Nonsingular zeros of polynomials defined over p-adic fields

We show that if F and G are polynomials defined over a p-adic field with gcd(F, G) = 1, then the problem of finding a nonzero nonsingular zero of F that is not a zero of G is equivalent to the problem of finding a nonsingular zero of the homogenization of F. In addition, we prove the existence of p-adic zeros of some polynomials of low degree that are not necessarily homogeneous. This extends some well-known results on the existence of p-adic zeros of homogeneous polynomials of low degree.