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 ??