A short proof of a fibre criterion for polynomials to belong to an ideal.
Krzysztof Nowak · Jagiellonian University Repository (Jagiellonian University) · 2001
on the occasion of his 60th birthday Abstract. The purpose of this paper is to give a short, purely algebraic proof of a fibre criterion for polynomials to belong to an ideal. W. Jarnicki–L. O’Carrol–T. Winiarski [2] present a method of expressing a given ideal I in the polynomial ring k[X1,..., Xn] as an intersection of zero-dimensional ideals, which is based on the theory of comprehensive Gröbner bases. A generalization of this result (with no conditions on fibres) is the following criterion for polynomials to belong to an ideal: Fibre Criterion. Let k be an algebraically closed field. For a polynomial f = f(X,Y) ∈ k[X,Y] in two sets of variables X = (X1,..., Xn) and Y = (Y1,..., Ym), an ideal I ⊂ k[X,Y] and any point a = (a1,..., am) ∈ km, set fa = fa(X): = f(X, a) ∈ k[X] and Ia: = {fa: f ∈ I} ⊂ k[X]. Suppose I ⊂ k[X,Y] is a primary ideal such that the ideal I ∩ k[Y] is radical (whence prime). Then in order for a polynomial f = f(X,Y) to belong to I, it is necessary and sufficient that fa ∈ Ia for every a ∈ km.