Face Vectors And Hilbert Functions
Chong, Kai Fong Ernest
This dissertation builds upon two well-known theorems: Macaulay's theorem on the Hilbert functions of graded ideals, and the Kruskal-Katona theorem on the face vectors of simplicial complexes. There are four parts in the dissertation. The first part covers the preliminaries, and also introduces the Frankl-F¨redi- u Kalai theorem, which is a result in extremal graph theory concerning r-colorable k-uniform hypergraphs. In Part II, we use the algebraic notion of Macaulay-Lex rings to show that Macaulay's theorem, the Kruskal-Katona theorem, and the Frankl-F¨redi-Kalai u theorem, are in fact three special cases of one main theorem. In particular, we introduce a class of quotient rings with combinatorial significance, which we call colored quotient rings, and we characterize all possible Macaulay-Lex colored quotient rings. This gives a simultaneous generalization of the Clements-Lindstr¨m o theorem and the Frankl-F¨redi-Kalai theorem. As part of our characterization, u we construct two new classes of Macaulay-Lex rings, thereby answering a question posed by Mermin-Peeva (2007). Using related ideas, we also show that with the exception of very special cases, the f -vectors of generalized colored simplicial complexes (resp. multicomplexes) are never characterized by "reverse-lexicographic" simplicial complexes (resp. multicomplexes). In Part III, we look at the famous Eisenbud-Green-Harris (EGH) conjecture. It proposes an extension of Macaulay's theorem to graded ideals containing regular sequences. We apply liaison theory to prove that the EGH conjecture is true for a certain subclass of graded licci ideals. As an important consequence, we show that the conjecture holds for Gorenstein ideals in the three-variable case. This is the first time liaison theory appears in the context of the EGH conjecture. In Part IV, we give complete numerical characterizations for the flag f -vectors of Cohen-Macaulay completely balanced complexes, and the fine f -vectors of generalized colored complexes. En route to these characterizations, we introduce the notion of Macaulay decomposability, which implies vertex-decomposability, and we generalize the notion of Macaulay representations. We also construct a poset P of such generalized Macaulay representations, and show that proving the Kruskal- Katona theorem is equivalent to finding chains in P, while characterizing the flag f -vectors of (non-generalized) colored complexes is equivalent to finding Boolean sublattices in P. A more general statement holds. Macaulay decomposability is also interesting by itself and we use it to extend results by Babson-Novik, Biermann-Van Tuyl, and Murai.
face vector; Hilbert function
Billera,Louis J.; Stillman,Michael Eugene
Ph. D., Mathematics
Doctor of Philosophy
dissertation or thesis