Results 1  10
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93
Upper bounds for the Betti numbers of a given Hilbert function
 Comm. Algebra
, 1993
"... 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 ..."
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Cited by 56 (0 self)
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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 kvectorspace. I has an essentially unique minimal graded free resolution
A Survey of Graph Pebbling
 Congr. Numer
, 1999
"... We survey results on the pebbling numbers of graphs as well as their historical connection with a numbertheoretic 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 gr ..."
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Cited by 31 (13 self)
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We survey results on the pebbling numbers of graphs as well as their historical connection with a numbertheoretic 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.
Generating Differential Invariants
, 2007
"... The equivariant method of moving frames is used to specify systems of generating differential invariants for finitedimensional Lie group actions. ..."
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Cited by 29 (17 self)
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The equivariant method of moving frames is used to specify systems of generating differential invariants for finitedimensional Lie group actions.
Thresholds for Families of Multisets, With an Application to Graph Pebbling
, 2000
"... In this paper we prove two multiset analogs of classical results. We prove a multiset analog of Lovász's version of the KruskalKatona Theorem and an analog of the Bollob asThomason threshold result. As a corollary we obtain the existence of pebbling thresholds for arbitrary graph sequences. In add ..."
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Cited by 22 (17 self)
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In this paper we prove two multiset analogs of classical results. We prove a multiset analog of Lovász's version of the KruskalKatona Theorem and an analog of the Bollob asThomason 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.
Standard bases and geometric invariant theory. I. Initial ideals and state polytopes
 J. Symbolic Comput
, 1988
"... We characterise the set of monomial ideals that can occur as the initial ideals of a given homogeneous ideal Q in a polynomial ring, as one varies the monomial order used within a fixed coordinate system. This set is canonically bijective to the set of vertices of the state polytope of Q, a convex p ..."
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Cited by 18 (3 self)
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We characterise the set of monomial ideals that can occur as the initial ideals of a given homogeneous ideal Q in a polynomial ring, as one varies the monomial order used within a fixed coordinate system. This set is canonically bijective to the set of vertices of the state polytope of Q, a convex polytope arising in the geometric invariant theory of the Hilbert point of Q. 1.
HBases for Polynomial Interpolation and System Solving
, 1999
"... 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 Hbases and Grobner bases and show parallels between the two conce ..."
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Cited by 14 (3 self)
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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 Hbases 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 Hbases 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 Hbasis, 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, Hbases are not unique. We give in section 6 an exact description of the structure of Hbases 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 Hbases in place of Grobner bases and then finally in section 8 how Hbases can be used to solve multivariate polynomial interpolation problems. H. M. Moller and T. Sauer / Hbases for interpolation and system solving 3 2. Hbases 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...
List Decoding of qary ReedMuller Codes
 IEEE Trans. Inform. Theory
, 2004
"... The qary ReedMuller codes RMq(u, m) of length n = qm are a generalization of ReedSolomon codes, which allow polynomials in m variables to encode the message. Using an idea of reducing the multivariate case to univariate case, randomized listdecoding algorithms for ReedMuller codes were given in ..."
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Cited by 13 (1 self)
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The qary ReedMuller codes RMq(u, m) of length n = qm are a generalization of ReedSolomon codes, which allow polynomials in m variables to encode the message. Using an idea of reducing the multivariate case to univariate case, randomized listdecoding algorithms for ReedMuller codes were given in [1] and [27]. The algorithm in [27] is an improvement of the algorithm in [1], it works for up to E < n(1 − √ 2u/q) errors but is applicable only to codes RMq(u, m) with u < q/2. In this paper, we will propose some deterministic listdecoding algorithms for qary ReedMuller codes. Viewing qary ReedMuller codes as codes from order domains, we present a listdecoding algorithm for qary ReedMuller codes, which is a straightforward generalization of the listdecoding algorithm of ReedSolomon codes in [9]. The algorithm works for up to n(1 − m+1 √ u/q) m − 1 errors, and it is applicable to codes RMq(u, m) with u < q. The algorithm can be implemented to run in time polynomial in the length of the codes. Following [12], we show that qary ReedMuller codes are subfield subcodes of ReedSolomon codes. We then present a second listdecoding algorithm for qary ReedMuller codes. This algorithm works for codes with any rates, and achieves an errorcorrection bound n(1 − √ (n − d)/n) − 1. So the second algorithm achieves a better errorcorrection bound than the algorithm in [27], since when u is small, n(1 − √ (n − d)/n) = n(1 − √ u/q). The implementation of the second algorithm requires O(n) field operations in Fq and O(n3) field operations in Fqm under some assumption. Also, we prove that qary ReedMuller codes can be described as onepoint AG codes. And using the algorithm of AG codes in [9], we give a third listdecoding
Consecutive cancellations in Betti numbers
 Proc. Amer. Math. Soc
"... 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 t ..."
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Cited by 12 (3 self)
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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
Socles of Buchsbaum modules, complexes and posets
 Advances in Math. 222
, 2009
"... 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 po ..."
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Cited by 11 (4 self)
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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 2kdimensional 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