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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 $f = {h_1}^2 + \dots + {h_m} ^2$?

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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 $h_1,\ldots,h_m$.

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    $\begingroup$ Another good example is $f(x) = x^2$. Although this is an even polynomial (hence the criterion in my answer above doesn't distinguish $f$ from a sum of squares of even polynomials) it is clear that if $f = h_1^2 + \cdots + h_m^2$ then at least one of $h_1,\ldots,h_m$ must have degree greater than zero. However since $h_1,\ldots,h_m$ involve only even degree monomials, the highest-degree monomial occurring in the sum $h_1^2 + \cdots + h_m^2$ must have degree at least 4 and must have a strictly positive coefficient. This precludes the possibility that $f = h_1^2 + \cdots + h_m^2$. $\endgroup$ Mar 15, 2023 at 3:26

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