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
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