Quasi-convex free polynomials

Proceedings of the American Mathematical Society - Tập 142 Số 8 - Trang 2581-2591
Sriram Balasubramanian1, Scott McCullough2
1Department of Mathematics and Statistics, Indian Institute of Science Education and Research (IISER) – Kolkata, Mohanpur Campus, Nadia District, Pin: 741246, West Bengal, India
2Department of Mathematics, The University of Florida, Box 118105, Gainesville, Florida 32611-8105

Tóm tắt

LetR⟨x⟩\mathbb R\langle x \rangledenote the ring of polynomials inggfreely noncommuting variablesx=(x1,…,xg)x=(x_1,\dots ,x_g). There is a natural involution∗*onR⟨x⟩\mathbb R\langle x \rangledetermined byxj∗=xjx_j^*=x_jand(pq)∗=q∗p∗(pq)^*=q^* p^*, and a free polynomialp∈R⟨x⟩p\in \mathbb R\langle x \rangleis symmetric if it is invariant under this involution. IfX=(X1,…,Xg)X=(X_1,\dots ,X_g)is aggtuple of symmetricn×nn\times nmatrices, then the evaluationp(X)p(X)is naturally defined and furtherp∗(X)=p(X)∗p^*(X)=p(X)^*. In particular, ifppis symmetric, thenp(X)∗=p(X)p(X)^*=p(X). The main result of this article says ifppis symmetric,p(0)=0p(0)=0and for eachnnand each symmetric positive definiten×nn\times nmatrixAAthe set{X:A−p(X)≻0}\{X:A-p(X)\succ 0\}is convex, thenpphas degree at most two and is itself convex, or−p-pis a hermitian sum of squares.

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Proceedings of the American Mathematical Society

Tập 142 Số 8

2581-2591

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