Many degree d nilpotent elements of quotients of polynomial rings and non-vanishing product

[Georgi Guninski]

Apologies for offtopic, but this is related to an open problem.

With latex formatting the question is on mathoverflow [1]

Let R be ring and let K=R[x_1,...x_k]/<f_1(x_1,...,x_k)...f_l(x_1,x_k)>

We are looking for subset S of K: S={a_1,a_2,...a_m} with the
following properties

1. For integer $d$ for all $i$ we have a_i^d=0
2. Product(S)=\prod a_i != 0

Q1: How large can $m=|S|$ be in terms of $d,k$.
Q2: Can we get something better than $(d-1)(k+1)$?

As a sanity check the following should return True:

sage: prod(S) != 0 and all(i^d==0 for i in S)

Some observations:

If R=Z/nZ with n with many prime factors there are many zero divisors
and polynomial f(x)=0 mod n may have many solution even if
degree(f)=2.


Thanks.

[1]:
Many degree d nilpotent elements of quotients of polynomial rings and
non-vanishing product

https://mathoverflow.net/q/504074/12481