Abstract
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.
Highlights
The theory of Grobner bases initially introduced over the fields is well developed over a great family of rings
We propose a method for computing a strong Grobner 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, given two ideals I = ⟨f1, . . . , fs⟩ and J = ⟨g1, . . . , gr⟩ of R, we propose an algorithm for computing a generating set for I ∩ J
Athough the ring V has no zero divisors, the ring A is noetherian and has zero divisors. We study this ring in detail and we propose an algorithm for computing strong Grobner basis for ideals of R
Summary
The theory of Grobner bases initially introduced over the fields (see [1]) is well developed over a great family of rings (see [2–8]). We are interested in the ring of dual valuation domain A = V + εV where V is a valuation domain. This ring is well known in the case of V = R as a typical example of. We study this ring in detail and we propose an algorithm for computing strong Grobner basis for ideals of R. We generalize Schreyer’s theorem which enables us to compute a strong Grobner basis for Syz(f1, . Fs} is a strong Grobner basis for some ideal I ⊂ R This important result is used later to compute a generating set for Syz(f1, . The goal of this paper is to compute the set of all the syzygies Syz(f1, . . . , fs, g1, . . . , gr) ⊂ Rs+r which will directly lead to computing a generating set for I ∩ J
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
