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). Assume that the partial $\partial F/\partial x_1 \not = 0$ for all points in $D$. For $\mu \in \mathbb{R}$ let us denote $W(\mu) = \{ \mathbf{x} \in \mathbb{R}^n : F(\mathbf{x}) = \mu \} \cap D.$ If $\{\mathbf{x} \in \mathbb{R}^n : F(\mathbf{x}) = \mu \}$ has a non-singular point inside $D$ then $W(\mu)$ has positive $(n-1)$-dimensional measure. We then define local coordinates $u_2, .., u_n$ on $W(\mu)$ so that the Jacobian $$ j = \det{ (\ \partial (F, u_2, ..., u_n) / \partial (x_1, ..., x_n) \ )} $$ is non-zero (in the paper they seem to pick $u_i = x_i$ for each $2 \leq i \leq n$). I am interested in the function $$ G(\mu) = \int_{W(\mu)} \frac{1}{j} \ d \mathbf{u}, $$ and I want to show that this function has a bounded derivative. (Here I can further assume that $D$ doesn't contain any "bad" points of $F$ to make the analysis easier.) Since there exists $C > 0$ such that once $|\mu| > C$ then $W(\mu)$ is an empty set, $G(\mu)$ has a compact support. I would greatly appreciate if someone could point out how I can achieve this.
This is basically 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 ); I tried to simplify the situation here. However, I am having difficulty understanding his arguments. Any comments/suggestions/explanations are appreciated. Thank you very much.
PS In fact I would greatly appreciate anything that would help me understand what is happening on p258 of this paper.
PPS $G(\mu)$ here is $\Psi_Q(\mu)$ in the paper defined on p258.
