Two Infinite Families of Regular Sequences of Power Sums in Three Variables

Let $$S=\mathbb{C}[x,y,z],p_m=x^m+y^m+z^m\mathrm{.}$$ For every $r\ge 1$, we prove that $$p_r,p_{r+1},p_{3r+1}\text{and}{p}_r,p_{r+1},p_{3r+2}$$ are regular sequences exactly when $r\equiv ̸1(\mathrm{m}\mathrm{o}\mathrm{d}3)$. This proves the Conca–Krattenthaler–Watanabe prediction for two infinite families. The proof is elementary. Four simple reductions modulo $(p_r,p_{r+1})$ reduce the problem to the two equations $x+y+z=0$ and $xy+xz+yz=0$.