Non-negative Polynomials, Sums of Squares & The Moment Problem


Non-negative polynomials are fundamental objects of study in real algebraic geometry and optimization. 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 present some of these certificates. We will consider the practicality of some popular algorithms for testing non-negativity of polynomials. We will also present a specific application of all these topics related to Quantum Information Theory, and some recent related work.

Dual to this, we have the classical Moment Problem from functional analysis, which has numerous applications to extremal problems and optimization in probability theory amongst others. In its simplest form it asks: given a sequence $(a_{n})_{n\in\mathbb{N}}$, does there exist a positive Borel measure $\mu$ on $\mathbb{R}$ such that $
a_{n} = \int_{\mathbb{R}}x^{n} \ d\mu 
$? This question has been extensively studied (and answered) by several famous mathematicians such as Hausdorff, Steiltjes and Hamburger. The Truncated Moment Problem extends this question to finite multisequences in several dimensions. There has been a lot of research done in this area by Curto and Fialkow, who have discovered several necessary conditions for a positive solution to this problem. 

In this talk we will be presenting necessary and sufficient conditions for a positive solution to the Truncated Moment Problem. We will also introduce the Truncated Tracial Moment Problem for non-commutative polynomials, and present some of our recent advanced in this area.