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.

No abstract

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

Buchberger's algorithm was already studied over many kinds of rings such as principal ideal rings, noetherian valuation rings with zero divisors, Dedekind rings with zero divisors, Gaussian rings,. .. . In this paper, we propose the Buchberger's algorithm over V[ε] satisfying to ε 2 = 0 where V is any noetherian valuation domain and we give some applications in Z p [ε] where p is a prime number and Z p = { a b | a ∈ Z, b / ∈ pZ}.