Let $f$ be a polynomial with real coefficients in several indeterminates $x\_1, \dots, x\_n$. Suppose that
$$ f = g^2 $$
for some polynomial $g$.


Is it true that we can find polynomials $h\_1, \dots, h\_m$ which only involve monomials of even degree such that $f = {h\_1}^2 + \dots + {h\_m} ^2$?



can the square of a polynomial be written as a sum of squares of polynomials with only even degree terms?
There do not necessarily exist polynomials $h\_1,\ldots,h\_m$ which only involve monomials of even degree such that $f = h\_1^2 + \cdots + h\_m^2$, even in the case when $f$ is a univariate polynomial.


A polynomial $h$ that only involves monomials of even degree is an even polynomial: it satisfies $h(x) = h(-x)$ for all $x$. A sum of squares of even polynomials must also be an even polynomial. However, the square of a polynomial need not be even: for example, if $g(x) = x+1$ then $f = g^2$ is not an even polynomial, and hence it can't be expressed as $f = h\_1^2 + \cdots + h\_m^2$ for even polynomials $h\_1,\ldots,h\_m$.


