polynomials which are nonnegative on Z^n

Let f be an n-variate polynomial with real coefficients, regarded as a
function from R^n to R, and such that f(x) >= 0 for all x in Z^n.

question (1):

Can f be unbounded below?

question (2):

If the answer to question (1) is yes, would the answer still be yes if
f was required to have integer coefficients?

quasi
.