For which forms $F$ is the following volume finite?

Question

Let $F \in \mathbb{R}[x_1, \cdots, x_n]$ be a homogeneous polynomial with degree $d \geq 2$. Put

$$V(F) = m(\{(x_1, \cdots, x_n) \in \mathbb{R}^n : |F(x_1, \cdots, x_n)| \leq 1\})$$

where $m$ denotes Lebesgue measure. For which $F$ is $V(F)$ finite?

Obviously, if $d$ is even and $F$ is positive definite (respectively negative definite), then $V(F)$ is finite because the set $\{\mathbf{x} \in \mathbb{R}^n : |F(\mathbf{x})| \leq 1\}$ is compact. However, this set is usually not compact; but nevertheless, there are many cases when it is finite. For example, when $F$ is a decomposable form (meaning it splits into linear factors over $\mathbb{C}$), then $V(F)$ is frequently finite, by a result of J.L. Thunder.

I am mostly interested in a classification theorem, if it is available.

Answer 1

Let $b(s)$ be the Bernstein-Sato polynomial of $F$ : it is a monic polynomial in $s$, and there is an algorithm computing it from $F$. By a result of Kashiwara, all roots of $F$ are negative rational numbers ; let $s_0$ be the largest root of $F$. Since $F$ is homogeneous of degree $d$, one has $s_0 \geq - \frac{n}{d}$. Then :

$V(F)$ is finite iff $s_0 = - \frac{n}{d}$ is a simple root of $b$.

This is easily seen from the link between the roots of $b$ and the poles of the (distribution-valued) meromorphic continuation of $F^s$, and from the identity $$ V(F) = c_n \int_{S^{n-1}} |F(\theta)|^{-\frac{n}{d}}d \sigma(\theta). $$