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Suppose I have $V\subset \mathbb{C}^n$ be the zero set of a polynomial $P(z_1, \dotsc, z_n),$ with bounded height of coefficients (where height is, to fix something, $|\log|a||$) and degree $d.$ Suppose I now have a ball $B=B(z_0, r) \subseteq \mathbb{C}^n.$ Is there an upper bound on $2n-2$ dimensional measure of $B\cap V?$

EDIT A quasi-answer: Wirtinger's formula (see Griffiths and Harris, p. 31) seems to indicate that the Fubini-Study volume of a $k$-dimensional sub variety $V$ of $\mathbb{P}^n$ equals $\deg(V) \mathrm{vol}(\mathbb{P}^k).$ For real algebraic varieties, there seems to be only a Cauchy-Crofton derived inequality, as suggested in the answer.

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sorry, what is a height of coefficient? – Fedor Petrov Aug 5 '11 at 17:42
I actually define it in the question in parentheses as the absolute value of the log of the modulus -- I am not sure if this is the most natural definition... – Igor Rivin Aug 5 '11 at 18:14
up vote 3 down vote accepted

There is an explicit upper bound based on a 2-d version of the Crofton formula. Namely, the area of $B \cap V$ is the integral of the number of points of intersection $W \cap (B \cap V)$ over the space of all affine 2-planes $W \subseteq \mathbb{R}^{2n}$. Since the real algebraic variety $V$ has degree $\leq d^2$ the number of points of intersection is at most $d^2$. So an upper bound is $d^2$ times the measure of the space of affine $2$-planes meeting $B$. It seems to me that, unless I have misunderstood, the bound on the coefficients is unnecssary.

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You are absolutely correct re bound on coefficients, and your argument is the best one can do for real algebraic sets, apparently. For complex, there is the Wirtinger formula I cite in my edit... – Igor Rivin Aug 6 '11 at 14:50
Well, it seems to me that the fundamental difference is not between real and complex varieties, but between affine and projective ones. Do you know of a relation between the volumes of intersection $B \cap V$ as in your question and the FS volume of the projective closure of $V$? – Stephen Aug 7 '11 at 3:37

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