Any polynomial $P(x,y)$ of degree at most $2009$ can be written uniquely
as a sum $\sum_{i=0}^{2009} P_i(x,y)$ in which $P_i(x,y)$ is a homogeneous
polynomial of degree $i$.
For $r>0$, let $C_r$ be the path $(r\cos \theta, r\sin \theta)$
for $0 \leq \theta \leq 2\pi$. Put $\lambda(P_i) = \oint_{C_1} P_i$; then
for $r>0$,
$$\oint_{C_r} P = \sum_{i=0}^{2009} r^i \lambda(P_i).$$
For fixed $P$, the right side is a polynomial in $r$, which vanishes for
all $r>0$ if and only if its coefficients vanish.
In other words,
$P$ is balanced
if and only if $\lambda(P_i) = 0$ for $i=0,\dots,2009$.

For $i$ odd, we have $P_i(-x,-y) = -P_i(x,y)$.
Hence $\lambda(P_i) = 0$, e.g.,
because the contributions to the integral from
$\theta$ and $\theta + \pi$ cancel.

For $i$ even, $\lambda(P_i)$ is a linear function of the coefficients of
$P_i$. This function is not identically zero, e.g., because for $P_i =
(x^2 + y^2)^{i/2}$, the integrand is always positive and so
$\lambda(P_i) > 0$. The kernel of $\lambda$ on the space of homogeneous
polynomials of degree $i$ is thus a subspace of codimension 1.

It follows that the dimension of $V$ is
$$(1 + \cdots + 2010) - 1005 = (2011 - 1) \times 1005 = 2020050.$$