A Sparse Version of Reznick’s Positivstellensatz

A Convex Positivstellensatz

We provide a specific representation of convex polynomials nonnegative on a convex (not necessarily compact) basic closed semi-algebraic set K ⊂ Rn. Namely, we prove that almost all of them belong to a specific subset of the quadratic module generated by the concave polynomials that define K. In particular, almost all nonnegative convex polynomials are sums of squares. Mathematics Subject Class...

Positivstellensatz, SDPs and Entangled Games

Today we are going to talk about an infinite hierarchy of semidefinite programmings which attempts to approximate the entangled value of multi-prover games [1]. This work is based on a recent result called non-commutative Positivstellensatz [2] about the representation of positive polynomials. In a one-round two-prover cooperative game, a verifier asks questions to two provers, Alice and Bob, w...

The convex Positivstellensatz in a free algebra

Given a monic linear pencil L in g variables, let PL = (PL(n))n∈N where PL(n) := { X ∈ Sn | L(X) 0 } , and Sn is the set of g-tuples of symmetric n × n matrices. Because L is a monic linear pencil, each PL(n) is convex with interior, and conversely it is known that convex bounded noncommutative semialgebraic sets with interior are all of the form PL. The main result of this paper establishes a ...

A Positivstellensatz for Non-commutative Polynomials

A non-commutative polynomial which is positive on a bounded semi-algebraic set of operators has a weighted sum of squares representation. This Positivstellensatz parallels similar results in the commutative case. A broader issue is to what extent does real semi-algebraic geometry extend to non-commutative polynomials? Our ”strict” Positivstellensatz is positive news, on the opposite extreme fro...

A Non-Commutative Positivstellensatz On Isometries