The square of a real-valued polynomial is clearly non-negative, and so the sum of the squares of polynomials is non-negative. What about the converse? Is a non-negative polynomial the sum of the squares of polynomials?

For polynomials in one variable, yes. For polynomials in several variables, no.

However, Emil Artin proved nearly a century ago that although a non-negative polynomial **cannot** in general be written as a sum of squares of **polynomials**, a non-negative polynomial **can** be written as a sum of squares of **rational functions**.

Several years ago I wrote about Motzkin’s polynomial,

an explicit example of a non-negative polynomial in two variables which cannot be written as the sum of the squares of polynomials. Artin’s theorem says it must be possible to write *M*(*x*, *y*) as the sum of the squares of rational functions. And indeed here’s one way:

Source: Mateusz Michaelek and Bernd Sturmfels. Invitation to Nonlinear Algebra. Graduate Studies in Mathematics 211.

Here’s a little Mathematica code to verify the example above.

m[x_, y_] := x^4 y^2 + x^2 y^4 + 1 - 3 x^2 y^2 a[x_, y_] := (x y (x^2 + y^2 - 2))/(x^2 + y^2) b[x_, y_] := (x a[x, y])^2 + (y a[x, y])^2 + a[x, y]^2 + ((x^2 - y^2)/(x^2 + y^2))^2 Simplify[m[x, y] - b[x, y]]

This returns 0.