In this lecture I will give an introduction to algebraic and semi-algebraic methods for proving the unsatisfiability of systems of real polynomial equations over the Boolean hypercube. The main goal of this talk is to compare the relative strength of these approaches using methods from proof complexity.

On the one hand, I will focus on the static semi-algebraic proof systems Sherali-Adams and sum-of-squares (a.k.a. Lasserre), which are based on linear and semi-definite programming relaxations. On the other hand, I will discuss polynomial calculus, which is a dynamic algebraic proof system that models Gröbner basis computations.

I am going to present two recent results on the relative strength between algebraic and semi-algebraic proof systems:¹

The first result is that sum-of-squares simulates polynomial calculus: any polynomial calculus refutation of degree d can be transformed into a sum-of-squares refutation of degree 2d and only polynomial increase in size. In contrast, the second result shows that this is not the case for Sherali-Adams: there are systems of polynomial equations that have polynomial calculus refutations of degree 3 and polynomial size, but require Sherali-Adams refutations of large degree and exponential size.

¹) this work was presented at STACS 2018; a preprint is available at https://eccc.weizmann.ac.il/report/2017/154/