  We consider the problem of finding the isolated common roots of a set of
polynomial functions defining a zero-dimensional ideal I in a ring R of
polynomials over C. Normal form algorithms provide an algebraic approach to
solve this problem. The framework presented in Telen et al. (2018) uses
truncated normal forms (TNFs) to compute the algebra structure of R/I and the
solutions of I. This framework allows for the use of much more general bases
than the standard monomials for R/I. This is exploited in this paper to
introduce the use of two special (nonmonomial) types of basis functions with
nice properties. This allows, for instance, to adapt the basis functions to the
expected location of the roots of I. We also propose algorithms for efficient
computation of TNFs and a generalization of the construction of TNFs in the
case of non-generic zero-dimensional systems. The potential of the TNF method
and usefulness of the new results are exposed by many experiments.
