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 $\varepsilon>0$, one has \[ \int_{[0,1)^k}\biggl| \sum_{1\le h\le X}\sum_{1\le z\le X}e(hzα_1+\ldots +hz^kα_k)\biggr|^