  The effectiveness of the perturbative renormalization group approach at fixed
space dimension d in the theory of critical phenomena is analyzed. Three models
are considered: the O(N) model, the cubic model and the antiferromagnetic model
defined on the stacked triangular lattice. We consider all models at fixed d=3
and analyze the resummation procedures currently used to compute the critical
exponents. We first show that, for the O(N) model, the resummation does not
eliminate all non-physical (spurious) fixed points (FPs). Then the dependence
of spurious as well as of the Wilson-Fisher FPs on the resummation parameters
is carefully studied. The critical exponents at the Wilson-Fisher FP show a
weak dependence on the resummation parameters. On the contrary, the exponents
at the spurious FP as well as its very existence are strongly dependent on
these parameters. For the cubic model, a new stable FP is found and its
properties depend also strongly on the resummation parameters. It appears to be
spurious, as expected. As for the frustrated models, there are two cases
depending on the value of the number of spin components. When N is greater than
a critical value Nc, the stable FP shows common characteristic with the
Wilson-Fisher FP. On the contrary, for N<Nc, the results obtained at the stable
FP are similar to those obtained at the spurious FPs of the O(N) and cubic
models. We conclude from this analysis that the stable FP found for N<Nc in
frustrated models is spurious. Since Nc>3, we conclude that the transitions for
XY and Heisenberg frustrated magnets are of first order.
