Sharp mean value estimates in the absence of translation-dilation invariance

We establish sharp mean value estimates for various exponential sums lacking translation-dilation invariance properties. For example, suppose that $k\ge 1$ and $s>0$. Then, for each $\epsilon >0$, one has $${\int }_{[0,1{)}^k}\mid \sum _{1\le h\le X}\sum _{1\le z\le X}e(hz{\alpha }_1+\dots +h{z}^k{\alpha }_k){\mid }^{2s}\mathrm{d}\mathit{\alpha }\ll X^{\epsilon }(X^s+X^{2s-k(k+3)\mathrm{/}2}),$$ confirming the main conjecture for a new family of exponential sums. Our methods are based on the application of very recent work of the author employing the nested efficient congruencing method.