Today I went to a talk delivered by a postdoc in the CS department about bilinear complexity. He raised this elementary result:
Theorem. Every polynomial in (that means, is a polynomial in with complex coefficients) can be expressed as a sum of two squares of complex polynomials.
The proof follows directly from the fact:
Another elementary thing is that:
is a sum of squares of two polynomials with integer coefficients.
Whether a real polynomial can be written as a sum of squares of real polynomials plays an important role in polynomial optimization. If I have time I will continue this post.