We give a systematic explanation of deriving a pointwise exponent sum estimate from mean value estimates. This follows the idea of Vinogradov. Then we describe how the idea works in the bombieri-iwaniec method. After that, we focus on the mean value problem that arises from the bombieri-iwaniec method. We show the power of decoupling theory. Also, we explain how the improvement was made by Cen-Gao-Zhang-Zhang in their new preprint.
