Solution of surface or two variable polynomial:
Suppose consider the expression $f(x,y)=\sum_{i,j=0}^{n} a_{ij}x^iy^j \in k[x,y]$ over a field $k$.
I am interested in the zeros of $f(x,y)$.
If we plot $f(x,y)$ in a $3D$ space, it will be a surface.
So the zeros of $f(x,y)$ are obtained by intersecting with $z=0$ plane.
Ofcourse, $z=0$ over the other two axes.
In other words, if $f(x,y)$ lie above the $xy$-plane, then no zeros of $f(x,y)$ are there in $k$.
Is it enough to observe ?
Am I missing something here ?