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Abstract: We introduce the notion of independent sequences with respect to a monomial order by using the least terms of polynomials vanishing at the sequence. Our main result shows that the Krull dimension of a Noetherian ring is equal to the supremum of the length of independent sequences. The proof has led to other notions of independent sequences, which have interesting applications. For example, we can show that is the maximum number of analytically independent elements in an arbitrary ideal J of a local ring R and that if are (not necessarily finitely generated) subalgebras of a finitely generated algebra over a Noetherian Jacobson ring. [Copyright &y& Elsevier]

Abstract: Given a linear action of a group G on a K-vector space V, we consider the invariant ring , where is the dual space. We are particularly interested in the case where and G is the group of all upper unipotent matrices or the group of all upper triangular matrices in . In fact, we determine for and . The result is a complete intersection for all values of n and q. We present explicit lists of generating invariants and their relations. This makes an addition to the rather short list of “doubly parametrized” series of group actions whose invariant rings are known to have a uniform description. [Copyright &y& Elsevier]