[Number Theory] Using Brahmagupta Identity and FTA to show Non-Negative Univariate Polynomial is SOS

Let's use the Fundamental Theorem Algebra (FTA) and the Brahmagupta Fibonacci identity to prove that every non-negative polynnomial $f(x) \in \mathbb{R}\left[ x \right]$ may be written as a sum of squares of polynomials.

Let $p \in \mathbb{R}[x] ~ \forall x \in \mathbb{R}$ and $p(x) \geq 0$. By the Fundamental Theorem of Algebra, a degree $n$ polynomial $p$ has exactly $n$ complex roots, and our polynomial $p$ is of the form constant times a product of positive quadratics. The graph of the polynomial lies entirely above the x-axis, or touches it. So in the first case, the roots of p are all complex, whereas in the latter, $p$ has real roots with even multiplicity. Hence our polynomial is of the form

$p = a~f_1^2f_2^2 \cdots f_n^2g_1 \cdots g_m,$

where $a$ is a real coefficient, the $f_i$'s have degree 1, and $g_j$'s have degree 2 and no real roots. We also note that by the complex conjugate root theorem, the roots of $p$ come in complex conjugate pairs. Also, $p(x)$ does not intersect the x-axis, so the degree of $p$ must be even.

A monic positive quadratic $g_j$, with no real roots, can be written as $(x+a)^2+b^2$ by completing the square. From here, an expression of $p$ as a sum of squares follows from the Brahmagupta-Fibonacci identity; the product of $g_j$'s, hence a product of the sums of squares, can be written as the sum of squares. For example, the product of two quadratics

$((x+a)^2+b^2)((x+c)^2+d^2)$

$= ((x+a)(x+c) + b^2d^2) + ((x+a)^2d^2-b^2(x+c)).$

If the degree of $p$ is 2, then

$p = (x-(a+ib))(x-(a-ib))$

$= x^2-2ax+a^2+b^2 = (x-a)^2+b^2$

where $a,b \in R[x],$ hence the sum of two squares.

If degree of $p$ is greater than or equal to 4, then we can write $p$ as the product of quadratics. By applying the Brahmagupta-Fibonacci identity as needed, the product of many sums of squares can be written as a sum of many squares.

Since the degree of $p$ is not odd, we have shown that a non-negative univariate polynomial is a sum of squares of polynomials. $\blacksquare$

Note that in the case that $p=a$, where $a\in \mathbb{R}$, $p$ is itself a square.