In this paper, polynomial equations with real coefficients and in one variable were considered which contained a small, positive but specified and fixed parameter ε₀ ≠ 0. By using the classical asymptotic method, roots of the polynomial equations have been constructed in the literature, which were proved to be valid for sufficiently small ε-values (or equivalently for ε → 0). In this paper, it was assumed that for some or all roots of a polynomial equation, the first few terms in a Taylor or Laurent series in a small parameter depending on ε exist and can be constructed. We also assumed that at least two approximations x₁ (ε) and x₂ (ε) for the real roots exist and can be constructed. For a complex root, we assumed that at least two real approximations a₁ (ε) and a₂ (ε) for the real part of this root, and that at least two real approximations b₁ (ε) and b₂ (ε) for the imaginary part of this root, exist and can be constructed. Usually it was not clear whether for ε = ε₀ the approximations were valid or not. It was shown in this paper how the classical asymptotic method in combination with the bisection method could be used to prove how accurate the constructed approximations of the roots were for a given interval in ε (usually including the specified and fixed value ε₀ ≠ 0). The method was illustrated by studying a polynomial equation of degree five with a small but fixed parameter ε₀ = 0.1. It was shown how (absolute and relative) error estimates for the real and imaginary parts of the roots could be obtained for all values of the small parameter in the interval (0, ε₀ ].

In this paper, we present a new approach on how the multiple time-scales perturbation method can be applied to differential-delay equations such that approximations of the solutions can be obtained which are accurate on long time-scales. It will be shown how approximations can be constructed which branch off from solutions of differential-delay equations at the unperturbed level (and not from solutions of ordinary differential equations at the unperturbed level as in the classical approach in the literature). This implies that infinitely many roots of the characteristic equation for the unperturbed differential-delay equation are taken into account and that the approximations satisfy initial conditions which are given on a time-interval (determined by the delay). Simple and more advanced examples are treated in detail to show how the method based on differential and difference operators can be applied.