Ordering monomial factors of polynomials in the product representation

The numerical construction of polynomials in the product representation (as used for instance in variants of the multiboson technique) can become problematic if rounding errors induce an imprecise or even unstable evaluation of the polynomial. We give criteria to quantify the effects of these rounding errors on the computation of polynomials approximating the function $1/s$. We consider polynomials both in a real variable $s$ and in a Hermitian matrix. By investigating several ordering schemes for the monomials of these polynomials, we finally demonstrate that there exist orderings of the monomials that keep rounding errors at a tolerable level.