We describe a method to express the susceptibility and higher derivatives of
the free energy in terms of the scaling variables (Wegner's nonlinear scaling
fields) associated with the high-temperature (HT) fixed point of Dyson
hierarchical model in arbitrary dimensions. We give a closed form solution of
the linearized problem. We check that up to order 7 in the HT expansion, all
the poles ("small denominators") that would naively appear in some positive
dimension are canceled by zeroes ("small numerators"). The requirement of
continuity in the dimension can be used to lift ambiguities which appear in
calculations at fixed dimension. We show that the existence of a HT phase in
the infinite volume limit for a continuous set of values of the dimension,
requires that this mechanism works to all orders. On the other hand, most poles
at negative values of the dimensional parameter (where the free energy density
is not well-defined, but RG flows can be studied) persist and reflect the fact
that for special negative values of the dimension, finite-size corrections
become leading terms. We show that the inverse problem is also free of small
denominator problems and that the initial values of the scaling variables can
be expressed in terms of the infinite volume limit of the susceptibility and
higher derivatives of the free energy. We discuss the existence of an infinite
number of conserved quantities (RG invariants) and their relevance for the
calculation of universal ratios of critical amplitudes.