We consider the class of polynomial differential equation x&= , 2(,)(,)(,)nnmnmPxyPxyPxy++++2(,)(,)(,)nnmnmyQxyQxyQxy++&=++. For where and are homogeneous polynomials of degree i. Inside this class of polynomial differential equation we consider a subclass of Darboux integrable systems. Moreover, under additional conditions we proved such Darboux integrable systems can have at most 1 limit cycle.

For a ring endomorphism $alpha$ and an $alpha$-derivation $delta$, we introduce a concept, so called skew $pi$-Armendariz ring, that is a generalization of both $pi$-Armendariz rings, and $(alpha,delta)$-compatible skew Armendariz rings. We first observe the basic properties of skew $pi$-Armendariz rings, and extend the class of skew $pi$-Armendariz rings through various ring extensions. We nex...

We construct sections of a structure presheaf of rational functions on a differential spectrum using only localization and projective limits. For this purpose we introduce a special form of a multiplicative system generated by one differential polynomial and call it D-localization. This technique allows us to obtain sections of a differential spectrum of a differential ring R without computatio...

The two papers in this series analyze quantum invariant differential operators for quantum symmetric spaces in the maximally split case. In this paper, we complete the proof of a quantum version of Harish-Chandra’s theorem: There is a Harish-Chandra map which induces an isomorphism between the ring of quantum invariant differential operators and a ring of Laurent polynomial invariants with resp...

In this paper, we present a new algorithm and an experimental implementation for factoring elements in the polynomial nth Weyl algebra, the polynomial nth shift algebra, and Zgraded polynomials in the nth q-Weyl algebra. The most unexpected result is that this noncommutative problem of factoring partial differential operators can be approached effectively by reducing it to the problem of solvin...

We highlight the role of primary decomposition of binomial ideals in a commutative polynomial ring, in the description of the holonomicity, the holonomic rank, and the shape of solutions of multivariate hypergeometric differential systems of partial differential equations. En honor a Mischa Cotlar, con afecto y admiración

We study primary submodules and decompositions from a differential computational point of view. Our main theoretical contribution is general structure theory representation theorem for an arbitrary finitely generated module over polynomial ring. characterize in terms operators punctual Quot schemes. Moreover, we introduce implement algorithm that computes minimal decomposition module.

Let K be a field of characteristic p > 0. It is proved that the group Autord(D(Ln)) of order preserving automorphisms of the ring D(Ln) of differential operators on a Laurent polynomial algebra Ln := K[x ±1 1 , . . . , x n ] is isomorphic to a skew direct product of groups Z n p⋊AutK(Ln) where Zp is the ring of p-adic integers. Moreover, the group Autord(D(Ln)) is found explicitly. Similarly, A...

In this paper we generalize the involutive methods and algorithms devised for polynomial ideals to differential ones generated by a finite set of linear differential polynomials in the differential polynomial ring over a zero characteristic differential field. Given a ranking of derivative terms and an involutive division, we formulate the involutivity conditions which form a basis of involutiv...

In this paper we generalize the involutive methods and algorithms devised for polynomial ideals to differential ones generated by a finite set of linear differential polynomials in the differential polynomial ring over a zero characteristic differential field. Given a ranking of derivative terms and an involutive division, we formulate the involutivity conditions which form a basis of involutiv...