prove that in $k[x_1,...,x_n]$ Ideal of polynomials vanishing on some $(a_1,a_2,...,a_n)$ will be $\langle x_1-a_1,...,x_2-a_2\rangle$.
$\ker \phi = (x_1-a_1, ..., x_n-a_n)$ for a ring homomorphism $\phi: R[x_1, ..., x_n] \to R$