Trace-positive complex polynomials in three unitaries

Abstract:

We consider the quadratic polynomials in three unitary generators, i.e. the elements of the group $*$-algebra of the free group with generators $u_1, u_2, u_3$ of the form $f=\sum _{j, k=1}^{3}\alpha _{jk}u_{j}^{*}u_{k}$, $\alpha _{jk} \in \mathbb {C}$. We prove that if $f$ is self-adjoint and $\mathrm {Tr}(f(U_{1}, U_2 ,U_{3}))\ge 0$ for arbitrary unitary matrices $U_{1}, U_2, U_3$, then $f$ is a sum of hermitian squares. To prove this statement we reduce it to the question whether a certain Tarski sentence is true. Tarski’s decidability theorem thus provides an algorithm to answer this question. We use an algorithm due to Lazard and Rouillier for computing the number of real roots of a parametric system of polynomial equations and inequalities implemented in Maple to check that the Tarski sentence is true.

As an application, we describe the set of parameters $a_1, a_2, a_3, a_4$ such that there are unitary operators $U_1, \ldots , U_4$ connected by the linear relation $a_1 U_1+a_2 U_2 +a_3 U_3 +a_4 U_4 =0$.