A Fibre Criterion for a Polynomial to Belong to an Ideal

In the paper we generalize a fibre criterion for a polynomial f to belong to a primary ideal I in the polynomial ring K[X, Y ]. We also investigate the general case where the ideal I is not primary. Let {X1, . . . , Xn} be any set of variables. We shall write K[X] instead of K[X1, . . . , Xn]. If f ∈ K[X,Y ], where X and Y are sets of variables, K is an algebraically closed field, Y = {Y1, . . . , Ym}, a ∈ Km, then fa := f(X1, . . . , Xn, a1, . . . , am). For a subset I of K[X,Y ] we define Ia = {fa | f ∈ I}. Of course, if I is an ideal then is Ia. We shall also write IY for I ∩K[Y ]. The following theorem was proved by Jarnicki-O’Carroll-Winiarski [2] (see also preprint, proposition 12): Let I be an ideal in K[X,Y ] such that I ∩ K[Y ] = (0), where K is an algebraically closed field. Assume that for all a ∈ Km the ideal Ia is proper and zero-dimentional. Then the following holds true: ∀f ∈ K[X,Y ] ∀a ∈ Km fa ∈ Ia =⇒ f ∈ I. (∗) We generalize the above to the following: Theorem. Let K be an algebraically closed field, I be a primary ideal in K[X,Y ]. Then the following conditions are equivalent: (1) ∀f ∈ K[X,Y ] ∀a ∈ Km fa ∈ Ia =⇒ f ∈ I, (2) IY is radical. We also investigate the case where the ideal I is not primary. The original proof by W. Jarnicki, L. O’Carroll and T. Winiarski uses comprehensive Gröbner bases and cannot be carried over to the general case. Our approach