  The system of $N$ classical particles on the line with the Weierstrass $\wp$
function as potential is known to be completely integrable. Recently D'Hoker
and Phong found a beautiful parameterization by the polynomial of degree $N$ of
the space of Riemann surfaces associated with this system. In the trigonometric
limit of the elliptic potential these Riemann surfaces degenerate into rational
curves. The D'Hoker-Phong polynomial in the limit describes the intersection
points of the rational curves. We found an explicit determinant representation
of the polynomial in the trigonometric case. We consider applications of this
result to the theory of Toeplitz determinants and to geometry of the spectral
curves. We also prove our earlier conjecture on the asymptotic behavior of the
ratio of two symplectic volumes when the number of particles tends to infinity.
