Quasi-convex free polynomials

Abstract

Let R ⟨ x ⟩ \mathbb R\langle x \rangle denote the ring of polynomials in g g freely noncommuting variables x = ( x 1 , … , x g ) x=(x_1,\dots ,x_g) . There is a natural involution ∗ * on R ⟨ x ⟩ \mathbb R\langle x \rangle determined by x j ∗ = x j x_j^*=x_j and ( p q ) ∗ = q ∗ p ∗ (pq)^*=q^* p^* , and a free polynomial p ∈ R ⟨ x ⟩ p\in \mathbb R\langle x \rangle is symmetric if it is invariant under this involution. If X = ( X 1 , … , X g ) X=(X_1,\dots ,X_g) is a g g tuple of symmetric n × n n\times n matrices, then the evaluation p ( X ) p(X) is naturally defined and further p ∗ ( X ) = p ( X ) ∗ p^*(X)=p(X)^* . In particular, if p p is symmetric, then p ( X ) ∗ = p ( X ) p(X)^*=p(X) . The main result of this article says if p p is symmetric, p ( 0 ) = 0 p(0)=0 and for each n n and each symmetric positive definite n × n n\times n matrix A A the set { X : A − p ( X ) ≻ 0 } \{X:A-p(X)\succ 0\} is convex, then p p has degree at most two and is itself convex, or − p -p is a hermitian sum of squares.