17 under Frank's mentorship, and give an outlook, why it still has potential for new discoveries.