Non-negative polynomials are fundamental objects of study in real algebraic geometry. Testing the non-negativity of polynomials is known to be NP hard. Because of this, one generally looks to satisfy easier certificates of non-negativity. A basic example of this is the gradient being zero, and the Hessian being positive definite.
In recent years more attention has been given to the Sum of Squares (SOS) certificates, and their corresponding algorithms related to semi-definite programming (SDP).
This talk will focus on presenting these certificates and their related SDP forms. We will consider the practicality of some popular algorithms for testing non-negativity of polynomials. Lastly, we'll present a specific application of all these topics related to Quantum Information Theory, and some recent related work.
Abhishek Bhardwaj is a PhD Student at the Mathematical Sciences Institute, ANU