An extended Vinogradov's mean value theorem

In this paper, we provide novel mean value estimates for exponential sums related to the extended main conjecture of Vinogradov's mean value theorem, by developing the Hardy-Littlewood circle method together with a refined shifting variables argument. Let $d\geq 2$ be a natural number and $\boldsymbolα=(α_d,\ldots, α_1)\in \mathbb{R}^d.$ Define the exponential sum \begin{equation*} f_d(\boldsymbolα;N):=\sum_{1 \leq n \leq N}e(α_d n^d + \cdots+ α_1 n). \end{equation*} For $p>0$, consider mean values of the exponential sums \begin{equation*} \mathcal{I}_{p,d}(u;N):=\int_{[0,1)\times [0,N^{-u})\times [0,1)^{d-2}}|f_d(\boldsymbolα;N)|^pd\boldsymbolα, \end{equation*} where we wrote $d\boldsymbolα=dα_1 dα_2\cdots dα_{d-1}dα_d.$ By making use of the aforementioned tools, we obtain the sharp upper bound for $\mathcal{I}_{p,d}(u;N)$, for $d=2,3$ and $0<u\leq 1$. Furthermore, for $d \geq 4$, we obtain analogous results depending on a small cap decoupling inequality for the moment curves in $\mathbb{R}^d.$