I will present problems arising from biochemical reaction networks, optimization, computer science, and complexity theory. These problems share the following characteristics:
1) they can be modeled by multivariate polynomials,
2) they demand different theorems than the ones offered by the traditional theory of computation and state-of-the-art theory of polynomials.
I will present recent results that blend algebra, probability, and computation to address some of these problems (and related conjectures). I plan to complete the discussion with a ground plan toward an algorithmic theory of real polynomials.