This is generalization of the univariate case and also related to open problem.

Let $n,k,m,B>1$ be positive integers and $f_1(x_1,...,x_m),f_k(x_1,...,x_m),...f_k(x_1,...,x_m)$ be polynomials with integer coefficients.

Let $K=(\mathbb{Z}/n \mathbb{Z})[x_1,...,x_m]/\langle f_1(x_i),f_2(x_i)...f_k(x_i)\rangle$.

Let $S$ be the set of degree 2 nilpotent elements of $K$ of degree at most $B$:

$$S=\{g : g \in K, g^2=0,\deg(g) \le B \}$$

I am trying to find large subset of $S'$ of $S$ such that $\prod S' \ne 0$.

Q1 How large can $S'$ be?

Lower bound for $|S'|$ is $m+1$ by setting $k=m$ and $n=n_0^2$ and $f_i=x_i^2$ and $S'=\{n_0,x_1,x_2,...x_m\}$.

Disadvantage of this construction is that working in this quotient is not efficient because of exponential growth of monomials.

This is generalization of the univariate case and also related to open problem.

Let $n,k,m,B>1$ be positive integers and $f_1(x_1,...,x_m),f_2(x_1,...,x_m),...f_k(x_1,...,x_m)$ be polynomials with integer coefficients.

Let $K=(\mathbb{Z}/n \mathbb{Z})[x_1,...,x_m]/\langle f_1(x_1,...,x_m),f_2(x_1,...,x_m),...f_k(x_1,...,x_m)\rangle$.

Let $S$ be the set of degree 2 nilpotent elements of $K$ of degree at most $B$:

$$S=\{g : g \in K, g^2=0,\deg(g) \le B \}$$

I am trying to find large subset of $S'$ of $S$ such that $\prod S' \ne 0$.

Q1 How large can $S'$ be?

Lower bound for $|S'|$ is $m+1$ by setting $k=m$ and $n=n_0^2$ and $f_i=x_i^2$ and $S'=\{n_0,x_1,x_2,...x_m\}$.

Disadvantage of this construction is that working in this quotient is not efficient because of exponential growth of monomials.

This is generalization of the univariate case and also related to open problem.

Let $n,k,m,B>1$ be positive integers and $f_1(x_1,...,x_m),f_k(x_1,...,x_m),...f_k(x_1,...,x_m)$ be polynomials with integer coefficients.

Let $K=(\mathbb{Z}/n \mathbb{Z})[x_1,...,x_m]/<f_1(x_i),f_2(x_i)...f_k(x_i)>$.

Let $S$ be the set of degree 2 nilpotent elements of $K$ of degree at most $B$:

$$S=\{g : g \in K, g^2=0,\deg(g) \le B \}$$

I am trying to find large subset of $S'$ of $S$ such that $\prod S' \ne 0$.

Q1 How large can $S'$ be?

Lower bound for $|S'|$ is $m+1$ by setting $k=m$ and $n=n_0^2$ and $f_i=x_i^2$ and $S'=\{n_0,x_1,x_2,...x_m\}$.

Disadvantage of this construction is that working in this quotient is not efficient because of exponential growth of monomials.

This is generalization of the univariate case and also related to open problem.

Let $n,k,m,B>1$ be positive integers and $f_1(x_1,...,x_m),f_k(x_1,...,x_m),...f_k(x_1,...,x_m)$ be polynomials with integer coefficients.

Let $K=(\mathbb{Z}/n \mathbb{Z})[x_1,...,x_m]/\langle f_1(x_i),f_2(x_i)...f_k(x_i)\rangle$.

Let $S$ be the set of degree 2 nilpotent elements of $K$ of degree at most $B$:

$$S=\{g : g \in K, g^2=0,\deg(g) \le B \}$$

I am trying to find large subset of $S'$ of $S$ such that $\prod S' \ne 0$.

Q1 How large can $S'$ be?

Lower bound for $|S'|$ is $m+1$ by setting $k=m$ and $n=n_0^2$ and $f_i=x_i^2$ and $S'=\{n_0,x_1,x_2,...x_m\}$.

Disadvantage of this construction is that working in this quotient is not efficient because of exponential growth of monomials.