Trace-positive complex polynomials in three unitaries

Journal article, 2010

We consider the quadratic polynomials in three unitary generators, i.e. the elements of the group *-algebra of the free group with generators u2, u3 of the form f = Sigma(3)(j,k=1) alpha(jk)u(j)(*)u(k), alpha(jk) is an element of C We prove that if f is self-adjoint and Tr(f(U-1, U-2, U-3)) >= 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 operatorsU(1),..., U-4 connected by the linear relation a(1)U(1) + a(2)U(2) + a(3)U(3) + a(4)U(4) = 0.