Explicit Bound on Collective Strength of Regular Sequences of Three Homogeneous Polynomials

TL;DR

This paper investigates explicit bounds on the collective strength of three homogeneous polynomials of degrees 2 and 3, establishing specific bounds for these cases and advancing understanding of regular sequences in polynomial rings.

Contribution

The paper provides explicit bounds for the collective strength of three homogeneous polynomials of degrees 2 and 3, refining previous general results.

Findings

For degree 2, the bound N(3,2) equals 2.

For degree 3, the bound N(3,3) is greater than 2.

Abstract

Let $f_1,\cdots,f_r\in k[x_1,\cdots,x_n]$ be homogeneous polynomial of degree $d$. Ananyan and Hochster (2016) proved that there exists a bound $N=N(r,d)$ where if collective strength of $f_1,\cdots,f_r\geq N$, then $f_1,\cdots,f_r$ are regular sequence. In this paper, we study the explicit bound $N(r,d)$ when $r=3$ and $d=2,3$ and show that $N(3,2)=2$ and $N(3,3)>2$.