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STANLEY DECOMPOSITIONS AND HILBERT DEPTH IN THE KOSZUL COMPLEX
, 909
"... Abstract. Stanley decompositions of multigraded modules M over polynomials rings have been discussed intensively in recent years. There is a natural notion of depth that goes with a Stanley decomposition, called the Stanley depth. Stanley conjectured that the Stanley depth of a module M is always at ..."
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Abstract. Stanley decompositions of multigraded modules M over polynomials rings have been discussed intensively in recent years. There is a natural notion of depth that goes with a Stanley decomposition, called the Stanley depth. Stanley conjectured that the Stanley depth of a module M is always at least the (classical) depth of M. In this paper we introduce a weaker type of decomposition, which we call Hilbert decomposition, since it only depends on the Hilbert function of M, and an analogous notion of depth, called Hilbert depth. Since Stanley decompositions are Hilbert decompositions, the latter set upper bounds to the existence of Stanley decompositions. The advantage of Hilbert decompositions is that they are easier to find. We test our new notion on the syzygy modules of the residue class field of K[X1,...,Xn] (as usual identified with K). Writing M(n, k) for the kth syzygy module, we show that the Hilbert depth of M(n, 1) is ⌊(n+1)/2⌋. Furthermore, we show that, for n> k ≥ ⌊n/2⌋, the Hilbert depth of M(n, k) is equal to n − 1. We conjecture that the same holds for the Stanley depth. For the range n/2> k> 1, it seems impossible to come up with a compact formula for the Hilbert depth. Instead, we provide very precise asymptotic results as n becomes large. 1.
ON THE STANLEY DEPTH OF SQUAREFREE VERONESE IDEALS
, 2009
"... Let K be a field and S = K[x1,...,xn]. In 1982, Stanley defined what is now called the Stanley depth of an Smodule M, denoted sdepth(M), and conjectured that depth(M) ≤ sdepth(M) for all finitely generated Smodules M. This conjecture remains open for most cases. However, Herzog, Vladoiu and Zhe ..."
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Let K be a field and S = K[x1,...,xn]. In 1982, Stanley defined what is now called the Stanley depth of an Smodule M, denoted sdepth(M), and conjectured that depth(M) ≤ sdepth(M) for all finitely generated Smodules M. This conjecture remains open for most cases. However, Herzog, Vladoiu and Zheng recently proposed a method of attack in the case when M = I/J with J ⊂ I being monomial Sideals. Specifically, their method associates M with a partially ordered set. In this paper we take advantage of this association by using combinatorial tools to analyze squarefree Veronese ideals in S. In particular, if In,d is the squarefree Veronese ideal generated by all squarefree monomials of ( n) degree d, we show that if 1 ≤ d ≤ n < 5d + 4, then sdepth(In,d) = d+1 / ( n) d + d, and ⌊ ( n) if d ≥ 1 and n ≥ 5d + 4, then d + 3 ≤ sdepth(In,d) ≤ d+1 / ( n) d + d.
Computing the Stanley depth
 J. Algebra
"... Abstract. Let Q and Q ′ be two monomial primary ideals of a polynomial algebra S over a field. We give an upper bound for the Stanley depth of S/(Q ∩Q ′ ) which is reached if Q,Q ′ are irreducible. Also we show that Stanley’s Conjecture holds for Q1 ∩ Q2, S/(Q1 ∩ Q2 ∩ Q3), (Qi)i being some irreducib ..."
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Abstract. Let Q and Q ′ be two monomial primary ideals of a polynomial algebra S over a field. We give an upper bound for the Stanley depth of S/(Q ∩Q ′ ) which is reached if Q,Q ′ are irreducible. Also we show that Stanley’s Conjecture holds for Q1 ∩ Q2, S/(Q1 ∩ Q2 ∩ Q3), (Qi)i being some irreducible monomial ideals of S.
RESEARCH STATEMENT
"... Broadly speaking, my research interests run the breadth of combinatorics, from weighted graph packing to preference aggregation for the Major League Baseball draft, with stops at the combinatorial calculation of algebraic invariants (such as Stanley depth), and the relationships between graph expans ..."
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Broadly speaking, my research interests run the breadth of combinatorics, from weighted graph packing to preference aggregation for the Major League Baseball draft, with stops at the combinatorial calculation of algebraic invariants (such as Stanley depth), and the relationships between graph expansion and selfish traffic routing. By inclination, and as a result of my training in the interdisciplinary Algorithms, Combinatorics, and Optimization program1, my research interests span a large number of combinatorial topics and involve a variety of different collaborators. Going forward, in addition to continuing to expand the breadth of my research and collaborations, I intend to continue work on three areas in which I have had particular research success; combinatorial Stanley depth of monomial ideals, dimensionlike measures for posets, and spectral graph theory and applications. In this section I provide a less technical overview of my efforts in these areas while providing a more detailed summary in the following sections. Since it was introduced, one of the fundamental problems in the study of Stanley depth has been to develop a finite time (not necessarily efficient, just finite) means of calculating the Stanley depth of a given module. Recently, a group of commutative algebraists has made progress on this problem by providing a correspondence between the Stanley depth of monomial ideals and a class