Let $F$ be a homogeneous form in $n$ variables with integer coefficients. Let $D$ be a closed box in $\mathbb{R}^n$ (product of closed and bounded intervals). I am interested in the function $$ G(\mu) = \int_{\{ F = \mu\} \cap D } 1 \ d \mathbf{x}, $$ and I want to show that this function has bounded derivatives. Here I can assume that $D$ doesn't contain any "bad" points of $F$ to make the analysis easier, for example singular points. I would greatly appreciate if someone could point out how I can achieve this.

This is a much simpler statement than what is proved on page 258 in B .J. Birch's "Forms in many variables"( https://www.jstor.org/stable/2414232?seq=1#page_scan_tab_contents ). However, I am having difficulty understanding his arguments. Any comments/suggestions/explanations are appreciated. Thank you very much.