# On Fr\"oberg-Macaulay conjectures for algebras

## TL;DR

This paper explores variants of Macaulay's theorem and Fr"oberg's conjecture for subalgebras of polynomial rings, presenting questions, partial results, and examples to understand their Hilbert functions.

## Contribution

It introduces new questions and partial results on the Hilbert functions of subalgebras generated by forms, extending classical conjectures from ideals to subalgebras.

## Key findings

- Partial results on Hilbert functions of subalgebras
- Examples illustrating the behavior of these functions
- Formulation of new questions related to the conjectures

## Abstract

Macaulay's theorem and Fr\"oberg's conjecture deal with the Hilbert function of homogeneous ideals in polynomial rings $S$ over a field $K$. In this short note we present some questions related to variants of Macaulay's theorem and Fr\"oberg's conjecture for $K$-subalgebras of polynomial rings. In details, given a subspace $V$ of forms of degree $d$ we consider the $K$-subalgebra $K[V]$ of $S$ generated by $V$. What can be said about Hilbert function of $K[V]$? The analogy with the ideal case suggests several questions. To state them we start by recalling Macaulay's theorem, Fr\"oberg's conjecture and Gotzmann's persistence theorem for ideals. Then we presents the variants for $K$-subalgebras along with some partial results and examples.