Fields for which there exist multivariable polynomials vanishing at single specified point

If $k$ is not algebraically closed, such a polynomial always exists (the opposite is also true and is mentioned in the post).

We may assume that $a_i=0$ for all $i$. Take an irreducible polynomial $g(x)$ of degree $d>1$, then for the homogeneous form $G(x,y)=y^dg(x/y)$ we have $G(x,y)=0$ if only if $x=y=0$. This solves the case $n=2$, for $n=3$ consider the polynomial $G(G(x,y),z)$, it takes zero value only when $x=y=z=0$, and so on.

I guess finite fields $k = \mathbb{F}_{q}$ satisfy this property, namely we can take $$ f = ((x_{1}-a_{1})^{q-1}-1) \dotsb ((x_{n}-a_{n})^{q-1}-1) - (-1)^{n} $$ for any $n$.

In fact we have a Lagrange interpolation formula for finite fields, see this answer.