New Dubrovin-type integrability theory applications of differential rings

Abstract

We present a new and effective approach to studying differential-algebraic relationships by means of specially constructed finitely-generated invariant subrings in differential rings. Based on their properties, we reanalyzed the Dubrovin integrability criterion for the Riemann type differential-functional constraints, perturbed by means of some elements from a suitably constructed differential ring. We also studied invariant finitely-generated ideals naturally related with constraints, generated by the corresponding Lie-algebraic endomorphic representations of derivations on differential ideals and which are equivalent to the corresponding differential-functional relationships on a generating function. The work in part generalizes the results devised before for proving integrability of the well known generalized hierarchy of the Riemann.