  We suggest a novel approach to polynomial processes solely based on a
polynomial action operator. With this approach, we can analyse such processes
on general state spaces, going far beyond Banach spaces. Moreover, we can be
very flexible in the definition of what "polynomial" means. We show that
"polynomial process" universally means "affine drift". Simple assumptions on
the polynomial action operators lead to stronger characterisations on the
polynomial class of processes.
  In our framework we do not need to specify polynomials explicitly but can
work with a general sequence of graded vector spaces of functions on the state
space. Elements of these graded vector spaces form the monomials by introducing
a sequence of vector space complements. The basic tool of our analysis is the
polynomial action operator, which is a semigroup of operators mapping
conditional expected values of monomials acting on the polynomial process to
monomials of the same or lower grade. Unlike the classical Euclidean case, the
polynomial action operator may not form a finite-dimensional subspace after a
finite iteration, a property we call locally finite. We study abstract
polynomial processes under both algebraic and topological assumptions on the
polynomial actions, and establish an affine drift structure. Moreover, we
characterize the covariance structure under similar but slightly stronger
conditions. A crucial part in our analysis is the use of the (algebraic or
topological) dual of the monomials of grade one, which serves as a
linearization of the state space of the polynomial process. Our general
framework covers polynomial processes with values in Banach spaces recently
studied by Cuchiero and Svaluto-Ferro.
