  We consider the problem of construction of determinant formulas for the
partition function of the six-vertex model with domain wall boundary
conditions. In pioneering works of Korepin and Izergin a determinant formula
was proposed and proved using a recursion relation. In later works, another
determinant formulas were given by Kostov for the rational case and by Foda and
Wheeler for the trigonometric case. Here, we develop an approach in which the
recursion relation is replaced by a system of algebraic equations with respect
to one set of spectral parameters. We prove that this system has a unique
solution. The result can be easily given as a determinant parametrized by an
arbitrary basis of polynomials. In particular, the choice of the basis of
Lagrange polynomials with respect to the second set of spectral parameters
leads to the Izergin-Korepin representation, and the choice of the monomial
basis leads to the Kostov and Foda-Wheeler representations.
