G ( y $_{) =}$   2 y$_{1}$ 0 3 y 2 3 0 0 2 y$_{2}$ 0 3 y 2 4 2 y$_{1}$ 2 y$_{2}$ 0 0   $^{.}$ ˜ G ( y )$_{1}$$_{=}$   2 y$_{1}$ 0 3 y 2 3 0 2 y$_{2}$ 0 2 y$_{1}$ 2 y$_{2}$ 0   $^{,}$$^{det}$ ( ˜ G ( y )$_{1}$ ) = − 12y$_{1}$y$_{2}$y 2 3 ˜ G ( y )$_{2}$ =    2 y$_{1}$ 0 0 0 2 y$_{2}$ 3 y 2 4 2 y$_{1}$ 2 y$_{2}$ 0   $^{,}$$^{det}$ ( ˜ G ( y )$_{2}$ ) = − 12 y$_{1}$y$_{2}$y 2 4 ˜ G ( y )$_{3}$$_{=}$   2 y$_{1}$ 3 y 2 3 0 0 0 3 y 2 4 2 y$_{1}$ 0 0   $^{,}$$^{det}$ ( ˜ G ( y )$_{3}$ ) = 18y$_{1}$y 2 $_{3}$y 2 4 ˜ G ( y )$_{4}$$_{=}$   0 3 y 2 3 0 2 y$_{2}$ 0 3 y 2 4 2 y$_{2}$ 0 0   $^{,}$$^{det}$ ( ˜ G ( y )$_{4}$ ) = 18y$_{2}$y 2 $_{3}$y 2 4 Hence ¯ α = 4 but det ( ˜ G ( y )$_{3}$ ) and det ( ˜ G ( y )$_{4}$ ) are homogeneous polyno- mials of degree 5 > ¯ α. Thus, ¯ α is the smallest possible degree of an homogeneous polynomial in 16 We have four possible q × q submatrices ( with q = 3 ):