Elementary techniques in a simple case of Vinogradov’s mean value theorem

FirstCaseofVMVT

 

In the above pdf, I explore the first nontrivial case of Vinogradov’s mean value theorem. That is, I seek to bound solutions to the simultaneous systems x_1^j + x_2^j + x_3^j = y_1^j + y_2^j + y_3^j for j = 1,2, where all the variables are integers in \{1 , \ldots , N\}. This is much easier (still nontrivial) than the general case, due to the existence of helpful symmetries.

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