Jeffrey Mermin

Abstract: Macaulay proved in 1927 that every Hilbert function in the polynomial ring R = k[x_1,...,x_n]  is attained by a lexicographic ideal.  We study the combinatorial and homological properties of lexicographic ideals and identify other settings in which lexicographic ideals attain every Hilbert function and other classes of ideals with similar properties.  We develop the theory of compression, which makes many of our arguments possible and leads to shorter new proofs of results of Macaulay, Bigatti, Hulett, and Pardue.  Using compression, we extend results of Green and Clements-Lindstrom and prove a special case of Evans’ Lex-Plus-Powers conjecture.