THE BETTI NUMBERS OF FORESTS

By Sean Jacques, Mordechai Katzman. 2005

53 citations recorded by the source corpus.

This paper produces a recursive formula of the Betti numbers of certain Stanley- Reisner ideals (graph ideals associated to forests). This gives a purely combinatorial definition of the projective dimension of these ideals, which turns out to be a new numerical invariant of forests. Finally, we propose a possible extension of this invariant to general graphs.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/c7f017a596614e5a20955bff13c0426827b07531

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In Algebra and Number Theory

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Citations in the corpus

Combinatorial symbolic powers (2006, methodology): “In fact, there is a close connection between applying this method for secant ideals and for differential powers.”
Algebraic Properties of the Path Ideal of a Tree (2009, result): “Our subclass contains the edge ideals of trees, and thus, our formula generalizes a result of Jacques and Katzman [12].”
Monomial ideals, edge ideals of hypergraphs, and their graded Betti numbers (2006, background): “Jacques [19] and Jacques and Katzman [20] first proved Corollary 5.10 in the special case that G is a forest, a subclass of chordal graphs.”
Resolutions of square-free monomial ideals via facet ideals: a survey (2006, methodology): “Katzman [33] shows that even if I(G) is an edge ideal, the graded Betti numbers are not independent of char(k).”
Splittable ideals and the resolutions of monomial ideals (2005, methodology): “An examination of the papers [3, 11, 17, 18, 19, 21, 27] reveals that these formulas provide the basis for most of the known results on the numbers βi,j(I(G)).”