Related to this.
The $p+1$ factorization algorithm works over $\mathbb{Z}/n\mathbb{Z}[x]/f(x)$ and hopes $p+1$ to be smooth.
We are trying to generalize this to multivariate case and also try to find nilpotent elements.
Let $p$ be prime and $k,m$ 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 $K=\mathbb{F}_p[x_1,...,x_m]/\langle f_1(x_1,...,x_m),f_2(x_1,...,x_m),...f_k(x_1,...,x_m)\rangle$.
Let $K^*$ be the multiplicative group of $K$ and let $\rho=|K^*|$. $\rho$ can be zero positive integer of infinity.
Q1 Is there simple closed form for $\rho$?
For $a \in K^*$, if $a^r$ is of bounded degree (which happens experimentally) then $a^r$ will become periodic.
Q2 for $p=5$ and $a$ given by the following sage code, is $a$ of finite order?
p=5;K.<x0,x1,x2>=GF(p)[]
Kquo=K.quotient([4*x0*x2^2 + 4*x1^2 + 3*x0*x2 + 3*x1*x2, x0*x1^2 + 2*x2^3 + 4*x0*x1 + 2*x0*x2 + 3*x2, 4*x0^2*x1 + x0*x2^2])
a=Kquo(4*x0^2*x2 + x0*x1*x2 + 2*x1^2*x2 + 2*x1*x2^2 + 2*x1^2 + 4*x2^2 + 2*x1)
Kquo
t=lift(a**(p*(p**6-1)**2));t
