On almost primes solutions to forms of odd degrees in many variables

Let $\mathcal{F}=\{f_1,\dots ,f_R\}$ be a family of forms of odd degrees at most $d$ in $s$ variables. We study the solutions to the system $f_1(\mathbf{x})=\dots =f_R(\mathbf{x})=0$ of the form $x_i=y_i{p}_i$ with $\mathrm{\mid }{y}_i\mathrm{\mid }\le Y_{\mathcal{F}}$ and $p_i$ being a prime for all $i\in [s]$ inside the box $[-N,N{]}^s$, for large $N$. We show that if the number of variables $s$ is sufficiently large with respect to the parameters $R$ and $d$, then there are at least $C_{\mathcal{F}}{N}^{s-D}\mathrm{/}(\log N{)}^s$ such solutions for some constants $C_{\mathcal{F}}>0$ and $D\in \mathbb{N}$, with $D$ depending only on the initial parameters $R$ and $d$.