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.

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Do you need some extra assumptions, e.g. with even degrees? It seems that e.g. is not a sum of squares.

For real polynomials, yes, also should assume the given polynomial is nonnegative.

I don’t understand. So how is be a sum of squares when regarded as a complex polynomial?

Ah… stupid me. Let me answer my own question:

.

Nice~