Length Of Polynomial Ascending Chains And Primitive Recursiveness

In a polynomial ring K[X 1 ; : : : ; Xn ] over a field, let I 0 ae I 1 ae \Delta \Delta \Delta ae I s be a strictly ascending chain of ideals, with the condition that every I i can be generated by elements of degree not greater than f(i). A. Seidenberg showed that there is a bound on the length s of such a chain depending only on n and f , which is recursive in f for every n and primitive recursive in f for n = 2. In this paper we give a better bound, expressed in a rather simple way in terms of f , which is attained when f is an increasing function. We prove that it is primitive recursive in f for all n. We also show that, on the contrary, there is no bound which is primitive recursive in n in general

Revlex Standard Bases Of Generic Complete Intersections

In this paper we study the Hilbert-Samuel function of a generic standard graded K-algebra K[X 1 ; : : : ; Xn ]=(g 1 ; : : : ; gm ) when refined by an (`)-adic filtration, ` being a linear form. From this we obtain a structure theorem which describes the stairs of a generic complete intersection for the degree-reverse-lexicographic order. We show what this means for generic standard (or Gröbner) bases for this order; in particular, we consider an "orderly filling up" conjecture, and we propose a strategy for the standard basis algorithm which could be useful in generic-like cases

An Ackermannian Polynomial Ideal

In this paper we answer the following question of Teo Mora ([Mora91]): Write down a monomial ideal starting with a monomial of degree d, adding a monomial of degree d + 1, another one of degree d + 2, and so on, with every new monomial added not being a multiple of the previous ones; which is the maximal degree one can reach with this construction? We also give some partial results for the opposite problem of finding the lowest bound for this degree, in the case of homogeneous ideals. The paper is organized as follows. In section 1 we state the result concerning Mora's question; sections 2 and 3 contain some preliminaries and the proof, while in section 4 an example is shown and some remarks and a generalization are made. The opposite problem is considered in section 5, where we give the results we have found and make a conjecture