On a Hybrid Version of the Vinogradov Mean Value Theorem

Given a family $\varphi= (\varphi_1, \ldots, \varphi_d)\in \mathbb{Z}[T]^d$ of $d$ distinct nonconstant polynomials, a positive integer $k\le d$ and a real positive parameter $\rho$, we consider the mean value $$M_{k, \rho} (\varphi, N) = \int_{\mathbf{x} \in [0,1]^k} \sup_{\mathbf{y} \in [0,1]^{d-k}} \left| S_{\varphi}(\mathbf{x}, \mathbf{y}; N) \right|^\rho d\mathbf{x}$$ of exponential sums $$S_{\varphi}( \mathbf{x}, \mathbf{y}; N) = \sum_{n=1}^{N} \exp\left(2 \pi i \left(\sum_{j=1}^k x_j \varphi_j(n)+ \sum_{j=1}^{d-k}y_j\varphi_{k+j}(n)\right)\right),$$ where $\mathbf{x} = (x_1, \ldots, x_k)$ and $\mathbf{y} =(y_1, \ldots, y_{d-k})$. The case of polynomials $\varphi_i(T) = T^i$, $i =1, \ldots, d$ and $k=d$ corresponds to the classical Vinaogradov mean value theorem. Here motivated by recent works of Wooley (2015) and the authors (2019) on bounds on $\sup_{\mathbf{y} \in [0,1]^{d-k}} \left| S_{\varphi}( \mathbf{x}, \mathbf{y}; N) \right|$ for almost all $\mathbf{x} \in [0,1]^k$, we obtain nontrivial bounds on $M_{k, \rho} (\varphi, N)$.