Zeros of polynomials over finite principal ideal rings

Abstract:

Let $R$ be a finite commutative ring with identity. For $f \in R[{X_1}, \cdots ,{X_n}]$ denote by $N(f)$ the number of zeros of $f$ in ${R^{(n)}}$. For integers $n,d \geq 1$ denote by ${A_{n,d}}$ the greatest common divisor of the integers $N(f);f \in R[{X_1}, \cdots ,{X_n}],\deg f = d$. J. Ax has shown that if $R$ is a field, then ${A_{n,d}} = |R{|^\alpha }$ where $\alpha$ is the integer satisfying $\alpha < n/d \leq \alpha + 1$. In this paper, ${A_{n,d}}$ is computed in the case that $R$ is a principal ideal ring.