Abstract: If D is a domain with quotient field K, then the ring of integer-valued polynomials in n indeterminates over D, Int(D^n), consists of those polynomials in K[x_1,...,x_n] which give a value in D for every argument in D^n. Skolem and Nullstellensatz properties are analogues of the weak Nullstellensatz and Hilbert's Nullstellensatz, respectively, for the ring of integer-valued polynomials in several indeterminates, Int(D^n).
More precisely, Int(D^n) is said to have the Nullstellensatz property if for all polynomials f_1, ..., f_m and f in Int(D^n) such that for all a in D^n, f(a) is in the ideal of D generated by f_1(a), ..., f_m(a), it is true that f lies in the radical of the ideal of Int(D^n) generated by f_1, ...,f_m.
Int(D^n) is said to have the Skolem property, if for every finite set of polynomials f_1, ...,f_m in Int(D^n) satisfying f_1(a)D + ... + f_m(a)D = D for all a in D^n, it is true that the ideal of Int(D^n) generated by f_1, ..., f_m is Int(D^n).
For Noetherian D we show the equivalence of Skolem property and Nullstellensatz property and characterize among Noetherian domains those for which Int(D^n) has the Nullstellensatz property. This extends a criterion previously shown for special cases by Brizolis and Chabert to all Noetherian domains D.
2000 Mathematics Subject Classification: Primary 13F20; Secondary 13B25, 11C08, 14A25.
PDF: S. Frisch, Nullstellensatz and Skolem properties for integer-valued polynomials