Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

share|improve this question
    
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

1 Answer 1

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.

share|improve this answer
    
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 $V$? –  S123 Aug 7 '11 at 3:37

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.