Number of polynomials in a finite commutative ring with n elements of degree d and of degree 2 in $R[x_1,x_2]$?

Question

Let R be a finite commutative ring with n elements.

a) What is the number of polynomials of degree $d$ in $R[x]$?

b) What is the number of polynomials of degree 2 in $R[x_1,x_2]$?

I think for part a, it would be $(n-1)*n^d$ since you have n options for all of the coefficients except the first term which must be nonzero.

I'm unsure of what to do for b

Answer 1

For question c), you have two kinds of coefficients:

• the constant term and those of linear terms, which can be arbitrary. This makes a contribution of $n^3$,
• the coefficients of the quadratic part in two variable, which has one constraint: they can't all be $0$. Hence there are $n^3-1$ possibility.

Summing up we have $n^3(n^3-1)$ valid sets of coefficients.