Let R: = k[X1,..., XN] be the polynomial ring in N indeterminates over a field k of characteristic 0 with deg(Xi) = 1 for i = 1,..., N, and let I be a homogeneous ideal of R. The Hilbert function of I is the function from N to N which associates to every natural number d the dimension of Id as a k-vectorspace. I has an essentially unique minimal graded free resolution

We survey results on the pebbling numbers of graphs as well as their historical connection with a number-theoretic question of Erdös and Lemke. We also present new results on two probabilistic pebbling considerations, first the random graph threshold for the property that the pebbling number of a graph equals its number of vertices, and second the pebbling threshold function for various natural graph sequences. Finally, we relate the question of the existence of pebbling thresholds to a strengthening of the normal property of posets, and show that the multiset lattice is not supernormal.

The equivariant method of moving frames is used to specify systems of generating differential invariants for finite-dimensional Lie group actions.

In this paper we prove two multiset analogs of classical results. We prove a multiset analog of Lovász's version of the Kruskal-Katona Theorem and an analog of the Bollob as-Thomason threshold result. As a corollary we obtain the existence of pebbling thresholds for arbitrary graph sequences. In addition, we improve both the lower and upper bounds for the `random pebbling' threshold of the sequence of paths.

Abstract: Let I be a homogeneous ideal in a polynomial ring over a field. By Macaulay’s Theorem, there exists a lexicographic ideal L with the same Hilbert function as I. We prove that the graded Betti numbers of I are obtained from those of L by a sequence of consecutive cancellations. Throughout the paper, S = k[x1,..., xn] is a polynomial ring over a field k, and I is a homogeneous ideal in S. First, we recall the definition of a lexicographic ideal. A monomial ideal M is called lexicographic if for every j ∈ N the space Mj is spanned by the first dim(Mj) monomials in the lexicographic order. By Macaulay’s Theorem [Ma], there exists a lexicographic ideal L with the same Hilbert function as I. Throughout the paper we denote this ideal by L. It was proved by Bigatti, Hulett, Pardue, that the graded Betti numbers βi,j(S/L) are greater or equal to the corresponding graded Betti numbers βi,j(S/I) (all graded Betti numbers are taken over S). The Hilbert function can be computed from the graded Betti numbers as follows (cf. [Ei]): dimk(S/I)j t j = j=0 dimk(S/L)j t j = j=0 � ∞ �n j=0 i=0 (−1)iβi,j(S/I) tj (1 − t) n � ∞ �n j=0 i=0 (−1)iβi,j(S/L) tj (1 − t) n These equalities imply that the graded Betti numbers βi,j(S/I) and βi,j(S/L) are related, as described below. Given a sequence of numbers {ci,j}, we obtain a new sequence by a cancellation as follows: fix a j, and choose i and i ′ so that one of the numbers is odd and the other is even; then replace ci,j

This paper is mainly addressed more to readers with background in Numerical Analysis than to those who are more familiar with Commutative Algebra. Therefore, we first explain at an elementary level the underlying common structure of H--bases and Grobner bases and show parallels between the two concepts. The dimensions of the vector spaces I " P d are described using the Hilbert function in section 3. There we also present modules of syzygies, a tool for analyzing linear dependencies among polynomials, and their dimensions. In section 4 we summarize the known methods for computing H--bases and turn then, in section 5, to regular sequences. There we show in thm. 5.3, that n polynomials in n variables are already an H--basis, if their homogeneous parts of maximal degree have no other zero in common than 0. In contrast to Grobner bases, which are under mild additional conditions uniquely determined by the given ideal and the term ordering, H--bases are not unique. We give in section 6 an exact description of the structure of H--bases for a given ideal. The last two sections are devoted to applications. First we show in section 7 that Stetter's Eigenmethod also works with H--bases in place of Grobner bases and then finally in section 8 how H--bases can be used to solve multivariate polynomial interpolation problems. H. M. Moller and T. Sauer / H--bases for interpolation and system solving 3 2. H--bases and Grobner bases Here and in the following sections we consider polynomials in n variables x 1 ; : : : ; x n with coefficients from a field K , say the field Q of rational numbers or the field R of real numbers. For short, we write P := K [x 1 ; : : : ; x n ] : Given a set F ae P, the set I := 8 ! : X f2F h f f j h f 2 P; only finitely many h f 6= 0 9 = ; is the i...

We study the multiplication of multivariate power series. We show that over large enough fields, the bilinear complexity of the product modulo a monomial ideal M is bounded by the product of the regularity of M by the degree of M. In some special cases, such as partial degree truncation, this estimate carries over to total complexity. This leads to complexity improvements for some basic algorithms with algebraic numbers, and some polynomial system solving algorithms.

The socle of a graded Buchsbaum module is studied and is related to its local cohomology modules. This algebraic result is then applied to face enumeration of Buchsbaum simplicial complexes and posets. In particular, new necessary conditions on face numbers and Betti numbers of such complexes and posets are established. These conditions are used to settle in the affirmative Kühnel’s conjecture for the maximum value of the Euler characteristic of a 2k-dimensional simplicial manifold on n vertices as well as Kalai’s conjecture providing a lower bound on the number of edges of a simplicial manifold in terms of its dimension, number of vertices, and the first Betti number. 1

Abstract. We discuss a particular problem of enumerating rational curves on a Grassmannian from several perspectives, including systems theory, real enumerative geometry, and symbolic computation. We also present a new transversality result, showing this problem is enumerative in all characteristics. While it is well-known how this enumerative problem arose in mathematical physics and also its importance to the development of quantum cohomology, it is less known how it arose independently in mathematical systems theory. We describe this second story. 1.

Abstract. In an earlier work the authors described a mechanism for lifting monomial ideals to reduced unions of linear varieties. When the monomial ideal is Cohen-Macaulay (including Artinian), the corresponding union of linear varieties is arithmetically Cohen-Macaulay. The first main result of this paper is that if the monomial ideal is Artinian then the corresponding union is in the Gorenstein linkage class of a complete intersection (glicci). This technique has some interesting consequences. For instance, given any (d+1)-times differentiable O-sequence H, there is a non-degenerate arithmetically Cohen-Macaulay reduced union of linear varieties with Hilbert function H which is glicci. In other words, any Hilbert function that occurs for arithmetically Cohen-Macaulay schemes in fact occurs among the glicci schemes. This is not true for licci schemes. Modifying our technique, the second main result is that any Cohen-Macaulay Borel-fixed monomial ideal is glicci. As a consequence, all arithmetically Cohen-Macaulay subschemes of projective