The main conjecture for a Vinogradov subsystem

Abstract

When $k\ge 2$ and $s>0$, we establish that for each $\varepsilon>0$, one has \[ \int_{[0,1)^{k-1}}\biggl| \sum_{1\le x\le X}e(\alpha_2x^2+\ldots +\alpha_k x^k)\biggr|^{2s}\,{\rm d}\boldsymbol \alpha \ll X^{s+\varepsilon}+X^{2s-(k^2+k-2)/2}, \] confirming the main conjecture for a new family of exponential sums beyond those of Vinogradov (translation-dilation invariant) type. Our methods are based on very recent work of the author employing the nested efficient congruencing method. Consequently, analogues of our results hold also in the setting of number fields and function fields.