  New approach to systems of polynomial recursions is developed based on the
Carleman linearization procedure. The article is divided into two main
sections: firstly, we focus on the case of uni-variable depth-one polynomial
recurrences. Subsequently, the systems of depth-one polynomial recurrence
relations are discussed. The corresponding transition matrix is constructed and
upper triangularized. Furthermore, the powers of the transition matrix are
calculated using the back substitution procedure. The explicit expression for a
solution to a broad family of recurrence relations is obtained. We investigate
to which recurrences the framework can be applied and construct a sufficient
conditions for the method to work. It is shown how introduction of auxiliary
variables can be used to reduce arbitrary depth systems to the depth-one system
of recurrences dealt with earlier. Finally, the limitations of the method are
discussed, outlining possible directions of future research.
