  After fixing a triangulation $L$ of a $k$-dimensional simplex that has no new
vertices on the boundary, we introduce a triangulation operation on all
simplicial complexes that replaces every $k$-face with a copy of $L$, via a
sequence of induced subdivisions. The operation may be performed in many ways,
but we show that the face numbers of the subdivided complex depend only on the
face numbers of the original complex, in a linear fashion. We use this linear
map to define a sequence of polynomials generalizing the Tchebyshev polynomials
of the first kind and show, that in many cases, but not all, the resulting
polynomials have only real roots, located in the interval $(-1,1)$. Some
analogous results are shown also for generalized Tchebyshev polynomials of the
higher kind, defined using summing over links of all original faces of a given
dimension in our generalized Tchebyshev triangulations. Generalized Tchebyshev
triangulations of the boundary complex of a cross-polytope play a central role
in our calculations, and for some of these we verify the validity of a
generalized lower bound conjecture by the second author.
