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Register a new userThe monomial ideal of independent sets associated to a graph
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Authors
Oana Olteanu
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Independent sets play a key role into the study of graphs and important problems arising in graph theory reduce to them. We define the monomial ideal of independent sets associated to a finite simple graph and describe its homological and algebraic invariants in terms of the combinatorics of the graph. We compute the minimal primary decomposition and characterize the Cohen--Macaulay ideals. Moreover, we provide a formula for computing the Betti numbers, which depends only on the coefficients of the independence polynomial of the graph.
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We introduce the combinatorial Lyubeznik resolution of monomial ideals. We prove that this resolution is isomorphic to the usual Lyubezbnik resolution. As an application, we give a combinatorial method to determine if an ideal is a Lyubeznik ideal. Furthermore, the minimality of the Lyubeznik resolution is characterized and we classify all the Lyubeznik symbols using combinatorial criteria. We get a combinatorial expression for the projective dimension, the length of Lyubeznik, and the arithmetical rank of a monomial ideal. We define the Lyubeznik totally ideals as those ideals that yield a minimal free resolution under any total order. Finally, we present that for a family of graphics, that their edge ideals are Lyubeznik totally ideals.
We prove a characterization of the j-multiplicity of a monomial ideal as the normalized volume of a polytopal complex. Our result is an extension of Teissiers volume-theoretic interpretation of the Hilbert-Samuel multiplicity for m-primary monomial ideals. We also give a description of the epsilon-multiplicity of a monomial ideal in terms of the volume of a region.
In this paper we develop a new technique to compute the Betti table of a monomial ideal. We present a prototype implementation of the resulting algorithm and we perform numerical experiments suggesting a very promising efficiency. On the way of describing the method, we also prove new constraints on the shape of the possible Betti tables of a monomial ideal.
Let $mathcal{A}={{bf a}_1,ldots,{bf a}_n}subsetBbb{N}^m$. We give an algebraic characterization of the universal Markov basis of the toric ideal $I_{mathcal{A}}$. We show that the Markov complexity of $mathcal{A}={n_1,n_2,n_3}$ is equal to two if $I_{mathcal{A}}$ is complete intersection and equal to three otherwise, answering a question posed by Santos and Sturmfels. We prove that for any $rgeq 2$ there is a unique minimal Markov basis of $mathcal{A}^{(r)}$. Moreover, we prove that for any integer $l$ there exist integers $n_1,n_2,n_3$ such that the Graver complexity of $mathcal{A}$ is greater than $l$.
Let $I_1,dots,I_n$ be ideals generated by linear forms in a polynomial ring over an infinite field and let $J = I_1 cdots I_n$. We describe a minimal free resolution of $J$ and show that it is supported on a polymatroid obtained from the underlying representable polymatroid by means of the so-called Dilworth truncation. Formulas for the projective dimension and Betti numbers are given in terms of the polymatroid as well as a characterization of the associated primes. Along the way we show that $J$ has linear quotients. In fact, we do this for a large class of ideals $J_P$, where $P$ is a certain poset ideal associated to the underlying subspace arrangement.
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