On a Hybrid Version of the Vinogradov Mean Value Theorem

Given a family $\phi =({\phi }_1,\dots ,{\phi }_d)\in \mathbb{Z}[T{]}^d$ of $d$ distinct nonconstant polynomials, a positive integer $k\le d$ and a real positive parameter $\rho$, we consider the mean value $$M_{k,\rho }(\phi ,N)={\int }_{\mathbf{x}\in [0,1{]}^k}\underset{\mathbf{y}\in [0,1{]}^{d-k}}{\sup }{\mid S_{\phi }(\mathbf{x},\mathbf{y};N)\mid }^{\rho }d\mathbf{x}$$ of exponential sums $$S_{\phi }(\mathbf{x},\mathbf{y};N)=\sum _{n=1}^N\exp \left(2\pi i(\sum _{j=1}^k{x}_j{\phi }_j(n)+\sum _{j=1}^{d-k}{y}_j{\phi }_{k+j}(n))\right),$$ where $$\mathbf{x}$ = (x_1, \ldots, x_k)$and$$\mathbf{y}$ =(y_1, \ldots, y_{d-k})$\mathrm{.}Thecaseofpolynomials$\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|$foralmostall$$\mathbf{x}$ \in [0,1]^k$, we obtain nontrivial bounds on $M_{k,\rho }(\phi ,N)$.