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Let M be a smooth manifold and A a Weil algebra. We discuss the differential forms in the Weil bundles MA,π,M, and we established a link between differential forms in MA and M as well as their cohomology. We also discuss the cohomology in.

We revisit the algebra of polynomial integrodifferential operators and we give a generalization of its relations. Generators of prime ideals of height 1, of the maximal ideal, and of the smallest ideal of In are discussed. A technique for obtaining a generating set for a given two-sided ideal of In is also discussed.

Let V be a valuation domain and let A=V+εV be a dual valuation domain. We propose a method for computing a strong Gröbner basis in R=A[x1,…,xn]; given polynomials f1,…,fs∈R, a method for computing a generating set for Syz(f1,…,fs)={(h1,…,hs)∈Rs∣h1f1+⋯+hsfs=0} is given; and, finally, given two ideals I=〈f1,…,fs〉 and J=〈g1,…,gr〉 of R, we propose an algorithm for computing a generating set for I∩J.