Subalgebras generated in degree two with minimal Hilbert function

TL;DR

This paper investigates subalgebras of polynomial rings generated in degree two that have minimal Hilbert functions, linking algebraic properties to combinatorial lattice path problems and proposing a conjecture about their generators.

Contribution

It introduces a combinatorial approach to characterize subalgebras with minimal Hilbert functions and conjectures their generators are initial Lex or RevLex segments.

Findings

Reduction of the problem to counting maximal lattice paths

Conjecture on generators being initial Lex or RevLex segments

Connection between algebraic properties and combinatorial structures

Abstract

What can be said about the subalgebras of the polynomial ring, with minimal or maximal Hilbert function? This question was discussed in a recent paper by M. Boij and A. Conca. In this paper we study the subalgebras generated in degree two with minimal Hilbert function. The problem to determine the generators of these algebras transfers into a combinatorial problem on counting maximal north-east lattice paths inside a shifted Ferrers diagram. We conjecture that the subalgebras generated in degree two with minimal Hilbert function are generated by an initial Lex or RevLex segment.