abstract algebra - Decomposition result for multivariate polynomial

(This answer was written before fedja's appeared, and has been edited to incorporate one of his observations.)

No. In particular, I claim that the image of $F(x, y) = x^2 + y^2$ is not of this form. Suppose there exist integer polynomials $f_1, ... f_n$ with the desired property. Since the image of an odd degree polynomial contains negative numbers, the polynomials $f_i$ must have degree at least $2$. However, $displaystyle sum_{i=1}^{n} sum_{x in mathbb{Z}} frac{1}{|f_i(x) + 1|}$ converges, but $displaystyle sum_{x, y in mathbb{Z}} frac{1}{x^2 + y^2 + 1}$ diverges (since, for example, it contains a subsequence which is essentially the sum of the reciprocals of the primes congruent to $1 bmod 4$).