Always true : $( \sum_{i=1}^d x_i^a )( \sum_{i=1}^d y_i^a ) = ( \sum_{i=1}^d z_i^a )$

All variables are integers $>-1$.

Consider $f(a) = d$ such that d is the smallest value $>1$ such that :

It is always true that

$$( \sum_{i=1}^d x_i^a )( \sum_{i=1}^d y_i^a ) = ( \sum_{i=1}^d z_i^a )$$

For example the famous Sum of two squares as a multiplicative norm. So $f(2) = 2$.

Keep in mind that nothing is negative ; So positive cubes , positive fifth powers etc.

This relates to diophantines and Warings problem ofcourse.

And maybe it relates to ring theory and matrices determinants.

So What is $f(a)$ like ? From the Waring problem we know there always exists a $d$ for every $a$.

What is $f(3),f(4),f(5)$ ? What are the asymptotics to $f(a)$ ?

I assume $f$ is strictly increasing ??

• $(a^2+b^2)(c^2+d^2) = |(a+ib)(c+id)|^2 = (ac-bd)^2+(ad+bc)^2$ so the $z_i$ are polynomials in $x_j,y_k$. Instead of norm you can think to determinants. The matrices of the form $\scriptstyle\begin{pmatrix} a & b \\ -b & a \end{pmatrix}$ are a subgroup of $GL_2(\mathbb{Q})$. That's why $\det(M_1)\det(M_2)=\det(M_1M_2)= \det(M_3)$. If you ask about the Waring's problem then it is more complicated I'd say. – reuns Aug 15 '17 at 16:38