Independence of generic forms and the Fr\"oberg conjecture

TL;DR

This paper proves the Fröberg conjecture for certain degrees and conditions involving generic forms and variable counts, advancing understanding of its validity.

Contribution

It establishes the conjecture's validity in the second non-trivial degree and up to degree 2d-1 under large variable assumptions.

Findings

Fröberg conjecture holds in the second non-trivial degree for generic forms of degree d>2.

Conjecture is true up to degree 2d-1 when the number of variables is large.

Provides new evidence supporting the conjecture in specific algebraic contexts.

Abstract

We show that the Fr\"oberg conjecture holds in the second non-trivial degree for an ideal generated by generic forms of degree $d>2$. We also show that the conjecture is true up to degree $2d-1$ provided that the number of variables is sufficiently large.