Lukas Katthän
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View article: The path-missing and path-free complexes of a directed graph
The path-missing and path-free complexes of a directed graph Open
We study two simplicial complexes arising from a directed graph $G = (V, E)$ with two chosen vertices $s$ and $t$: the *path-free complex*, consisting of all subsets $F \subseteq E$ that contain no path from $s$ to $t$, and the *path-missi…
View article: Which series are Hilbert series of graded modules over standard multigraded polynomial rings?
Which series are Hilbert series of graded modules over standard multigraded polynomial rings? Open
Consider a polynomial ring R with the ‐grading where the degree of each variable is a standard basis vector. In other words, R is the homogeneous coordinate ring of a product of n projective spaces. In this setting, we characterize the for…
View article: Linear Maps in Minimal Free Resolutions of Stanley-Reisner Rings
Linear Maps in Minimal Free Resolutions of Stanley-Reisner Rings Open
In this short note we give an elementary description of the linear part of the minimal free resolution of a Stanley-Reisner ring of a simplicial complex Δ . Indeed, the differentials in the linear part are simply a compilation of restricti…
View article: The Lecture Hall cone as a toric deformation
The Lecture Hall cone as a toric deformation Open
The Lecture Hall cone is a simplicial cone whose lattice points naturally correspond to Lecture Hall partitions. The celebrated Lecture Hall Theorem of Bousquet-Mélou and Eriksson states that a particular specialization of its multivariate…
View article: The symmetric signature of cyclic quotient singularities
The symmetric signature of cyclic quotient singularities Open
The symmetric signature is an invariant of local domains which was recently\nintroduced by Brenner and the first author in an attempt to find a replacement\nfor the $F$-signature in characteristic zero. In the present note we compute\nthe …
View article: Spanning Lattice Polytopes and the Uniform Position Principle
Spanning Lattice Polytopes and the Uniform Position Principle Open
A lattice polytope $P$ is called IDP if any lattice point in its $k$th dilate is a sum of $k$ lattice points in $P$. In 1991 Stanley proved a strong inequality in Ehrhart theory for IDP lattice polytopes. We show that his conclusion holds …
View article: When is a polynomial ideal binomial after an ambient automorphism?
When is a polynomial ideal binomial after an ambient automorphism? Open
Can an ideal I in a polynomial ring k[x] over a field be moved by a change of coordinates into a position where it is generated by binomials $x^a - cx^b$ with c in k, or by unital binomials (i.e., with c = 0 or 1)? Can a variety be moved i…
View article: The Homology of Connective Morava $E$-theory with coefficients in $\mathbb{F}_p$
The Homology of Connective Morava $E$-theory with coefficients in $\mathbb{F}_p$ Open
Let $e_n$ be the connective cover of the Morava $E$-theory spectrum $E_n$ of height $n$. In this paper we compute its homology $H_*(e_n;\mathbb{F}_p)$ for any prime $p$ and $n \leq 4$ up to possible multiplicative extensions. In order to a…
View article: Finding binomials in polynomial ideals
Finding binomials in polynomial ideals Open
We describe an algorithm which finds binomials in a given ideal $I\subset\mathbb{Q}[x_1,\dots,x_n]$ and in particular decides whether binomials exist in $I$ at all. Binomials in polynomial ideals can be well hidden. For example, the lowest…
View article: Stanley depth and the lcm-lattice
Stanley depth and the lcm-lattice Open
In this paper we show that the Stanley depth, as well as the usual depth, are essentially determined by the lcm-lattice. More precisely, we show that for quotients of monomial ideals , both invariants behave monotonic with respect to certa…
View article: Ehrhart Theory of Spanning Lattice Polytopes
Ehrhart Theory of Spanning Lattice Polytopes Open
© 2018 Oxford University Press. All rights reserved. The key object in the Ehrhart theory of lattice polytopes is the numerator polynomial of the rational generating series of the Ehrhart polynomial, called h-polynomial. In this article, w…
View article: Linearity in minimal resolutions of monomial ideals
Linearity in minimal resolutions of monomial ideals Open
Let $S = k[x_1, \dotsc, x_n]$ be a polynomial ring over a field $k$ and let $M$ be a graded $S$-module with minimal free resolution $\mathbb{F}_\bullet$. Its linear part $lin(\mathbb{F}_\bullet)$ is obtained by deleting all non-linear entr…
View article: Hilbert series of modules over positively graded polynomial rings
Hilbert series of modules over positively graded polynomial rings Open
In this note, we give examples of formal power series satisfying certain conditions that cannot be realized as Hilbert series of finitely generated modules. This answers to the negative a question raised in a recent article by the second a…
View article: Which series are Hilbert series of graded modules over polynomial rings?
Which series are Hilbert series of graded modules over polynomial rings? Open
Let $S$ be a multigraded polynomial ring such that the degree of each variable is a unit vector; so $S$ is the homogeneous coordinate ring of a product of projective spaces. In this setting, we characterize the formal Laurent series which …
View article: Betti Posets and the Stanley Depth
Betti Posets and the Stanley Depth Open
Let $S$ be a polynomial ring and let $I \subseteq S$ be a monomial ideal. In this short note, we propose the conjecture that the Betti poset of $I$ determines the Stanley projective dimension of $S/I$ or $I$. Our main result is that this c…