A numerical property of Hilbert functions and lex segment ideals

TL;DR

This paper introduces fractal expansions and functions that characterize O-sequences, encode information on lex segment ideals, and classify Hilbert functions of bigraded algebras, providing new tools for understanding algebraic structures.

Contribution

It presents fractal expansions and functions as novel methods to analyze Hilbert functions and lex segment ideals, offering a new classification approach.

Findings

Fractal expansions characterize O-sequences.

Fractal functions classify Hilbert functions of bigraded algebras.

New insights into lex segment ideals through fractal sequences.

Abstract

We introduce the fractal expansions, sequences of integers associated to a number. We show that these sequences characterize the O-sequences and encode some information on lex segment ideals. Moreover, we introduce a numerical functions called fractal functions, and we show that fractal functions can be used to classify the Hilbert functions of bigraded algebras.