Algebraic geometric measure theory

2011-08-05T17:38:29Z

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.

Answer by SPG for Algebraic geometric measure theory

2011-08-05T18:51:27Z

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.

Algebraic geometric measure theory - MathOverflow

Algebraic geometric measure theory

Answer by SPG for Algebraic geometric measure theory