# Expression of a form as a sum of powers.

I have a small question about one short sentence appearing in page : $376$ of the following electronic textbook : https://scholar.harvard.edu/files/joeharris/files/000-final-3264.pdf

• The short sentence says :

If $f$ and $g$ are general polynomials of degree $d=2m$ in one variable over a field of characteristic $0$, how many linear combination of $f$ and $g$ are expressible as a sum of $m$ $d^{\text{th}}$ powers of linear forms ?

• Question :

What do we mean by saying that a linear combination of $f$ and $g$ is expressible as a sum of $m$ $d^{\text{th}}$ powers of linear forms ? Could you translate to me mathemtically what do we mean by : a sum of $m$ $d^{\text{th}}$ powers of linear forms ?

• It means $af + bg = \sum_{i=1}^m l_i^d$ for some linear forms $l_i$. – Sasha May 16 '18 at 19:21