where lm(I) denotes the ideal generated by the leading monomials of the elements of I. This condition has already been studied in Bayer et al. (1991) and it has been shown that (1.1) holds for any ideal and any term order if and only if π is flat. In this paper we study condition (1.1) under the additional assumption that R′ is not a general Noetherian commutative ring with identity but a field...

where θ = 1 + √ d 2 if d mod 4 = 1 and θ = √ d if d mod 4 = 2, 3. The purpose of this paper is to compute the extended gcd of to quadratic integers in ring Z[θ]. We assume throughout that Z[θ] is principal ideal ring, but not necessarily an euclidean ring. If [a, b+ cθ] is the module {ax+(b+ cθ)y, x, y ∈ Z}, it can be shown [3] that I is an ideal of Z[θ] if and only if I = [a, b+ cθ]; where a, ...

The regularity part follows from the primary decompositions part, so the heart of this paper is the analysis of the primary decompositions. In [S], this was proved for the primary components of height at most one over the ideal. This paper proves the existence of such a k but does not provide a formula for it. In the paper [SS], Karen E. Smith and myself find explicit k for ordinary and Frobeni...

Let A be an abelian variety over an algebraically closed field. The Chow group CH0(A) of zero-dimensional cycles modulo rational equivalence forms a ring under the Pontryagin product operation with respect to which the subset I of degree zero cycles is an ideal. The nth power of I is shown to agree with the nth power of the algebraic equivalence relation as defined by H. Saito. A result is also...

Introduction 2 Lecture 1. Hermite and Smith normal forms 3 Lecture 2. Integral similarity and the Latimer-MacDuffee-Taussky theorem 8 Lecture 3. Ideal class numbers, integral matrices nonderogatory modulo every prime and maximal orders of number fields 12 Lecture 4. Factorizations of integer matrices as products of elementary matrices, involutions etc 20 Lecture 5. Additive commutators. Solving...

We show that a problem of deciding whether a formula for a multivariate polynomial of n variables over a finite field of characteristic 2 has degree n when reduced modulo a certain Boolean ideal belongs to ⊕ P. When the formula is allowed to have succinct representations as sums of monomials, the problem becomes ⊕ P-complete. key words: degree of multivariate polynomials, finite field of charac...