Non-singlet structure functions beyond the next-to-next-to-leading order

We study the evolution of the flavour non-singlet deep-inelastic structure functions F-2,F-NS and F-3 at the next-to-next-tu-next-to-leading order ((N LO)-L-3) of massless perturbative QCD. The present information on the corre sponding three-loop coefficient functions is used to derive approximate exp ressions of these quantities which prove completely sufficient for values x > 10(-2) of the Bjorken variable. The inclusion of the N3LO corrections re duces the theoretical uncertainty of alpha (s) determinations from non-sing let scaling violations arising from the truncation of the perturbation seri es to less than 1%. We also study the predictions of the soft-gluon resumma tion, of renormalization-scheme optimizations by the principle of minimal s ensitivity (PMS) and the effective charge (ECH) method, and of the Fade sum mation for the structure-function evolution kernels. The PMS, ECH and Fade approaches are found to facilitate a reliable estimate of the corrections b eyond (NLO)-L-3. (C) 2001 Elsevier Science B.V. All rights reserved.