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

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 $N(3,3)>2$.