POISSON BRACKETS DETERMINED BY JACOBIANS

Fix $n-2$ elements $h_1,{\cdots},h_{n-2}$ of the quotient field B of the polynomial algebra $\mathbb{C}[x_1,x_2,{\cdots},x_n]$. It is proved that B is a Poisson algebra with Poisson bracket defined by $\{f,g\}=det(Jac(f,g,h_1,{\cdots},h_{n-2})$ for any $f,g{\in}B$, where det(Jac) is the determinant of a Jacobian matrix.