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Let $f(x)$ be a polynomial function with real coefficients such that $f(x)\geq 0 \;\forall x\in\Bbb R$. Prove that there exist polynomials $A(x),B(x)$ with real coeficients such that $f(x)=A^2(x)+B^2(x)\;\forall x\in\Bbb R$

I don't know how to approach this, apart from some cases of specific polynomials that turned out really ugly. Any hints to point me to the right direction?

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    $\begingroup$ What is the source of this problem???????? $\endgroup$ – Will Jagy Nov 9 '14 at 1:39
  • $\begingroup$ @WillJagy A mathematics journal. $\endgroup$ – UserX Nov 9 '14 at 1:42
  • $\begingroup$ What mathematics journal and specific article would that be? $\endgroup$ – Will Jagy Nov 9 '14 at 1:42
  • $\begingroup$ @WillJagy Άσκηση 52 here $\endgroup$ – UserX Nov 9 '14 at 1:46
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Consider roots of $f(x)$, as $f(x)\geq0,\forall x\in\mathbb{R}$, so $f(x)$ can be rewritten as following: $$f(x)=a^2(x-a_1)^2\cdots(x-a_k)^2[(x-\alpha_1)(x-\bar{\alpha_1})]\cdots[(x-\alpha_l)(x-\bar{\alpha_l})]$$ Where $a,a_1,\cdots,a_k\in\mathbb{R},\alpha_1,\cdots,\alpha_l\in\mathbb{C}$.

Denote $g(x)=a(x-a_1)\cdots(x-a_k),h(x)=(x-\alpha_1)\cdots(x-\alpha_l)=h_1(x)+ih_2(x)$, then \begin{align*} f(x)&=g^2(x) \, h(x) \, \bar{h}(x)\\ &=g^2(x) \, [h_1(x)+ih_2(x)] \, [h_1(x)-ih_2(x)]\\ &=(g(x)h_1(x))^2+(g(x)h_2(x))^2 \end{align*}

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Assume that the leading coefficient of $f$ is $1$. If $f(x)=x^k$ for some $k$, then $k$ is even and the result follows. By completing the square we see that the result holds if the degree of $f$ is less than or equal to $2$. If $f$ has a real root, we may assume this root is at 0 by replacing $x$ with $x-a$. We can see that this must be a multiple root by the fact that $f(x)\geq 0$ for all $x$, so by factoring out $x^2$ we reduce the problem to a polynomial of lower degree. We may thus assume that $\deg(f)>2$, the result holds for polynomials smaller degree, and that $f$ has a nonzero complex root. The nonzero roots of $f$ are $z_1,z_1',z_2,z_2',\ldots,z_n,z_n'$ where $z_i$ is the complex conjugate of $z_i'$ for all $i$. Thus there is a quadratic polynomial $x^2+bx+c=(x-z_1)(x-z_1')$ and a polynomial $p(x)$ of degree two less than the degree of $f$ such that $$f(x)=(x^2+bx+c)p(x)$$

By induction, $p(x)=A(x)^2+B(x)^2$ for some polynomials $A(x)$ and $B(x)$. Thus $$f(x)=(x^2+bx+c)(A(x)^2+B(x)^2)$$ We also have that $$f(x)=((x+\frac{b}{2})^2+c-\frac{b^2}{4})(A(x)^2+B(x)^2)$$ so $$f(x)=\left((x+\frac{b}{2})A(x)+\sqrt{c-\frac{b^2}{4}}B(x)\right)^2+\left((x+\frac{b}{2})B(x)-\sqrt{c-\frac{b^2}{4}}A(x)\right)^2$$ by the Brahmagupta–Fibonacci identity, so the result follows by induction provided $c\geq b^2/4$. However, if this were not the case then $x^2+bx+c$ would have a real root, so we are done.

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  • $\begingroup$ Maybe I missed this in the first paragraph but...how do we know that $p(x)$ has to be nonnegative? At first I thought it has to be nonnegative because $x^2+bx+c$ has to be nonnegative, since if it wasn't it would have real non-repeated roots, and thus $f(x)$ would be nonnegative. But that's not the case, since we could write $f(x) = (x-1)^4(x+1)^4 = (x-1)(x+1)p(x)$ $\endgroup$ – Ovi Sep 18 '18 at 5:09
  • $\begingroup$ @ovi Your example has real roots and hence can be $\endgroup$ – Matt Samuel Sep 18 '18 at 8:02
  • $\begingroup$ @ovi reduced to lower degree as shown. $\endgroup$ – Matt Samuel Sep 18 '18 at 8:03
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Survey article by Bruce Reznick called Some Concrete Aspects of Hilbert's 17th Problem, includes your case in the paragraph on Before 1900:

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