How can we prove Schwartz Zippel PIT is applicable to natural polynomials?
Suppose that $P$ is a multivariate polynomial with integer coefficients. Then $P$ vanishes on all natural inputs iff it vanishes on all integer inputs. You can prove this by showing that if $P$ is not the zero polynomial then there is a natural input on which it is non-zero. This, in turn, you can prove by induction on the number of variables.

The base case is when $P$ is a univariate polynomial. In this case it is not hard to check that $P$ is non-zero for sufficiently large integer input. Suppose next that $P$ is a $(d+1)$-variate polynomial. Choose one variable $x$ which appears in some monomial of $P$, and write $P$ as a univariate polynomial whose coefficients are $d$-variate polynomials. Suppose that the highest power of $x$ that appears in $P$ is $x^n$, and that the corresponding coefficient is the $d$-variate polynomial $Q$ in the other variables. The induction hypothesis shows that there is an input on which $Q \neq 0$. Applying the base case again, we find a non-zero input for $P$.