Simple Relation between the Lowest-Order Element of Ideal 〈G,H〉 and the Last Element of Polynomial Remainder Sequence

摘要

Let G and H be relatively prime polynomials in K[x,u], where K is a number field and (u) = (u₁, . . . , u_ℓ), with ℓ ≥ 2. Let GB(G,H) be the reduced Gröbner basis of ideal hG,Hi, w.r.t. the elimination order for x. Let r̂ be the lowest-order element of GB(G,H), and Ã, B ∈ K[x,u] be the cofactors of r̂: ÃG+ e BH = r̂. Let PRS(G,H) be the polynomial remainder sequence w.r.t. x, with initial polynomials G and H. Let Pk ∈ K[u] be the last element of PRS(G,H), and A_k,_Bk ∈ K[x,u] be cofactors of P_k: A_kG + B_kH = P_k. Ak and B_k satisfy degree conditions: deg_x(Ak) <; deg_x(H) and deg_x(B_k) <; deg_x(G), while e à and B in general do not. Computing  and B by reducing the degrees w.r.t. x, of à and B, respectively, and by normalizing Pk suitably, we show that (P_k,A_k,B_k) = c ( r̂, Â, B), where (P,A,B) denotes a tuple and c ∈ K. We then present a new method for solving the multivariate Diophantine equation Φ G + ψH = F for given F ∈ K[x,u] and unknowns Φ , ψ ∈ K(u)[x]. Furthermore, we show a new method for solving the ideal-membership problem for 〈G,H〉, without computing any Gröbner basis.