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Juergen Herzog

Publications and source records attributed to Juergen Herzog.

At least 19 recordsLinked to original sources
arXiv2021

Algebraically rigid simplicial complexes and graphs

We call a simplicial complex algebraically rigid if its Stanley-Reisner ring admits no nontrivial infinitesimal deformations, and call it inseparable if does not allow any deformation to other simplicial complexes. Algebraically rigid simplicial complexes are inseparable. In this paper we study inseparability and rigidity of Stanley-Reisner rings, and apply the general theory to letterplace ideals as well as to edge ideals of graphs. Classes of algebraically rigid simplicial complexes and graphs are identified.

math.AC↗
arXiv2021

On the index of powers of edge ideals

The index of a graded ideal measures the number of linear steps in the graded minimal free resolution of the ideal. In this paper we study the index of powers and squarefree powers of edge ideals. Our results indicate that the index as a function of the power of an edge ideal $I$ is strictly increasing if $I$ has linear relations. Examples show that this need not to be the case for monomial ideals generated in degree greater than two.

math.AC↗
arXiv2014

The coordinate ring of a simple polyomino

In this paper it is shown that a polyomino is balanced if and only if it is simple. As a consequence one obtains that the coordinate ring of a simple polyomino is a normal Cohen-Macaulay domain.

math.AC↗
arXiv2012

Determinantal Facet Ideals

We consider ideals generated by general sets of $m$-minors of an $m\times n$-matrix of indeterminates. The generators are identified with the facets of an $(m-1)$-dimensional pure simplicial complex. The ideal generated by the minors corresponding to the facets of such a complex is called a determinantal facet ideal. Given a pure simplicial complex $Δ$, we discuss the question when the generating minors of its determinantal facet ideal $J_Δ$ form a Gröbner basis and when $J_Δ$ is a prime ideal.

math.AC↗
arXiv2011

Ideals generated by adjacent 2-minors

Ideals generated by adjacent 2-minors are studied. First, the problem when such an ideal is a prime ideal as well as the problem when such an ideal possesses a quadratic Gröbner basis is solved. Second, we describe explicitly a primary decomposition of the radical ideal of an ideal generated by adjacent 2-minors, and challenge the question of classifying all ideals generated by adjacent 2-minors which are radical ideals. Finally, we discuss connectedness of contingency tables in algebraic statistics.

math.AC↗
arXiv2010

Monomial ideals and toric rings of Hibi type arising from a finite poset

In this paper we study monomial ideals attached to posets, introduce generalized Hibi rings and investigate their algebraic and homological properties. The main tools to study these objects are Groebner basis theory, the concept of sortability due to Sturmfels and the theory of weakly polymatroidal ideals.

math.AC↗
arXiv2010

Gröbner bases of syzygies and Stanley depth

Let F. be a any free resolution of a Z^n-graded submodule of a free module over the polynomial ring K[x_1, ..., x_n]. We show that for a suitable term order on F., the initial module of the p'th syzygy module Z_p is generated by terms m_ie_i where the m_i are monomials in K[x_{p+1}, ..., x_n]. Also for a large class of free resolutions F., encompassing Eliahou-Kervaire resolutions, we show that a Gröbner basis for Z_p is given by the boundaries of generators of F_p. We apply the above to give lower bounds for the Stanley depth of the syzygy modules Z_p, in particular showing it is at least p+1. We also show that if I is any squarefree ideal in K[x_1, ..., x_n], the Stanley depth of I is at least of order the square root of 2n.

math.AC↗
arXiv2009

The Betti polynomials of powers of an ideal

For an ideal $I$ in a regular local ring or a graded ideal $I$ in the polynomial ring we study the limiting behavior of the Betti numbers of S/I^k as k goes to infinity. By Kodiyalam's result it is known that in each homological degree the Betti number is a polynomial for large k. We call these polynomials the Kodiyalam polynomials and encode the limiting behavior in their generating polynomial. It is shown that the limiting behavior depends only on the coefficients on the Kodiyalam polynomials in the highest possible degree. For these we exhibit lower bounds in special cases and conjecture that the bounds are valid in general. We also show that the Kodiyalam polynomials have weakly descending degrees and identify a situation where the polynomials have all highest possible degree.

math.AC↗
arXiv2009

Binomial edge ideals and conditional independence statements

We introduce binomial edge ideals attached to a simple graph $G$ and study their algebraic properties. We characterize those graphs for which the quadratic generators form a Gröbner basis in a lexicographic order induced by a vertex labeling. Such graphs are chordal and claw-free. We give a reduced squarefree Gröbner basis for general $G$. It follows that all binomial edge ideals are radical ideals. Their minimal primes can be characterized by particular subsets of the vertices of $G$. We provide sufficient conditions for Cohen--Macaulayness for closed and nonclosed graphs. Binomial edge ideals arise naturally in the study of conditional independence ideals. Our results apply for the class of conditional independence ideals where a fixed binary variable is independent of a collection of other variables, given the remaining ones. In this case the primary decomposition has a natural statistical interpretation

math.AC↗
arXiv2009

Powers of componentwise linear ideals

We give criteria for graded ideals to have the property that all their powers are componentwise linear. Typical examples to which our criteria can be applied include the vertex cover ideals of certain finite graphs.

math.AC↗
arXiv2008

Finite Generation of Algebras Associated to Powers of Ideals

We study generalized symbolic powers and form ideals of powers of ideals and compare their growth with the growth of ordinary powers, and we discuss the question of when the graded rings attached to symbolic powers or to form ideals of powers are finitely generated.

math.AC↗
arXiv2008

Unmixed bipartite graphs and sublattices of the Boolean lattices

The correspondence between unmixed bipartite graphs and sublattices of the oolean lattice is discussed. By using this correspondence, we show the existence of squarefree quadratic initial ideals of toric ideals arising from minimal vertex covers of unmixed bipartite graphs.

math.AC↗
arXiv2008

Skeletons of monomial ideals

In analogy to the skeletons of a simplicial complex and their Stanley--Reisner ideals we introduce the skeletons of an arbitrary monomial ideal $I\subset S=K[x_1,...,x_n]$. This allows us to compute the depth of $S/I$ in terms of its skeleton ideals. We apply these techniques to show that Stanley's conjecture on Stanley decompositions of $S/I$ holds provided it holds whenever $S/I$ is Cohen--Macaulay. We also discuss a conjecture of Soleyman-Jahan and show that it suffices to prove his conjecture for monomial ideals with linear resolution.

math.AC↗
arXiv2007

Bounds for Hilbert coefficients

We compute the Hilbert coefficients of a graded module with pure resolution and discuss lower and upper bounds for these coefficients for arbitrary graded modules.

math.AC↗