Proof of a symmetrized trace conjecture for the Abelian Born-Infeld Lagrangian

In this paper we prove a conjecture regarding the form of the Born-Infeld Lagrangian with a U(1)²ⁿ gauge group after the elimination of the auxiliary fields. We show that the Lagrangian can be written as a symmetrized trace of Lorentz invariant bilinears in the field strength. More generally, we prove a theorem regarding certain solutions of unilateral matrix equations of arbitrary order. For solutions which have perturbative expansions in the matrix coefficients, the solution and all its positive powers are sums of terms which are symmetrized in all the matrix coefficients and of terms which are commutators.