A polynomial that should have (real) roots at $x_1, x_2, ...$ can easily be constructed with the factored form of a (univariate) polynomial $p(x) = (x-x_1) (x-x_2) ...$.
How about the multivariate case? The multivariate polynomial should have the roots at $(x_1, y_1), (x_2, y_2), ...$. How does the polynomial look like?