  We consider intersections of n diagonal forms of degrees k 1 < $\bullet$
$\bullet$ $\bullet$ < kn, and we prove an asymptotic formula for the number of
rational points of bounded height on these varieties. The proof uses the
Hardy-Littlewood method and recent breakthroughs on the Vinogradov system. We
also give a sharper result for one specific value of (k 1 ,. .. , kn), using a
technique due to Wooley and an estimate on exponential sums derived from a
recent approach in the van der Corput's method.
