The cubic case of the main conjecture in Vinogradov's mean value theorem

Abstract

We apply a variant of the multigrade efficient congruencing method to estimate Vinogradov's integral of degree $3$ for moments of order $2s$, establishing strongly diagonal behaviour for $1\le s\le 6$. Consequently, the main conjecture is now known to hold for the first time in a case of degree exceeding $2$.

arXiv admin note: substantial text overlap with arXiv:1401.2932

References in corpus (3)

• Translation invariance, exponential sums, and Waring's problem
• Mean value estimates for odd cubic Weyl sums
• Rational solutions of pairs of diagonal equations, one cubic and one quadratic

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