Neither nor. 1. A polynomial vanishing on a set $ A\times B$ with $ A,B$ infinite subsets of $ \mathbb R$ vanishes everywhere. 2. Assuming that the half circle line belongs to the unit disk (wlog), $ S$ is contained in the zero-set of $ f(X,Y)=X^2+Y^2$. This implies that of power of $ f$ divides $ p$. Since $ f$ is irreducible, $ f$ divides $ p$. Thus $ p$ must vanish on the complete circle line which is assumed not to be.

Third Edition wrote: Neither nor. 1. A polynomial vanishing on a set $ A\times B$ with $ A,B$ infinite subsets of $ \mathbb R$ vanishes everywhere. 2. Assuming that the half circle line belongs to the unit disk (wlog), $ S$ is contained in the zero-set of $ f(X,Y)=X^2+Y^2$. This implies that of power of $ f$ divides $ p$. Since $ f$ is irreducible, $ f$ divides $ p$. Thus $ p$ must vanish on the complete circle line which is assumed not to be. Well, it's no clear from context if with "an square" you mean the solid figure or its perimeter, and the same with "circle" or "circumference". More important, the problem does not say anything about the coordinates X, Y being aligned with any of the figures, they may be slanted. Anyway, if the author of the problem refers to "solid figures", the poin (1) above solves both cases. If the problem is about the perimeters, then both a straight lines -the ones containing the sides of the square- and circumferences have rational-function parameterizations: $(at+b, ct+d)$ and $(x_0+r\frac{2t}{1+t^2},y_0+r\frac{1-t^2}{1+t^2})$ being $t$ the parameter and $a$, $b$, $c$, $d$, $x_0$, $y_0$, $r$ real number constants. So we have polynomial equations in $t$ with an infinite number of solutions, so should be identically null. Id est: - a polynomial subset of the plane containing a segment of line should contain the whole line - a polynomial subset of the plane containing an arc of circumference should contain the whole circumference. Q.E.D.