Denote the graded rings $R:=\mathbb{R}[x_{1},\dots x_{n}]$ and $S:=R[x_{0}]$ adding the homogenizing variable $x_{0}.$ Consider $h\in S$ a homogenous polynomial of degree $d$ with leading coefficient $1$ when seen as a polynomial in the variable $x_{0}$ with coefficients in the ring $R.$ Form the graded $R$-module $M:=S/(h).$ Now suppose that the set $\{u_{1},\dots,u_{r}\}\subseteq M_{e},$ for some integer $e,$ generates $M_{\geq e}.$
Can we always choose representatives $a_{i}+b_{i}h$ of $u_{i},$ for each $i\in\{1,\dots,r\},$ and a homogeneous polynomial $g\in S$ such that $N:=S/(gh)$ and the corresponding classes $\overline{u}_{i}$ of these representatives verify that $\{\overline{u}_{1},\dots\overline{u}_{r}\}\subseteq N_{f}$ and that set is a basis of $N_{\geq f}$ (for some $f$)? How could you do this?
What if we want to fix some previously chosen representatives first?
