representing polynomial range values as sums of 2 squares

A few more polynomial range conjectures ...

Let f be a polynomial in n variables, n>=1, with integer coefficients.
Regard f as a function from Z^n to Z.

Conjecture 1:

If f is nonconstant, then range(f) contains infinitely many elements
which can be represented as the sum of 2 squares of integers.

Conjecture 2:

If every element of range(f) is a sum of 2 squares of integers, then f
is a sum of 2 squares of polynomials with integer coefficients.

quasi
.

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