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Showing 1 to 10 of 10 for “"monomial ideals"”.
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Powers of Monomial Ideals
We study monomial ideals in polynomial rings in two variables x, y over a field K. We determine various monomial ideals I such that Ik = (Mn)k where M is the maximal ideal generated by x, y and k is the least such integer. This is related to the well-known notion of Ratliff-Rush closures of an …
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Monomial Ideals, N-Lists, and Smallest Graded Betti Numbers
… an Artinian Hilbert function H , consider all ideals I ⊂ R such that H(R/I) = H . Then the graded betti numbers which correspond to these ideals form a finite set which we partially order component-wise. It is known that this set has a largest element, but may fail to have a smallest …
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Algebraic Reliability. Monomial ideals applied to multi-state system reliability
… un acercamiento algebraico basado en ideales monomiales. Consideramos un sistema como un conjunto de componentes junto con una función de estructura. Tanto las componentes como el sistema se dicen binarios si u ́nicamente pueden alcanzar dos niveles o estados de funcionamiento diferentes: 0 es …
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Combinatorial Koszul Homology: computations and Applications
… of the Koszul homology and Betti numbers of monomial ideals, the main goals od this thesis are the following: Analyze the Koszul homology of monomial ideals and apply it to describe the structure of monomial ideals. Describe algorithms to perform efficient computations of the homological …
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The Combinatorics of Involutive Bases: Theory, algorithms and applications
In this thesis, we conduct an in-depth study of monomial ideals and other related structures in commutative algebra. Our primary tool for exploring the connections between these algebraic objects and combinatorics is the use of involutive bases, along with other types of Gröbner bases that exhibit …
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Koszul Complexes, Local Cohomology, Universal Resolutions, and Cohomological Support Varieties
… closure of certain local cohomology modules of ideals generated by regular sequences. We then explore various universal resolutions. Within this, we first provide an overview of DG algebras and universal resolutions. We then provide a survey concerning the relationship between some relatively …
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Monomization of power Ideals and parking functions
… in terms of graphs. Using the symmetry of these ideals, we can find monomial ideals which preserve much of the structure of the zonotopal algebras while being computationally very efficient, in particular far faster than Gröbner basis techniques. We extend this monomization theory from the known …
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Some Duality Results in Homological Algebra
… examine local cohomology involving square-free monomial ideals. We provide an alternate proof of one of Mustataˇ's constructions for computing H˙IR and use it to provide a simpler proof of E. Miller's duality result relating H˙IR to H˙m R/I , also showing that the latter result …
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Unique Signed Minimal Wiring Diagrams and the Stanley-Reisner Correspondence
… diagrams rely on the primary decomposition of ideals in polynomial rings.</p> <p>Stanley-Reisner theory provides a one-to-one correspondence between squarefree monomial ideals and abstract simplicial complexes. In this work, we use this correspondence to determine conditions under which a given …
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Derived categories of coherent sheaves on rational homogeneous manifolds
… Brion (thm. 4.1.1) and cellular resolutions of monomial ideals a la Bayer/Sturmfels. We give a new proof of the theorem of Beilinson on $\mathbb{P}^n$ in order to show that this approach might work in general. We also prove theorem 4.2.5 which gives a concrete description of certain functors …