## Volume of a polynomial in many variables

Let $f$ be a complex polynomial of $n$ variables.

Is the asymptotic behavior of the measure of the set

$$(x_1,...,x_n) \in [-1;1]^n: |f(x_1,...,x_n)| < V$$

known for $V$ small ?

The so-called Igusa zeta-function is essentially the Mellin transform in $V$ of this set. Me and a friend of mine are wondering if it is possible to understand this measure directly without having recourse to Igusa's zeta function (one proof of the analytic continuation uses for example Hironaka's theorem).

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Maybe I don't understand the question. If, say, $f=x_1+x_2+10$, then the measure is zero for $V$ small, indeed, for $V\le8$. – Gerry Myerson Apr 18 2012 at 11:38