# can the square of a polynomial be written as a sum of squares of polynomials with only even degree terms?

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

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 $$h_1,\ldots,h_m$$.
• 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 $f = h_1^2 + \cdots + h_m^2$. Mar 15, 2023 at 3:26