Algorithmic Properties of Polynomial Rings

In this paper we investigate how algorithms for computing heights, radicals, unmixed and primary decompositions of ideals can be lifted from a Noetherian commutative ring R to polynomial rings over R. It is a standard problem in mathematics to study which properties of a mathematical structure are preserved in derived structures. A typical result of this kind is the Hilbert Basis Theorem which says that polynomial rings over Noetherian commutative rings are Noetherian. In this paper we prove that certain algorithmic properties of coefficient rings are preserved in polynomial rings. The proofs are based on lifting techniques. Studying algorithmic properties of polynomial rings by means of lifting techniques has a long history and different lifting algorithms have been proposed. In our approach two fundamental and, compared to classical methods, efficient algorithmic tools are used in this lifting process: Gröbner bases (Buchberger 1965, 1970) and an algorithm which can be considered as a generalization of Ritt's prime decomposition algorithm for radicals (Ritt (1950), Wu (1984)). In order to ensure the existence of these algorithms (see Trinks (1978), Zacharias (1978), Schaller (1978)) we assume that linear equations are solvable in the coefficient ring R, i.e. ideal membership is decidable and bases of syzygy modules are computable in R. Note that solvability of linear equations itself is an algorithmic property which is preserved in polynomial rings. In this paper we study whether this is also true for the algorithmic properties (1) heights of ideals are computable, (2) radicals of ideals are computable, (3) unmixed decompositions of ideals are computable, (4) primary decompositions of ideals are computable. More precisely, let R be a Noetherian commutative ring with identity and assume that linear equations are solvable in R. We want to know for each of these four algorithmic properties whether it holds in R[x 1 ,. .. , x n ] if it holds in R. We give complete solutions for the four lifting problems by proving the following results: (1) If heights of ideals are computable in R then heights of ideals are computable in R[x 1 ,. .. , x n ]. We show that only one Gröbner basis in R[x 1 ,. .. , x n ] with respect to an arbitrary order and the heights of some ideals in R have to be computed in order to determine the height of an ideal in R[x 1 ,. (2) We give necessary …