On the representation of friable integers by linear forms

Let $P^{+}(n)$ denote the largest prime of the integer $n$. Using the \begin{align*}\Psi_{F_1\cdots F_t}\left($\mathcal{K}$\cap[-N,N]^d,N^{1/u}\right):=\#\left{$\mathcal{K}$\in {$\mathbf{N}$}\cap[-N,N]^d:\vphantom{P^+(F_1(\boldsymbol{n})\cdots F_t(\boldsymbol{n}))\leq N^{1/u}}\right.\left.P^+(F_1(\boldsymbol{n})\cdots F_t(\boldsymbol{n}))\leq N^{1/u}\right}\end{align*} where $(F\mathrm{\_}1,\dots ,F\mathrm{\_}t)$ is a system of affine-linear forms of $\mathbf{Z}[X\mathrm{\_}1,\dots ,X\mathrm{\_}d]$ no two of which are affinely related and $\mathcal{K}$ is a convex body. This improves upon Balog, Blomer, Dartyge and Tenenbaum’s work~\cite{BBDT12} in the case of product of linear forms.