On Fr\"oberg-Macaulay conjectures for algebras

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.