Let s, k and n be positive integers and define rs,k(n) to be the number of solutions of the diophantine equation
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0025579300007439/resource/name/S0025579300007439_eqn1.gif?pub-status=live)
in positive integers xi. In 1922, using their circle method, Hardy and Littlewood [2] established the asymptotic formula
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0025579300007439/resource/name/S0025579300007439_eqn2.gif?pub-status=live)
whenever s≥(k−2)2k−1 + 5. Here
, the singular series, relates the local solubility of (1.1). For each k we define
to be the smallest value of s0 such that for all s ≥ s0 we have (1.2), the asymptotic formula in Waring's problem. The main result of this memoir is the following theorem which improves upon bounds of previous authors when k≤9.