  Wave function ansatz based on the backflow transformation are widely used to
parametrize anti-symmetric multivariable functions for many-body quantum
problems. We study the geometric aspects of such ansatz, in particular we show
that in general totally antisymmetric polynomials cannot be efficiently
represented by backflow transformation ansatz at least in the category of
polynomials. In fact, one needs a linear combination of at least $O(N^{3N-3})$
determinants to represent a generic totally antisymmetric polynomial. Our proof
is based on bounding the dimension of the source of the ansatz from above and
bounding the dimension of the target from below.
