Polynomially bounded solutions to the Loewner differential equation in several complex variables

We determine the form of polynomially bounded solutions to the Loewner differential equation that is satisfied by univalent subordination chains of the form f ( z , t ) = e t A z + ⋯ , where A ∈ L ( C n , C n ) has the property m ( A ) > 0 . Here m ( A ) = min { R 〈 A ( z ) , z 〉 : ‖ z ‖ = 1 } . We also give sufficient conditions for g ( z , t ) = L ( f ( z , t ) ) to be polynomially bounded, where f ( z , t ) is an A-normalized polynomially bounded Loewner chain solution to the Loewner differential equation.