This article answers the second part of problem 2 in Mel Nathanson's article "Matrix Scaling, Explicit Sinkhorn limits, and Arithmetic" The exact value of the Sinkhorn limit for a certain symmetric matrix of size, 3 By Shalsoh B. Ekhad Consider the following symmetric matrix [K 1 1] [ ] [1 L 1] [ ] [1 1 M] Let , z, be the smallest real positive root of the polynomial equation 2 2 2 2 2 L M (L M - 1) + (-4 K L M + 4 K L M + 2 L M - L - M) z + (6 K L M 2 2 2 - 6 K L M - 2 K L M - 2 K L M - 2 K L M + 3 K L + 3 K M - 3 L M + 2 L 2 3 2 2 3 2 2 2 2 2 + 2 M - 1) z + (-4 K L M + 4 K L M + 4 K L M + 4 K L M - 2 K L M 2 2 3 - 3 K L - 3 K M - 2 K L M + 2 K - L - M + 2) z 4 + (K M - 1) K (K L - 1) (K L M - K - L - M + 2) z = 0 and in Maple notation L*M*(L*M-1)+(-4*K*L^2*M^2+4*K*L*M+2*L*M-L-M)*z+(6*K^2*L^2*M^2-6*K^2*L*M-2*K*L^2 *M-2*K*L*M^2-2*K*L*M+3*K*L+3*K*M-3*L*M+2*L+2*M-1)*z^2+(-4*K^3*L^2*M^2+4*K^3*L*M +4*K^2*L^2*M+4*K^2*L*M^2-2*K^2*L*M-3*K^2*L-3*K^2*M-2*K*L*M+2*K-L-M+2)*z^3+(K*M-\ 1)*K*(K*L-1)*(K*L*M-K-L-M+2)*z^4 = 0 and in LaTex LM \left( LM-1 \right) + \left( -4\,K{L}^{2}{M}^{2}+4\,KML+2\,LM-L-M \right) z+ \left( 6\,{K}^{2}{L}^{2}{M}^{2}-6\,{K}^{2}LM-2\,KM{L}^{2}-2 \,KL{M}^{2}-2\,KML+3\,LK+3\,KM-3\,LM+2\,L+2\,M-1 \right) {z}^{2}+ \left( -4\,{K}^{3}{L}^{2}{M}^{2}+4\,{K}^{3}LM+4\,{K}^{2}{L}^{2}M+4\,{K }^{2}L{M}^{2}-2\,{K}^{2}LM-3\,{K}^{2}L-3\,{K}^{2}M-2\,KML+2\,K-L-M+2 \right) {z}^{3}+ \left( KM-1 \right) K \left( LK-1 \right) \left( KML -K-L-M+2 \right) {z}^{4}=0 NULL Then the diagonal matrix X such that S=XMX is doubly stochastic has in its d\ iagonal 4 3 4 4 3 3 3 3 3 4 2 3 3 2 4 1/2 2 L K M L K M 4 L K M 3 L K M 9 L K M z [1, ---------- - -------- - ---------- + ---------- - ---------- %1 %1 %1 %1 %1 3 3 2 2 3 2 / 3 3 2 4 2 2 L K M L 2 L M M 3 M | 2 K M 16 L K M + ---------- - ---- + --- + ---- - ---- - --- + |- ------- + ----------- %1 %1 %1 %1 %1 %1 \ %1 L %1 2 3 3 2 2 2 2 4 K M 4 K M 38 L K M 2 K M 2 K M 3 K M 24 L M K - ---- + ----- + --------- - ------ - ------- + ------ - ---------- %1 L %1 L %1 %1 L %1 L %1 L %1 4 2 4 5 3 4 4 2 4 4 2 4 4 30 L K M 4 L K M 6 L K M 2 L K M L K M 11 L K M - ---------- + -------- + --------- - ---------- - ------- - ----------- %1 %1 %1 %1 %1 %1 2 2 5 3 4 3 3 4 3 3 3 3 2 4 16 L M K 15 L K M 8 L M K 26 L K M 8 L K M 7 K M + ----------- - ----------- + --------- - ---------- + ---------- + ------ %1 %1 %1 %1 %1 %1 5 3 4 4 5 2 2 2 2 5 4 3 2 4 K M 2 K M K M 14 L K 2 K 4 K L K L K L K - ----- - ------- - ----- - ------- + ---- - ---- - ----- + ----- - ----- %1 %1 %1 %1 %1 L %1 L %1 %1 %1 4 3 3 2 3 3 2 2 2 3 2 5 L K 5 L K 10 L K 6 L K 6 L K 20 K M 18 K M + ------ - ------- + -------- - ------ + ------- - ------- + -------- %1 %1 %1 %1 %1 %1 %1 3 2 2 3 3 2 4 4 3 3 4 2 10 K M 4 K M 4 K M K M 2 K M 3 K M 11 K 4 K - ------- - ------- + ------- - ----- - ------- - ------- + ----- + --- %1 %1 %1 %1 %1 %1 %1 %1 3 3 5 2 5 3 4 4 2 5 5 2 6 K 2 L K M 10 L K M 7 L K M 3 L M K 2 L K M - ---- + --------- - ----------- + --------- + --------- + --------- %1 %1 %1 %1 %1 %1 5 3 4 2 2 4 3 4 3 4 6 4 4 6 3 6 L K M 10 L K M 28 L K M 2 L K M 2 L K M L K M - -------- + ----------- + ----------- + --------- - ---------- + -------- %1 %1 %1 %1 %1 %1 3 6 4 3 6 3 3 6 2 2 6 3 4 5 3 4 5 2 L K M 4 L K M 2 L K M 2 L K M 4 L K M L K M + -------- + ---------- - ---------- - ---------- + ---------- - -------- %1 %1 %1 %1 %1 %1 3 5 4 2 5 4 2 6 2 3 5 3 5 4 4 3 8 L K M 5 L K M 2 L K M 10 L K M L K M K M + ---------- - ---------- - ---------- - ----------- + ------- + ----- %1 %1 %1 %1 %1 %1 L 3 4 3 4 2 6 6 2 3 5 2 2 3 6 L K M K M L M K L K M 7 L K M K L K L + --------- + ----- + ------- + ------- - ---------- - --- - ----- %1 %1 L %1 %1 %1 %1 %1 2 2 3 3 2 3 2 2 3 4 K M 2 K M 7 K M K M 28 L K M 4 L K M 12 L K M - ------ + ----- + ------- + ---- - ----------- - ------- - ---------- %1 %1 %1 %1 %1 %1 %1 3 2 3 2 2 2 2 2 3 2 2 2 2 4 L K M L K M 8 L K M 8 L K M 5 L K M 23 L K M - --------- - ------- - ---------- - --------- + --------- + ---------- %1 %1 %1 %1 %1 %1 2 3 3 2 2 2\ / 3 3 18 L K M 2 L K M L K M 2 L K M | 3 | 3 K M + ----------- + -------- + ------ + --------| z + |- ------- %1 %1 %1 %1 / \ %1 L 2 3 4 2 4 2 2 3 2 2 2 16 L K M 28 L K M 2 K M 2 K M 19 L K M K M 6 K M + ----------- - ----------- + ------ + ----- + --------- - ----- + ------ %1 %1 %1 L %1 L %1 %1 L %1 L 2 4 4 2 4 3 3 4 4 3 15 L M K 11 L K M 18 L K M 11 L K M 3 L K M - ---------- - ---------- + --------- - ---------- - ---------- %1 %1 %1 %1 %1 4 4 2 3 4 4 2 4 4 2 2 5 3 4 3 2 L K M 19 L K M 14 L K M 6 L M K 10 L K M - ---------- - ----------- + ----------- + ---------- + ----------- %1 %1 %1 %1 %1 3 3 2 4 3 3 3 3 3 2 2 15 L K M M K L 14 L K M 8 L K M 16 L K 4 K - ---------- + ------- + ----------- + ---------- - ------- - ---- %1 %1 %1 %1 %1 %1 L 2 4 3 3 2 3 3 2 2 2 3 2 2 L K 4 L K 10 L K 11 L K 8 L K 13 K M 8 K M + ------- - ------- + -------- - ------- + ------- - ------- + ------- %1 %1 %1 %1 %1 %1 %1 3 2 2 3 3 4 3 3 4 4 2 3 19 K M 16 K M 6 K M 4 K M 4 K M 2 K M L M - ------- - -------- + ------- + ------- + ------- + ------- - ---- + ---- %1 %1 %1 %1 %1 %1 %1 %1 2 2 2 5 3 4 4 2 5 4 M 2 M 4 12 K 14 K 6 L K M 4 L K M 3 L M K - ---- + --- - ---- + ----- + ---- + ---------- - --------- - --------- %1 %1 %1 L %1 %1 %1 %1 %1 5 2 3 4 2 2 4 3 4 3 4 3 2 3 L K M 18 L K M 30 L K M 16 L K M 4 L K M - --------- + ----------- + ----------- - ---------- - ---------- %1 %1 %1 %1 %1 4 3 4 5 4 4 5 3 3 5 4 3 5 3 5 L K M 6 L K M 3 L K M 3 L K M 12 L K M + --------- + ---------- - ---------- - ---------- - ----------- %1 %1 %1 %1 %1 3 4 3 5 2 2 2 3 2 11 L K M 6 L K M 6 K L 14 K L K L 8 K M 4 K M - ---------- + ---------- + ------ - ------ - ----- - ------ - ----- %1 %1 %1 %1 %1 %1 %1 2 3 3 2 2 2 3 4 K M L K L M 2 L M 4 L M M 4 M 3 K M + ------- - ---- + ---- + ------ - ----- - ---- + ---- + ------ %1 %1 %1 %1 %1 %1 L %1 L %1 2 3 2 3 2 2 2 2 3 2 4 3 38 L K M 4 L K M 8 L K M 3 L K M 5 L K M K M - ----------- + ---------- - ---------- - --------- - ------- - ---- %1 %1 %1 %1 %1 %1 L 2 3 3 2 3 2 2 2 2 2 3 12 L K M 38 L K M 8 L K M 8 L K M 10 L K M + ---------- + ---------- - --------- - ---------- + ---------- %1 %1 %1 %1 %1 2 2 2 2 3 2 2 3 2 3 3 12 L K M 21 L K M L K M 4 L K M 3 L K M 24 L K M + ---------- + ---------- - ------ - --------- - -------- - ----------- %1 %1 %1 %1 %1 %1 2 2 2 \ / 2 3 4 2 4 2 4 L K M 5 L K M 13 L K M 4 | 2 | 18 L K M 6 L K M - -------- + -------- + --------- + ----| z + |- ----------- - ---------- %1 %1 %1 %1 / \ %1 %1 2 4 3 2 3 2 4 4 2 2 K M 6 K M 2 L K M 18 L K M 3 K M 3 L M K 3 L K M + ------ - ----- + -------- - --------- + ------- + --------- + --------- %1 L %1 L %1 %1 %1 L %1 %1 3 3 4 4 3 3 4 4 3 4 3 3 3 2 4 5 L K M 3 L K M 3 L K M 12 L K M 19 L K M M K L - --------- + ---------- + ---------- + ----------- + ---------- - ------- %1 %1 %1 %1 %1 %1 4 3 3 3 3 2 2 2 3 2 2 2 3 2 4 L K M 14 L K M 6 L K L K 6 L K 18 K M K M - ---------- - ----------- + ------ - ----- - ------- + ------- - ----- %1 %1 %1 %1 %1 %1 %1 2 2 3 3 2 4 2 3 2 6 K M 6 K M 2 K M L 2 L 2 M 3 M 4 M 2 - ------- - ------- - ------- + ---- - --- - ---- + ---- + --- + ---- %1 %1 %1 %1 %1 %1 %1 %1 %1 L 3 4 2 2 4 3 4 3 2 4 4 4 3 4 6 K 6 L K M 6 L K M 6 L K M 6 L K M 6 L K M - --- - ---------- - ---------- + ---------- - ---------- + --------- %1 %1 %1 %1 %1 %1 3 2 3 2 2 3 2 2 3 2 L K M 5 K L 8 K L 2 K L 11 K M 14 K M 12 K M - ---------- - ------ + ----- + ------- + ------- + ------ - -------- %1 %1 %1 %1 %1 %1 %1 3 3 2 2 3 2 2 2 3 2 L K L M L M 2 L M 2 L M 6 L M 2 M 5 M 14 K M + ------ - ---- + ----- + ------ + ------ - ------ + ---- - ---- - ------- %1 %1 %1 %1 %1 %1 %1 L %1 L %1 4 2 2 2 2 4 2 3 2 3 2 2 2 2 3 2 L K M 12 L K M 18 L K M 9 L K M 3 L K M + ---------- - ----------- + ----------- - ---------- + ---------- %1 %1 %1 %1 %1 2 4 3 3 4 3 2 3 3 2 7 L K M 20 L K M 8 L K M 2 K M 18 L K M 12 L K M + --------- + ----------- - ------- + ------ + ---------- + ---------- %1 %1 %1 %1 L %1 %1 3 2 2 2 2 2 3 3 2 2 3 2 2 L K M 24 L K M 27 L K M 4 L K M 6 L K M 12 L K M - ------- + ----------- + ---------- + --------- - --------- + ---------- %1 %1 %1 %1 %1 %1 2 2 3 2 2 3 2 3 3 2 26 L K M 5 L K M 4 L K M 13 L K M 32 L K M 27 L K M - ---------- - -------- + --------- + --------- - ----------- - --------- %1 %1 %1 %1 %1 %1 2 2 \ 2 2 3 3 3 6 L K M 18 L K M 1 | K M 6 K M 3 K M L K L M + -------- - --------- - ----| z + ---- - ----- + ------- - ---- + ---- %1 %1 %1 / %1 %1 %1 %1 %1 2 2 3 2 2 2 3 2 2 3 2 L M L M L M 4 L M 2 L M M 2 M L M L M + ------- - ---- - ---- + ------ - ----- - ---- + ---- - ----- - ----- %1 %1 %1 %1 %1 %1 L %1 L %1 %1 3 4 2 2 2 2 4 2 3 2 3 2 2 2 2 3 5 K M 3 L K M 9 L K M 2 L K M 3 L K M 12 L K M + ------ - ---------- + ---------- + ---------- + ---------- + ----------- %1 %1 %1 %1 %1 %1 2 4 4 2 3 3 2 4 3 3 4 3 3 L K M L K M 5 L K M 2 L K M L K M 6 L K M K M - --------- - ------- + --------- + --------- - -------- + ------- - ---- %1 %1 %1 %1 %1 %1 %1 L 2 3 3 2 3 2 2 2 2 2 3 3 2 L K M L K M 3 L K M 6 L K M 9 L K M 4 L K M - ------- - ------- + --------- - ---------- - --------- - --------- %1 %1 %1 %1 %1 %1 2 3 2 2 2 2 3 2 2 3 2 L K M 6 L K M 3 L K M 6 L K M 6 L K M 12 L K M - --------- - --------- - --------- + -------- - --------- - --------- %1 %1 %1 %1 %1 %1 2 3 3 2 2 2 3 2 4 2 L K M 6 L K M 5 L K M 13 L K M K L 3 M K L + ---------- + -------- - -------- + ---------, ---- + --------- %1 %1 %1 %1 %1 M %1 4 3 4 4 3 3 3 3 3 4 2 3 3 2 4 2 L K M L K M 4 L K M 9 L K M 3 L K M - ---------- + -------- + ---------- + ---------- - ---------- %1 %1 %1 %1 %1 3 3 2 2 2 3 / 2 2 2 2 2 L K M L 3 L M 2 M L |2 L K 3 L K K L - ---------- + ---- + --- + ---- - --- - ---- + |------- - ------ + ---- %1 %1 %1 %1 %1 %1 \ %1 M %1 M %1 M 3 3 3 4 3 3 4 5 4 2 4 2 4 K L 2 L K 2 L K K L K L M K L 16 L K M - ----- + ------ + ------- - ----- - ----- - ------- - ----------- %1 M %1 M %1 M %1 M %1 M %1 %1 3 2 4 4 2 4 3 3 5 3 38 L K M 30 L M K 24 L K M 4 L K M 26 L K M 2 L K M - --------- + ---------- + ---------- - -------- + ---------- - --------- %1 %1 %1 %1 %1 %1 4 4 2 4 4 2 4 4 2 2 5 3 4 3 11 L K M 7 L K M 2 L K M 16 L M K 15 L K M + ----------- - --------- + ---------- - ----------- + ----------- %1 %1 %1 %1 %1 3 4 3 3 2 2 4 3 4 5 3 4 4 2 2 L M K 18 L K M K L 2 L K K L 2 K L 2 K - --------- - ----------- + ----- + ------- + ----- + ------- - ---- %1 %1 %1 %1 %1 %1 %1 M 4 5 2 2 2 5 4 3 4 3 3 4 K 5 K M K M 20 L K L K 3 L K 7 L K 4 L K + ---- - ------ + ----- + ------- + ----- + ------- - ------ - ------- %1 M %1 %1 %1 %1 %1 %1 %1 2 3 3 2 2 2 3 2 3 2 2 18 L K 10 L K 4 L K 14 K M 10 K M 6 K M 6 K M - -------- + ------- + ------- + ------- - -------- + ------ - ------- %1 %1 %1 %1 %1 %1 %1 3 3 3 4 4 2 2 3 3 5 2 5 3 5 K M K M K M 11 K 4 K 6 K 6 L K M 7 L K M + ------- - ----- + ----- - ----- - --- + ---- - --------- + ---------- %1 %1 %1 %1 %1 %1 %1 %1 4 4 2 5 5 2 5 3 4 2 2 4 3 L K M 2 L M K 3 L K M 6 L K M 28 L K M 10 L K M + ------- - --------- - --------- + -------- - ----------- - ----------- %1 %1 %1 %1 %1 %1 4 3 4 3 4 6 4 4 6 3 3 6 4 3 6 3 8 L K M 6 L K M 2 L K M L K M L K M 4 L K M - --------- - --------- + ---------- - -------- - -------- - ---------- %1 %1 %1 %1 %1 %1 3 6 2 2 6 3 4 5 3 4 5 2 3 5 4 2 L K M 2 L K M 8 L K M 5 L K M 4 L K M + ---------- + ---------- - ---------- + ---------- - ---------- %1 %1 %1 %1 %1 2 5 4 2 6 2 3 5 3 2 6 6 2 3 5 2 L K M 2 L K M 10 L K M L M K L K M 10 L K M + -------- + ---------- + ----------- - ------- - ------- + ----------- %1 %1 %1 %1 %1 %1 2 2 3 2 3 3 2 3 2 4 K L 2 K L 7 K L K M K M L K 28 L K M 4 L K M + ------ - ----- - ------- + --- + ----- - ---- + ----------- + ------- %1 %1 %1 %1 %1 %1 %1 %1 2 3 3 2 3 2 2 2 2 2 3 2 2 4 L K M 12 L K M 8 L K M 8 L K M L K M 23 L K M + --------- + ---------- + --------- + ---------- + ------- - ---------- %1 %1 %1 %1 %1 %1 2 2 2 3 3 2 2 2\ / 2 2 5 L K M 8 L K M 2 L K M 2 L K M L K M | 3 |L K - --------- - ---------- - -------- - -------- - ------| z + |----- %1 %1 %1 %1 %1 / \%1 M 2 2 3 3 2 3 4 2 4 2 3 6 L K 2 K L 2 K L 3 L K 4 L K M 28 L K M 19 L K M - ------ - ------ - ----- + ------- + ---------- + ----------- - --------- %1 M %1 M %1 M %1 M %1 %1 %1 2 4 4 2 4 3 3 4 4 3 11 L M K 15 L K M 18 L K M 15 L K M 19 L K M + ---------- + ---------- - --------- + ---------- + ----------- %1 %1 %1 %1 %1 4 4 2 4 4 3 4 4 2 4 4 2 2 5 14 L K M 4 L K M 3 L K M 2 L K M 6 L M K - ----------- + --------- + ---------- + ---------- - ---------- %1 %1 %1 %1 %1 3 4 3 3 4 3 3 3 2 4 3 3 3 10 L K M 16 L M K 11 L K M K L 3 M K L 14 L K M - ----------- + ---------- + ---------- + ---- + --------- - ----------- %1 %1 %1 %1 M %1 %1 3 3 2 3 4 2 4 3 2 4 3 3 24 L K M 4 L K 4 K 13 L K 4 L K 2 L K 6 L K + ----------- - ------- + ---- + ------- - ------- - ------- - ------- %1 %1 %1 M %1 %1 %1 %1 2 3 3 2 2 2 3 2 3 2 2 8 L K 19 L K 16 L K 16 K M 10 K M 11 K M 8 K M - ------- + ------- + -------- + ------- - -------- + ------- - ------- %1 %1 %1 %1 %1 %1 %1 3 3 4 2 2 3 2 4 K M 2 K M 4 L 2 L M L 12 K 14 K 4 + ------- - ------- + ---- - --- + ---- - ---- - ----- - ---- + ---- %1 %1 %1 %1 %1 %1 %1 %1 %1 M 2 5 3 2 5 5 2 3 4 2 2 4 3 6 L K M 3 L M K 3 L K M 30 L K M 18 L K M - ---------- + --------- + --------- - ----------- - ----------- %1 %1 %1 %1 %1 4 3 2 4 3 4 5 4 4 5 3 3 5 4 16 L K M 11 L K M 6 L K M 3 L K M 3 L K M - ----------- + ---------- - ---------- + ---------- + ---------- %1 %1 %1 %1 %1 3 5 3 3 4 3 5 2 2 2 3 2 12 L K M 5 L K M 6 L K M 8 K L 4 K L 4 K L 6 K M + ----------- - --------- - ---------- + ------ + ----- - ------- - ------ %1 %1 %1 %1 %1 %1 %1 2 3 2 3 2 2 3 14 K M K M L 4 L 3 L K 2 L M L M 4 L M K M + ------ + ----- + ---- - ---- - ------ - ------ - ---- + ----- + ---- %1 %1 %1 M %1 M %1 %1 %1 %1 %1 2 3 2 3 2 2 2 2 3 2 4 2 3 38 L K M 8 L K M 4 L K M L K M 5 L K M 38 L K M + ----------- + ---------- - ---------- - ------- + ------- - ---------- %1 %1 %1 %1 %1 %1 3 2 3 2 2 2 2 2 3 2 2 12 L K M 10 L K M 8 L K M 8 L K M 21 L K M - ---------- - ---------- + ---------- + --------- - ---------- %1 %1 %1 %1 %1 2 2 3 2 2 3 2 3 3 2 12 L K M 3 L K M 4 L K M L K M 8 L K M 4 L K M - ---------- + -------- + --------- + ------ - ---------- + -------- %1 %1 %1 %1 %1 %1 2 2 \ / 2 4 2 3 13 L K M 5 L K M 4 | 2 | 2 K L 6 K L 2 M L K 3 K L - --------- - -------- - ----| z + |- ------ + ----- - -------- - ------- %1 %1 %1 / \ %1 M %1 M %1 %1 M 2 3 4 2 4 2 3 2 4 4 2 3 3 6 L K M 6 L K M 18 L K M 3 L M K 3 L K M 19 L K M - ---------- + ---------- + --------- - --------- - --------- - ---------- %1 %1 %1 %1 %1 %1 4 4 3 3 4 4 3 4 3 3 3 3 2 4 3 L K M 3 L K M 12 L K M 5 L K M 2 K L 7 M K L - ---------- - ---------- - ----------- + --------- - ------ - --------- %1 %1 %1 %1 %1 M %1 4 3 3 3 3 2 2 4 2 3 3 2 3 20 L K M 32 L K M 2 K L 18 L K 6 L K L K - ----------- + ----------- + ------- - ------- + ------- + ----- %1 %1 %1 %1 %1 %1 2 2 2 3 2 2 2 2 2 3 6 L K 6 K M K M 6 K M 3 L 4 L M 2 M 2 L + ------- - ------ + ----- + ------- - ---- - --- - ---- + --- + ---- %1 %1 %1 %1 %1 %1 %1 %1 %1 3 4 2 2 4 3 4 3 2 4 3 6 K 2 6 L K M 6 L K M 18 L K M 6 L K M + --- - ---- + ---------- + ---------- + ----------- - --------- %1 %1 M %1 %1 %1 %1 4 4 4 3 2 3 2 2 3 2 6 L K M 2 L K M 11 K L 14 K L 12 K L 5 K M 8 K M + ---------- + ---------- - ------- - ------ + -------- + ------ - ----- %1 %1 %1 %1 %1 %1 %1 2 3 2 3 3 2 2 3 2 2 K M 2 L 5 L 14 L K 2 L M L M L M 6 L M - ------- - ---- + ---- + ------- - ------ - ----- + ---- + ------ %1 %1 M %1 M %1 %1 %1 %1 %1 2 3 4 2 2 2 2 4 2 3 2 3 2 2 2 L M 2 K M 12 L K M 2 L K M 18 L K M 3 L K M - ------ - ------ + ----------- - ---------- - ----------- - ---------- %1 %1 %1 %1 %1 %1 2 2 3 2 4 3 3 4 2 3 3 2 9 L K M L K M 4 L K M 8 L K M 12 L K M 18 L K M + ---------- + ------- + ---------- + ------- - ---------- - ---------- %1 %1 %1 %1 %1 %1 3 2 2 2 2 2 3 3 2 2 3 2 2 27 L K M 24 L K M L K M 6 L K M 4 L K M 26 L K M - ---------- - ----------- + ------- + --------- - --------- + ---------- %1 %1 %1 %1 %1 %1 2 2 3 2 2 3 2 3 3 2 12 L K M 13 L K M 4 L K M 5 L K M 14 L K M 27 L K M - ---------- - --------- - --------- + -------- + ----------- + --------- %1 %1 %1 %1 %1 %1 2 2 \ 2 2 3 2 18 L K M 6 L K M 1 | K L 6 K L 3 K L L 2 L + --------- - -------- + ----| z - ---- + ----- - ------- + ---- - ---- %1 %1 %1 / %1 %1 %1 %1 M %1 M 3 3 2 2 3 2 2 3 2 2 3 5 L K L M 2 L M L M 4 L M L M 2 L M L M L M - ------ + ---- - ------- - ---- - ------ + ---- + ----- + ----- + ----- %1 %1 %1 %1 %1 %1 %1 %1 %1 3 4 2 2 2 2 4 2 3 2 3 2 2 2 2 3 K M 9 L K M 3 L K M 2 L K M 12 L K M 3 L K M + ---- - ---------- + ---------- - ---------- - ----------- - ---------- %1 %1 %1 %1 %1 %1 4 2 3 3 2 4 3 3 4 2 3 3 2 2 L K M 5 L K M L K M L K M 6 L K M L K M L K M - --------- - --------- + ------- + -------- - ------- + ------- + ------- %1 %1 %1 %1 %1 %1 %1 3 2 2 2 2 2 3 3 2 2 3 2 2 9 L K M 6 L K M 3 L K M 2 L K M 4 L K M 3 L K M + --------- + ---------- - --------- + --------- + --------- + --------- %1 %1 %1 %1 %1 %1 2 2 3 2 2 3 2 3 3 2 6 L K M 12 L K M 6 L K M 6 L K M 2 L K M 6 L K M + --------- + --------- + --------- - -------- - ---------- - -------- %1 %1 %1 %1 %1 %1 2 2 13 L K M 5 L K M - --------- + --------] %1 %1 4 3 3 4 4 2 2 4 4 3 2 %1 := 4 K L M - 4 K L M - 4 K L M + 4 K L M + K L M - 3 K L M 2 3 4 4 2 2 4 3 3 3 2 + 3 K L M - K L M - L M + L M + 4 K L M - 4 K L M + 2 L M 2 3 3 2 2 3 3 3 2 - 2 L M - K L - K L M + K L M + K M + 2 L M - 2 L M - 4 L M 2 2 2 + 4 L M - L + M + 2 L - 2 M and in Maple notation z^(1/2)*[1, 2/(4*K*L^4*M^3-4*K*L^3*M^4-4*K*L^4*M^2+4*K*L^2*M^4+K*L^4*M-3*K*L^3* M^2+3*K*L^2*M^3-K*L*M^4-L^4*M^2+L^2*M^4+4*K*L^3*M-4*K*L*M^3+2*L^3*M^2-2*L^2*M^3 -K*L^3-K*L^2*M+K*L*M^2+K*M^3+2*L^3*M-2*L*M^3-4*L^2*M+4*L*M^2-L^2+M^2+2*L-2*M)*L ^4*K^3*M^4-1/(4*K*L^4*M^3-4*K*L^3*M^4-4*K*L^4*M^2+4*K*L^2*M^4+K*L^4*M-3*K*L^3*M ^2+3*K*L^2*M^3-K*L*M^4-L^4*M^2+L^2*M^4+4*K*L^3*M-4*K*L*M^3+2*L^3*M^2-2*L^2*M^3- K*L^3-K*L^2*M+K*L*M^2+K*M^3+2*L^3*M-2*L*M^3-4*L^2*M+4*L*M^2-L^2+M^2+2*L-2*M)*L^ 4*K^3*M^3-4/(4*K*L^4*M^3-4*K*L^3*M^4-4*K*L^4*M^2+4*K*L^2*M^4+K*L^4*M-3*K*L^3*M^ 2+3*K*L^2*M^3-K*L*M^4-L^4*M^2+L^2*M^4+4*K*L^3*M-4*K*L*M^3+2*L^3*M^2-2*L^2*M^3-K *L^3-K*L^2*M+K*L*M^2+K*M^3+2*L^3*M-2*L*M^3-4*L^2*M+4*L*M^2-L^2+M^2+2*L-2*M)*L^3 *K^3*M^3+3/(4*K*L^4*M^3-4*K*L^3*M^4-4*K*L^4*M^2+4*K*L^2*M^4+K*L^4*M-3*K*L^3*M^2 +3*K*L^2*M^3-K*L*M^4-L^4*M^2+L^2*M^4+4*K*L^3*M-4*K*L*M^3+2*L^3*M^2-2*L^2*M^3-K* L^3-K*L^2*M+K*L*M^2+K*M^3+2*L^3*M-2*L*M^3-4*L^2*M+4*L*M^2-L^2+M^2+2*L-2*M)*L^4* K^2*M^3-9/(4*K*L^4*M^3-4*K*L^3*M^4-4*K*L^4*M^2+4*K*L^2*M^4+K*L^4*M-3*K*L^3*M^2+ 3*K*L^2*M^3-K*L*M^4-L^4*M^2+L^2*M^4+4*K*L^3*M-4*K*L*M^3+2*L^3*M^2-2*L^2*M^3-K*L ^3-K*L^2*M+K*L*M^2+K*M^3+2*L^3*M-2*L*M^3-4*L^2*M+4*L*M^2-L^2+M^2+2*L-2*M)*L^3*K ^2*M^4+2/(4*K*L^4*M^3-4*K*L^3*M^4-4*K*L^4*M^2+4*K*L^2*M^4+K*L^4*M-3*K*L^3*M^2+3 *K*L^2*M^3-K*L*M^4-L^4*M^2+L^2*M^4+4*K*L^3*M-4*K*L*M^3+2*L^3*M^2-2*L^2*M^3-K*L^ 3-K*L^2*M+K*L*M^2+K*M^3+2*L^3*M-2*L*M^3-4*L^2*M+4*L*M^2-L^2+M^2+2*L-2*M)*L^3*K^ 3*M^2-1/(4*K*L^4*M^3-4*K*L^3*M^4-4*K*L^4*M^2+4*K*L^2*M^4+K*L^4*M-3*K*L^3*M^2+3* K*L^2*M^3-K*L*M^4-L^4*M^2+L^2*M^4+4*K*L^3*M-4*K*L*M^3+2*L^3*M^2-2*L^2*M^3-K*L^3 -K*L^2*M+K*L*M^2+K*M^3+2*L^3*M-2*L*M^3-4*L^2*M+4*L*M^2-L^2+M^2+2*L-2*M)*L^2+2/( 4*K*L^4*M^3-4*K*L^3*M^4-4*K*L^4*M^2+4*K*L^2*M^4+K*L^4*M-3*K*L^3*M^2+3*K*L^2*M^3 -K*L*M^4-L^4*M^2+L^2*M^4+4*K*L^3*M-4*K*L*M^3+2*L^3*M^2-2*L^2*M^3-K*L^3-K*L^2*M+ K*L*M^2+K*M^3+2*L^3*M-2*L*M^3-4*L^2*M+4*L*M^2-L^2+M^2+2*L-2*M)*L+1/(4*K*L^4*M^3 -4*K*L^3*M^4-4*K*L^4*M^2+4*K*L^2*M^4+K*L^4*M-3*K*L^3*M^2+3*K*L^2*M^3-K*L*M^4-L^ 4*M^2+L^2*M^4+4*K*L^3*M-4*K*L*M^3+2*L^3*M^2-2*L^2*M^3-K*L^3-K*L^2*M+K*L*M^2+K*M ^3+2*L^3*M-2*L*M^3-4*L^2*M+4*L*M^2-L^2+M^2+2*L-2*M)*M^3-1/(4*K*L^4*M^3-4*K*L^3* M^4-4*K*L^4*M^2+4*K*L^2*M^4+K*L^4*M-3*K*L^3*M^2+3*K*L^2*M^3-K*L*M^4-L^4*M^2+L^2 *M^4+4*K*L^3*M-4*K*L*M^3+2*L^3*M^2-2*L^2*M^3-K*L^3-K*L^2*M+K*L*M^2+K*M^3+2*L^3* M-2*L*M^3-4*L^2*M+4*L*M^2-L^2+M^2+2*L-2*M)*M^2-3/(4*K*L^4*M^3-4*K*L^3*M^4-4*K*L ^4*M^2+4*K*L^2*M^4+K*L^4*M-3*K*L^3*M^2+3*K*L^2*M^3-K*L*M^4-L^4*M^2+L^2*M^4+4*K* L^3*M-4*K*L*M^3+2*L^3*M^2-2*L^2*M^3-K*L^3-K*L^2*M+K*L*M^2+K*M^3+2*L^3*M-2*L*M^3 -4*L^2*M+4*L*M^2-L^2+M^2+2*L-2*M)*M+(-2/(4*K*L^4*M^3-4*K*L^3*M^4-4*K*L^4*M^2+4* K*L^2*M^4+K*L^4*M-3*K*L^3*M^2+3*K*L^2*M^3-K*L*M^4-L^4*M^2+L^2*M^4+4*K*L^3*M-4*K *L*M^3+2*L^3*M^2-2*L^2*M^3-K*L^3-K*L^2*M+K*L*M^2+K*M^3+2*L^3*M-2*L*M^3-4*L^2*M+ 4*L*M^2-L^2+M^2+2*L-2*M)/L*K^3*M^3+16/(4*K*L^4*M^3-4*K*L^3*M^4-4*K*L^4*M^2+4*K* L^2*M^4+K*L^4*M-3*K*L^3*M^2+3*K*L^2*M^3-K*L*M^4-L^4*M^2+L^2*M^4+4*K*L^3*M-4*K*L *M^3+2*L^3*M^2-2*L^2*M^3-K*L^3-K*L^2*M+K*L*M^2+K*M^3+2*L^3*M-2*L*M^3-4*L^2*M+4* L*M^2-L^2+M^2+2*L-2*M)*L^2*K^4*M^2-1/(4*K*L^4*M^3-4*K*L^3*M^4-4*K*L^4*M^2+4*K*L ^2*M^4+K*L^4*M-3*K*L^3*M^2+3*K*L^2*M^3-K*L*M^4-L^4*M^2+L^2*M^4+4*K*L^3*M-4*K*L* M^3+2*L^3*M^2-2*L^2*M^3-K*L^3-K*L^2*M+K*L*M^2+K*M^3+2*L^3*M-2*L*M^3-4*L^2*M+4*L *M^2-L^2+M^2+2*L-2*M)/L*K*M^2+4/(4*K*L^4*M^3-4*K*L^3*M^4-4*K*L^4*M^2+4*K*L^2*M^ 4+K*L^4*M-3*K*L^3*M^2+3*K*L^2*M^3-K*L*M^4-L^4*M^2+L^2*M^4+4*K*L^3*M-4*K*L*M^3+2 *L^3*M^2-2*L^2*M^3-K*L^3-K*L^2*M+K*L*M^2+K*M^3+2*L^3*M-2*L*M^3-4*L^2*M+4*L*M^2- L^2+M^2+2*L-2*M)/L*K*M+38/(4*K*L^4*M^3-4*K*L^3*M^4-4*K*L^4*M^2+4*K*L^2*M^4+K*L^ 4*M-3*K*L^3*M^2+3*K*L^2*M^3-K*L*M^4-L^4*M^2+L^2*M^4+4*K*L^3*M-4*K*L*M^3+2*L^3*M ^2-2*L^2*M^3-K*L^3-K*L^2*M+K*L*M^2+K*M^3+2*L^3*M-2*L*M^3-4*L^2*M+4*L*M^2-L^2+M^ 2+2*L-2*M)*L*K^3*M-2/(4*K*L^4*M^3-4*K*L^3*M^4-4*K*L^4*M^2+4*K*L^2*M^4+K*L^4*M-3 *K*L^3*M^2+3*K*L^2*M^3-K*L*M^4-L^4*M^2+L^2*M^4+4*K*L^3*M-4*K*L*M^3+2*L^3*M^2-2* L^2*M^3-K*L^3-K*L^2*M+K*L*M^2+K*M^3+2*L^3*M-2*L*M^3-4*L^2*M+4*L*M^2-L^2+M^2+2*L -2*M)/L*K^3*M-2/(4*K*L^4*M^3-4*K*L^3*M^4-4*K*L^4*M^2+4*K*L^2*M^4+K*L^4*M-3*K*L^ 3*M^2+3*K*L^2*M^3-K*L*M^4-L^4*M^2+L^2*M^4+4*K*L^3*M-4*K*L*M^3+2*L^3*M^2-2*L^2*M ^3-K*L^3-K*L^2*M+K*L*M^2+K*M^3+2*L^3*M-2*L*M^3-4*L^2*M+4*L*M^2-L^2+M^2+2*L-2*M) /L*K^2*M^2+3/(4*K*L^4*M^3-4*K*L^3*M^4-4*K*L^4*M^2+4*K*L^2*M^4+K*L^4*M-3*K*L^3*M ^2+3*K*L^2*M^3-K*L*M^4-L^4*M^2+L^2*M^4+4*K*L^3*M-4*K*L*M^3+2*L^3*M^2-2*L^2*M^3- K*L^3-K*L^2*M+K*L*M^2+K*M^3+2*L^3*M-2*L*M^3-4*L^2*M+4*L*M^2-L^2+M^2+2*L-2*M)/L* K^2*M-24/(4*K*L^4*M^3-4*K*L^3*M^4-4*K*L^4*M^2+4*K*L^2*M^4+K*L^4*M-3*K*L^3*M^2+3 *K*L^2*M^3-K*L*M^4-L^4*M^2+L^2*M^4+4*K*L^3*M-4*K*L*M^3+2*L^3*M^2-2*L^2*M^3-K*L^ 3-K*L^2*M+K*L*M^2+K*M^3+2*L^3*M-2*L*M^3-4*L^2*M+4*L*M^2-L^2+M^2+2*L-2*M)*L^2*M* K^4-30/(4*K*L^4*M^3-4*K*L^3*M^4-4*K*L^4*M^2+4*K*L^2*M^4+K*L^4*M-3*K*L^3*M^2+3*K *L^2*M^3-K*L*M^4-L^4*M^2+L^2*M^4+4*K*L^3*M-4*K*L*M^3+2*L^3*M^2-2*L^2*M^3-K*L^3- K*L^2*M+K*L*M^2+K*M^3+2*L^3*M-2*L*M^3-4*L^2*M+4*L*M^2-L^2+M^2+2*L-2*M)*L*K^4*M^ 2+4/(4*K*L^4*M^3-4*K*L^3*M^4-4*K*L^4*M^2+4*K*L^2*M^4+K*L^4*M-3*K*L^3*M^2+3*K*L^ 2*M^3-K*L*M^4-L^4*M^2+L^2*M^4+4*K*L^3*M-4*K*L*M^3+2*L^3*M^2-2*L^2*M^3-K*L^3-K*L ^2*M+K*L*M^2+K*M^3+2*L^3*M-2*L*M^3-4*L^2*M+4*L*M^2-L^2+M^2+2*L-2*M)*L*K^4*M+6/( 4*K*L^4*M^3-4*K*L^3*M^4-4*K*L^4*M^2+4*K*L^2*M^4+K*L^4*M-3*K*L^3*M^2+3*K*L^2*M^3 -K*L*M^4-L^4*M^2+L^2*M^4+4*K*L^3*M-4*K*L*M^3+2*L^3*M^2-2*L^2*M^3-K*L^3-K*L^2*M+ K*L*M^2+K*M^3+2*L^3*M-2*L*M^3-4*L^2*M+4*L*M^2-L^2+M^2+2*L-2*M)*L*K^5*M^3-2/(4*K *L^4*M^3-4*K*L^3*M^4-4*K*L^4*M^2+4*K*L^2*M^4+K*L^4*M-3*K*L^3*M^2+3*K*L^2*M^3-K* 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^4-L^4*M^2+L^2*M^4+4*K*L^3*M-4*K*L*M^3+2*L^3*M^2-2*L^2*M^3-K*L^3-K*L^2*M+K*L*M^ 2+K*M^3+2*L^3*M-2*L*M^3-4*L^2*M+4*L*M^2-L^2+M^2+2*L-2*M)*L^3*K^4*M^3+8/(4*K*L^4 *M^3-4*K*L^3*M^4-4*K*L^4*M^2+4*K*L^2*M^4+K*L^4*M-3*K*L^3*M^2+3*K*L^2*M^3-K*L*M^ 4-L^4*M^2+L^2*M^4+4*K*L^3*M-4*K*L*M^3+2*L^3*M^2-2*L^2*M^3-K*L^3-K*L^2*M+K*L*M^2 +K*M^3+2*L^3*M-2*L*M^3-4*L^2*M+4*L*M^2-L^2+M^2+2*L-2*M)*L^3*M*K^4-26/(4*K*L^4*M ^3-4*K*L^3*M^4-4*K*L^4*M^2+4*K*L^2*M^4+K*L^4*M-3*K*L^3*M^2+3*K*L^2*M^3-K*L*M^4- L^4*M^2+L^2*M^4+4*K*L^3*M-4*K*L*M^3+2*L^3*M^2-2*L^2*M^3-K*L^3-K*L^2*M+K*L*M^2+K *M^3+2*L^3*M-2*L*M^3-4*L^2*M+4*L*M^2-L^2+M^2+2*L-2*M)*L*K^3*M^3+8/(4*K*L^4*M^3-\ 4*K*L^3*M^4-4*K*L^4*M^2+4*K*L^2*M^4+K*L^4*M-3*K*L^3*M^2+3*K*L^2*M^3-K*L*M^4-L^4 *M^2+L^2*M^4+4*K*L^3*M-4*K*L*M^3+2*L^3*M^2-2*L^2*M^3-K*L^3-K*L^2*M+K*L*M^2+K*M^ 3+2*L^3*M-2*L*M^3-4*L^2*M+4*L*M^2-L^2+M^2+2*L-2*M)*L^3*K^3*M^2+7/(4*K*L^4*M^3-4 *K*L^3*M^4-4*K*L^4*M^2+4*K*L^2*M^4+K*L^4*M-3*K*L^3*M^2+3*K*L^2*M^3-K*L*M^4-L^4* 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M^2+L^2*M^4+4*K*L^3*M-4*K*L*M^3+2*L^3*M^2-2*L^2*M^3-K*L^3-K*L^2*M+K*L*M^2+K*M^3 +2*L^3*M-2*L*M^3-4*L^2*M+4*L*M^2-L^2+M^2+2*L-2*M)*L^3*M^2+1/(4*K*L^4*M^3-4*K*L^ 3*M^4-4*K*L^4*M^2+4*K*L^2*M^4+K*L^4*M-3*K*L^3*M^2+3*K*L^2*M^3-K*L*M^4-L^4*M^2+L ^2*M^4+4*K*L^3*M-4*K*L*M^3+2*L^3*M^2-2*L^2*M^3-K*L^3-K*L^2*M+K*L*M^2+K*M^3+2*L^ 3*M-2*L*M^3-4*L^2*M+4*L*M^2-L^2+M^2+2*L-2*M)*L^2*M^3+1/(4*K*L^4*M^3-4*K*L^3*M^4 -4*K*L^4*M^2+4*K*L^2*M^4+K*L^4*M-3*K*L^3*M^2+3*K*L^2*M^3-K*L*M^4-L^4*M^2+L^2*M^ 4+4*K*L^3*M-4*K*L*M^3+2*L^3*M^2-2*L^2*M^3-K*L^3-K*L^2*M+K*L*M^2+K*M^3+2*L^3*M-2 *L*M^3-4*L^2*M+4*L*M^2-L^2+M^2+2*L-2*M)*K*M^3-9/(4*K*L^4*M^3-4*K*L^3*M^4-4*K*L^ 4*M^2+4*K*L^2*M^4+K*L^4*M-3*K*L^3*M^2+3*K*L^2*M^3-K*L*M^4-L^4*M^2+L^2*M^4+4*K*L ^3*M-4*K*L*M^3+2*L^3*M^2-2*L^2*M^3-K*L^3-K*L^2*M+K*L*M^2+K*M^3+2*L^3*M-2*L*M^3-\ 4*L^2*M+4*L*M^2-L^2+M^2+2*L-2*M)*L^4*K^2*M^2+3/(4*K*L^4*M^3-4*K*L^3*M^4-4*K*L^4 *M^2+4*K*L^2*M^4+K*L^4*M-3*K*L^3*M^2+3*K*L^2*M^3-K*L*M^4-L^4*M^2+L^2*M^4+4*K*L^ 3*M-4*K*L*M^3+2*L^3*M^2-2*L^2*M^3-K*L^3-K*L^2*M+K*L*M^2+K*M^3+2*L^3*M-2*L*M^3-4 *L^2*M+4*L*M^2-L^2+M^2+2*L-2*M)*L^2*K^2*M^4-2/(4*K*L^4*M^3-4*K*L^3*M^4-4*K*L^4* M^2+4*K*L^2*M^4+K*L^4*M-3*K*L^3*M^2+3*K*L^2*M^3-K*L*M^4-L^4*M^2+L^2*M^4+4*K*L^3 *M-4*K*L*M^3+2*L^3*M^2-2*L^2*M^3-K*L^3-K*L^2*M+K*L*M^2+K*M^3+2*L^3*M-2*L*M^3-4* L^2*M+4*L*M^2-L^2+M^2+2*L-2*M)*L^2*K^3*M^2-12/(4*K*L^4*M^3-4*K*L^3*M^4-4*K*L^4* M^2+4*K*L^2*M^4+K*L^4*M-3*K*L^3*M^2+3*K*L^2*M^3-K*L*M^4-L^4*M^2+L^2*M^4+4*K*L^3 *M-4*K*L*M^3+2*L^3*M^2-2*L^2*M^3-K*L^3-K*L^2*M+K*L*M^2+K*M^3+2*L^3*M-2*L*M^3-4* L^2*M+4*L*M^2-L^2+M^2+2*L-2*M)*L^3*K^2*M^2-3/(4*K*L^4*M^3-4*K*L^3*M^4-4*K*L^4*M ^2+4*K*L^2*M^4+K*L^4*M-3*K*L^3*M^2+3*K*L^2*M^3-K*L*M^4-L^4*M^2+L^2*M^4+4*K*L^3* M-4*K*L*M^3+2*L^3*M^2-2*L^2*M^3-K*L^3-K*L^2*M+K*L*M^2+K*M^3+2*L^3*M-2*L*M^3-4*L ^2*M+4*L*M^2-L^2+M^2+2*L-2*M)*L^2*K^2*M^3-2/(4*K*L^4*M^3-4*K*L^3*M^4-4*K*L^4*M^ 2+4*K*L^2*M^4+K*L^4*M-3*K*L^3*M^2+3*K*L^2*M^3-K*L*M^4-L^4*M^2+L^2*M^4+4*K*L^3*M -4*K*L*M^3+2*L^3*M^2-2*L^2*M^3-K*L^3-K*L^2*M+K*L*M^2+K*M^3+2*L^3*M-2*L*M^3-4*L^ 2*M+4*L*M^2-L^2+M^2+2*L-2*M)*L^4*K*M^2-5/(4*K*L^4*M^3-4*K*L^3*M^4-4*K*L^4*M^2+4 *K*L^2*M^4+K*L^4*M-3*K*L^3*M^2+3*K*L^2*M^3-K*L*M^4-L^4*M^2+L^2*M^4+4*K*L^3*M-4* K*L*M^3+2*L^3*M^2-2*L^2*M^3-K*L^3-K*L^2*M+K*L*M^2+K*M^3+2*L^3*M-2*L*M^3-4*L^2*M +4*L*M^2-L^2+M^2+2*L-2*M)*L^3*K*M^3+1/(4*K*L^4*M^3-4*K*L^3*M^4-4*K*L^4*M^2+4*K* L^2*M^4+K*L^4*M-3*K*L^3*M^2+3*K*L^2*M^3-K*L*M^4-L^4*M^2+L^2*M^4+4*K*L^3*M-4*K*L *M^3+2*L^3*M^2-2*L^2*M^3-K*L^3-K*L^2*M+K*L*M^2+K*M^3+2*L^3*M-2*L*M^3-4*L^2*M+4* L*M^2-L^2+M^2+2*L-2*M)*L^2*K*M^4+1/(4*K*L^4*M^3-4*K*L^3*M^4-4*K*L^4*M^2+4*K*L^2 *M^4+K*L^4*M-3*K*L^3*M^2+3*K*L^2*M^3-K*L*M^4-L^4*M^2+L^2*M^4+4*K*L^3*M-4*K*L*M^ 3+2*L^3*M^2-2*L^2*M^3-K*L^3-K*L^2*M+K*L*M^2+K*M^3+2*L^3*M-2*L*M^3-4*L^2*M+4*L*M ^2-L^2+M^2+2*L-2*M)*L^3*K^3*M^4-6/(4*K*L^4*M^3-4*K*L^3*M^4-4*K*L^4*M^2+4*K*L^2* M^4+K*L^4*M-3*K*L^3*M^2+3*K*L^2*M^3-K*L*M^4-L^4*M^2+L^2*M^4+4*K*L^3*M-4*K*L*M^3 +2*L^3*M^2-2*L^2*M^3-K*L^3-K*L^2*M+K*L*M^2+K*M^3+2*L^3*M-2*L*M^3-4*L^2*M+4*L*M^ 2-L^2+M^2+2*L-2*M)*L*K*M+1/(4*K*L^4*M^3-4*K*L^3*M^4-4*K*L^4*M^2+4*K*L^2*M^4+K*L ^4*M-3*K*L^3*M^2+3*K*L^2*M^3-K*L*M^4-L^4*M^2+L^2*M^4+4*K*L^3*M-4*K*L*M^3+2*L^3* M^2-2*L^2*M^3-K*L^3-K*L^2*M+K*L*M^2+K*M^3+2*L^3*M-2*L*M^3-4*L^2*M+4*L*M^2-L^2+M ^2+2*L-2*M)*L^2*K^3*M+1/(4*K*L^4*M^3-4*K*L^3*M^4-4*K*L^4*M^2+4*K*L^2*M^4+K*L^4* M-3*K*L^3*M^2+3*K*L^2*M^3-K*L*M^4-L^4*M^2+L^2*M^4+4*K*L^3*M-4*K*L*M^3+2*L^3*M^2 -2*L^2*M^3-K*L^3-K*L^2*M+K*L*M^2+K*M^3+2*L^3*M-2*L*M^3-4*L^2*M+4*L*M^2-L^2+M^2+ 2*L-2*M)*L*K^3*M^2+9/(4*K*L^4*M^3-4*K*L^3*M^4-4*K*L^4*M^2+4*K*L^2*M^4+K*L^4*M-3 *K*L^3*M^2+3*K*L^2*M^3-K*L*M^4-L^4*M^2+L^2*M^4+4*K*L^3*M-4*K*L*M^3+2*L^3*M^2-2* L^2*M^3-K*L^3-K*L^2*M+K*L*M^2+K*M^3+2*L^3*M-2*L*M^3-4*L^2*M+4*L*M^2-L^2+M^2+2*L -2*M)*L^3*K^2*M+6/(4*K*L^4*M^3-4*K*L^3*M^4-4*K*L^4*M^2+4*K*L^2*M^4+K*L^4*M-3*K* L^3*M^2+3*K*L^2*M^3-K*L*M^4-L^4*M^2+L^2*M^4+4*K*L^3*M-4*K*L*M^3+2*L^3*M^2-2*L^2 *M^3-K*L^3-K*L^2*M+K*L*M^2+K*M^3+2*L^3*M-2*L*M^3-4*L^2*M+4*L*M^2-L^2+M^2+2*L-2* M)*L^2*K^2*M^2-3/(4*K*L^4*M^3-4*K*L^3*M^4-4*K*L^4*M^2+4*K*L^2*M^4+K*L^4*M-3*K*L ^3*M^2+3*K*L^2*M^3-K*L*M^4-L^4*M^2+L^2*M^4+4*K*L^3*M-4*K*L*M^3+2*L^3*M^2-2*L^2* M^3-K*L^3-K*L^2*M+K*L*M^2+K*M^3+2*L^3*M-2*L*M^3-4*L^2*M+4*L*M^2-L^2+M^2+2*L-2*M )*L*K^2*M^3+2/(4*K*L^4*M^3-4*K*L^3*M^4-4*K*L^4*M^2+4*K*L^2*M^4+K*L^4*M-3*K*L^3* M^2+3*K*L^2*M^3-K*L*M^4-L^4*M^2+L^2*M^4+4*K*L^3*M-4*K*L*M^3+2*L^3*M^2-2*L^2*M^3 -K*L^3-K*L^2*M+K*L*M^2+K*M^3+2*L^3*M-2*L*M^3-4*L^2*M+4*L*M^2-L^2+M^2+2*L-2*M)*L ^3*K*M^2+4/(4*K*L^4*M^3-4*K*L^3*M^4-4*K*L^4*M^2+4*K*L^2*M^4+K*L^4*M-3*K*L^3*M^2 +3*K*L^2*M^3-K*L*M^4-L^4*M^2+L^2*M^4+4*K*L^3*M-4*K*L*M^3+2*L^3*M^2-2*L^2*M^3-K* L^3-K*L^2*M+K*L*M^2+K*M^3+2*L^3*M-2*L*M^3-4*L^2*M+4*L*M^2-L^2+M^2+2*L-2*M)*L^2* K*M^3+3/(4*K*L^4*M^3-4*K*L^3*M^4-4*K*L^4*M^2+4*K*L^2*M^4+K*L^4*M-3*K*L^3*M^2+3* K*L^2*M^3-K*L*M^4-L^4*M^2+L^2*M^4+4*K*L^3*M-4*K*L*M^3+2*L^3*M^2-2*L^2*M^3-K*L^3 -K*L^2*M+K*L*M^2+K*M^3+2*L^3*M-2*L*M^3-4*L^2*M+4*L*M^2-L^2+M^2+2*L-2*M)*L^2*K^2 *M+6/(4*K*L^4*M^3-4*K*L^3*M^4-4*K*L^4*M^2+4*K*L^2*M^4+K*L^4*M-3*K*L^3*M^2+3*K*L ^2*M^3-K*L*M^4-L^4*M^2+L^2*M^4+4*K*L^3*M-4*K*L*M^3+2*L^3*M^2-2*L^2*M^3-K*L^3-K* L^2*M+K*L*M^2+K*M^3+2*L^3*M-2*L*M^3-4*L^2*M+4*L*M^2-L^2+M^2+2*L-2*M)*L*K^2*M^2+ 12/(4*K*L^4*M^3-4*K*L^3*M^4-4*K*L^4*M^2+4*K*L^2*M^4+K*L^4*M-3*K*L^3*M^2+3*K*L^2 *M^3-K*L*M^4-L^4*M^2+L^2*M^4+4*K*L^3*M-4*K*L*M^3+2*L^3*M^2-2*L^2*M^3-K*L^3-K*L^ 2*M+K*L*M^2+K*M^3+2*L^3*M-2*L*M^3-4*L^2*M+4*L*M^2-L^2+M^2+2*L-2*M)*L^3*K*M+6/(4 *K*L^4*M^3-4*K*L^3*M^4-4*K*L^4*M^2+4*K*L^2*M^4+K*L^4*M-3*K*L^3*M^2+3*K*L^2*M^3- K*L*M^4-L^4*M^2+L^2*M^4+4*K*L^3*M-4*K*L*M^3+2*L^3*M^2-2*L^2*M^3-K*L^3-K*L^2*M+K *L*M^2+K*M^3+2*L^3*M-2*L*M^3-4*L^2*M+4*L*M^2-L^2+M^2+2*L-2*M)*L^2*K*M^2-6/(4*K* L^4*M^3-4*K*L^3*M^4-4*K*L^4*M^2+4*K*L^2*M^4+K*L^4*M-3*K*L^3*M^2+3*K*L^2*M^3-K*L *M^4-L^4*M^2+L^2*M^4+4*K*L^3*M-4*K*L*M^3+2*L^3*M^2-2*L^2*M^3-K*L^3-K*L^2*M+K*L* M^2+K*M^3+2*L^3*M-2*L*M^3-4*L^2*M+4*L*M^2-L^2+M^2+2*L-2*M)*L*K*M^3-2/(4*K*L^4*M ^3-4*K*L^3*M^4-4*K*L^4*M^2+4*K*L^2*M^4+K*L^4*M-3*K*L^3*M^2+3*K*L^2*M^3-K*L*M^4- L^4*M^2+L^2*M^4+4*K*L^3*M-4*K*L*M^3+2*L^3*M^2-2*L^2*M^3-K*L^3-K*L^2*M+K*L*M^2+K *M^3+2*L^3*M-2*L*M^3-4*L^2*M+4*L*M^2-L^2+M^2+2*L-2*M)*L^2*K^3*M^3-6/(4*K*L^4*M^ 3-4*K*L^3*M^4-4*K*L^4*M^2+4*K*L^2*M^4+K*L^4*M-3*K*L^3*M^2+3*K*L^2*M^3-K*L*M^4-L ^4*M^2+L^2*M^4+4*K*L^3*M-4*K*L*M^3+2*L^3*M^2-2*L^2*M^3-K*L^3-K*L^2*M+K*L*M^2+K* M^3+2*L^3*M-2*L*M^3-4*L^2*M+4*L*M^2-L^2+M^2+2*L-2*M)*L*K^2*M-13/(4*K*L^4*M^3-4* K*L^3*M^4-4*K*L^4*M^2+4*K*L^2*M^4+K*L^4*M-3*K*L^3*M^2+3*K*L^2*M^3-K*L*M^4-L^4*M ^2+L^2*M^4+4*K*L^3*M-4*K*L*M^3+2*L^3*M^2-2*L^2*M^3-K*L^3-K*L^2*M+K*L*M^2+K*M^3+ 2*L^3*M-2*L*M^3-4*L^2*M+4*L*M^2-L^2+M^2+2*L-2*M)*L^2*K*M+5/(4*K*L^4*M^3-4*K*L^3 *M^4-4*K*L^4*M^2+4*K*L^2*M^4+K*L^4*M-3*K*L^3*M^2+3*K*L^2*M^3-K*L*M^4-L^4*M^2+L^ 2*M^4+4*K*L^3*M-4*K*L*M^3+2*L^3*M^2-2*L^2*M^3-K*L^3-K*L^2*M+K*L*M^2+K*M^3+2*L^3 *M-2*L*M^3-4*L^2*M+4*L*M^2-L^2+M^2+2*L-2*M)*L*K*M^2] and the Sinkhorn limit is 4 3 3 4 3 2 4 2 3 4 2 2 3 3 2 [[K z, - (2 K L M - K L M - K L M - 2 K L M - 5 K L M 3 2 3 4 2 4 2 3 3 3 2 2 3 3 - 5 K L M + K L M + K L M + 2 K L M + 10 K L M + 2 K L M 3 2 3 2 2 3 2 2 2 2 3 3 2 + K L M + K L M + 4 K L M + 10 K L M + 4 K L M - K L 3 3 2 2 3 2 2 2 2 2 3 2 2 - 4 K L M - K M - K L - 14 K L M - 14 K L M - K M + K L 2 2 2 3 2 2 3 2 2 + 4 K L M + K M - K L - 6 K L M - 6 K L M - K M + 3 K L + 3 K M 2 2 2 2 + 5 K L + 16 K L M + 5 K M - 5 K L - 5 K M + L + 2 L M + M - 2 K - 4 L 3 4 3 3 4 3 2 4 2 3 4 2 2 - 4 M + 4) K z /(%3) + (6 K L M - 3 K L M - 3 K L M - 6 K L M 3 3 2 3 2 3 4 2 4 2 3 3 - 8 K L M - 8 K L M + 3 K L M + 3 K L M + 2 K L M 3 2 2 3 3 3 2 3 2 2 3 2 2 2 + 12 K L M + 2 K L M + 8 K L M + 8 K L M + 2 K L M + 6 K L M 2 3 3 2 3 3 2 2 3 2 2 + 2 K L M - 2 K L - 12 K L M - 2 K M + K L - 4 K L M 2 2 2 3 2 2 2 2 2 2 2 - 4 K L M + K M - 5 K L - 8 K L M - 5 K M + 2 K L M + 2 K L M 2 2 2 2 2 + 7 K L + 7 K M - 2 K L - 8 K L M - 2 K M + 6 K L + 6 K M - L - 2 L M 2 2 2 3 2 2 3 2 2 2 - M - 4 K + 4 L + 4 M - 4) z /(%3) + (2 K L M - K L M - K L M 2 2 2 2 2 2 2 3 2 2 - 2 K L M - 4 K L M + K L + K L M - K L + 2 K L M + 3 K L M 2 2 2 2 + 3 K L + 2 K L M - K M + L M + L M - 4 K L - 2 L M - L - M + 2) M/(%3 3 3 3 3 3 2 3 2 3 3 2 2 2 3 2 ) - (6 K L M - 3 K L M - 3 K L M - 6 K L M - 3 K L M 2 2 3 3 2 3 2 2 3 2 2 2 2 3 - 7 K L M + 3 K L M + 3 K L M - K L M + 4 K L M + 3 K L M 2 2 2 2 2 3 3 2 2 3 2 2 + 10 K L M + 9 K L M - K M + K L M - K L M + 2 K L M - K L 2 2 2 2 2 2 2 - 12 K L M - K M + 4 K L M + 2 K L M - 3 K L - 11 K L M - 4 K M 2 2 2 2 + L M + L M + 4 K L + 6 K M - L - 4 L M - M + 3 L + 3 M - 2) z/(%3), ( 4 3 3 4 3 2 4 2 3 4 2 2 3 3 2 3 2 3 2 K L M - K L M - K L M - 2 K L M - 5 K L M - 5 K L M 4 2 4 2 3 3 3 2 2 3 3 3 2 + K L M + K L M + 2 K L M + 10 K L M + 2 K L M + K L M 3 2 2 3 2 2 2 2 3 3 2 3 3 2 + K L M + 4 K L M + 10 K L M + 4 K L M - K L - 4 K L M - K M 2 3 2 2 2 2 2 3 2 2 2 2 2 - K L - 14 K L M - 14 K L M - K M + K L + 4 K L M + K M 3 2 2 3 2 2 2 - K L - 6 K L M - 6 K L M - K M + 3 K L + 3 K M + 5 K L + 16 K L M 2 2 2 3 + 5 K M - 5 K L - 5 K M + L + 2 L M + M - 2 K - 4 L - 4 M + 4) K z /(%3 4 3 3 4 3 2 4 2 3 4 2 2 3 3 2 ) - (6 K L M - 3 K L M - 3 K L M - 6 K L M - 8 K L M 3 2 3 4 2 4 2 3 3 3 2 2 3 3 - 8 K L M + 3 K L M + 3 K L M + 2 K L M + 12 K L M + 2 K L M 3 2 3 2 2 3 2 2 2 2 3 3 2 + 8 K L M + 8 K L M + 2 K L M + 6 K L M + 2 K L M - 2 K L 3 3 2 2 3 2 2 2 2 2 3 2 2 - 12 K L M - 2 K M + K L - 4 K L M - 4 K L M + K M - 5 K L 2 2 2 2 2 2 2 2 - 8 K L M - 5 K M + 2 K L M + 2 K L M + 7 K L + 7 K M - 2 K L 2 2 2 - 8 K L M - 2 K M + 6 K L + 6 K M - L - 2 L M - M - 4 K + 4 L + 4 M - 4 2 2 2 3 2 2 2 2 3 2 2 2 2 ) z /(%3) - (2 K L M - K L M - K L M - 2 K L M - 4 K L M 2 2 2 2 2 3 2 2 + K L M + K M + 3 K L M + 2 K L M - K M - K L + 2 K L M + 3 K M 2 2 3 3 3 + L M + L M - 4 K M - 2 L M - L - M + 2) L/(%3) + (6 K L M 3 3 2 3 2 3 3 2 2 2 3 2 2 2 3 - 3 K L M - 3 K L M - 6 K L M - 7 K L M - 3 K L M 3 2 3 2 2 3 2 2 2 2 3 2 3 + 3 K L M + 3 K L M + 3 K L M + 4 K L M - K L M - K L 2 2 2 2 3 2 2 3 2 2 2 + 9 K L M + 10 K L M + 2 K L M - K L M + K L M - K L - 12 K L M 2 2 2 2 2 2 2 2 - K M + 2 K L M + 4 K L M - 4 K L - 11 K L M - 3 K M + L M + L M 2 2 4 3 3 + 6 K L + 4 K M - L - 4 L M - M + 3 L + 3 M - 2) z/(%3)], [- (2 K L M 4 3 2 4 2 3 4 2 2 3 3 2 3 2 3 4 2 - K L M - K L M - 2 K L M - 5 K L M - 5 K L M + K L M 4 2 3 3 3 2 2 3 3 3 2 3 2 + K L M + 2 K L M + 10 K L M + 2 K L M + K L M + K L M 2 3 2 2 2 2 3 3 2 3 3 2 2 3 + 4 K L M + 10 K L M + 4 K L M - K L - 4 K L M - K M - K L 2 2 2 2 2 3 2 2 2 2 2 3 - 14 K L M - 14 K L M - K M + K L + 4 K L M + K M - K L 2 2 3 2 2 2 - 6 K L M - 6 K L M - K M + 3 K L + 3 K M + 5 K L + 16 K L M 2 2 2 3 + 5 K M - 5 K L - 5 K M + L + 2 L M + M - 2 K - 4 L - 4 M + 4) K z /(%3 4 3 3 4 3 2 4 2 3 4 2 2 3 3 2 ) + (6 K L M - 3 K L M - 3 K L M - 6 K L M - 8 K L M 3 2 3 4 2 4 2 3 3 3 2 2 3 3 - 8 K L M + 3 K L M + 3 K L M + 2 K L M + 12 K L M + 2 K L M 3 2 3 2 2 3 2 2 2 2 3 3 2 + 8 K L M + 8 K L M + 2 K L M + 6 K L M + 2 K L M - 2 K L 3 3 2 2 3 2 2 2 2 2 3 2 2 - 12 K L M - 2 K M + K L - 4 K L M - 4 K L M + K M - 5 K L 2 2 2 2 2 2 2 2 - 8 K L M - 5 K M + 2 K L M + 2 K L M + 7 K L + 7 K M - 2 K L 2 2 2 - 8 K L M - 2 K M + 6 K L + 6 K M - L - 2 L M - M - 4 K + 4 L + 4 M - 4 2 2 3 2 2 3 2 2 2 2 2 2 2 ) z /(%3) + (2 K L M - K L M - K L M - 2 K L M - 4 K L M 2 2 2 3 2 2 2 2 + K L + K L M - K L + 2 K L M + 3 K L M + 3 K L + 2 K L M - K M 2 2 3 3 3 + L M + L M - 4 K L - 2 L M - L - M + 2) M/(%3) - (6 K L M 3 3 2 3 2 3 3 2 2 2 3 2 2 2 3 - 3 K L M - 3 K L M - 6 K L M - 3 K L M - 7 K L M 3 2 3 2 2 3 2 2 2 2 3 2 2 + 3 K L M + 3 K L M - K L M + 4 K L M + 3 K L M + 10 K L M 2 2 2 3 3 2 2 3 2 2 2 + 9 K L M - K M + K L M - K L M + 2 K L M - K L - 12 K L M 2 2 2 2 2 2 2 2 - K M + 4 K L M + 2 K L M - 3 K L - 11 K L M - 4 K M + L M + L M 2 2 4 4 4 + 4 K L + 6 K M - L - 4 L M - M + 3 L + 3 M - 2) z/(%3), (2 K L M 4 3 4 4 4 2 4 3 3 4 2 4 3 4 3 3 3 4 - 2 K L M - K L M - 4 K L M + K L M - 5 K L M - 5 K L M 4 3 2 4 2 3 3 3 3 3 2 4 4 3 + 2 K L M + 2 K L M + 10 K L M + 4 K L M + K L M 4 2 2 4 3 3 4 3 3 2 3 2 3 3 4 + 2 K L M - K L M + 2 K L M + 10 K L M + 2 K L M - 2 K L M 2 4 2 2 3 3 2 2 4 4 2 3 3 + 4 K L M + 10 K L M + 4 K L M - 2 K L M - 6 K L M 3 2 2 2 3 2 2 2 3 2 4 3 3 3 2 - 14 K L M - 12 K L M - 16 K L M - 2 K L M - K L - 2 K L M 3 3 2 4 2 3 2 2 2 2 3 2 4 4 + K M - K L - 8 K L M - 6 K L M + 2 K L M + K M - K L M 3 2 2 3 4 3 2 3 2 3 - 6 K L M - 6 K L M - K L M + 2 K L + 4 K L M + 4 K L 2 2 2 2 2 3 3 2 2 3 + 20 K L M + 14 K L M - 2 K M + 4 K L M + 16 K L M + 6 K L M 2 2 2 3 2 2 3 2 2 - 2 K L - 4 K L M + 2 K L + 4 K L M - 2 K L M + L M + 2 L M 3 2 2 2 2 2 2 + L M - 4 K L - 2 K M - 8 K L - 18 K L M - 2 K M - 4 L M - 4 L M 2 2 3 + 7 K L + 3 K M - L + 2 L M - M + 2 K + 4 L + 4 M - 4) K z /(%1) - ( 4 4 4 4 4 3 4 3 4 4 4 2 4 3 3 6 K L M + 2 K L M - 8 K L M - 4 K L M - 12 K L M 4 2 4 3 4 3 3 3 4 4 3 2 4 2 3 + 4 K L M - 8 K L M - 8 K L M + 4 K L M + 8 K L M 3 4 2 3 3 3 3 2 4 4 3 4 2 2 - 4 K L M + 12 K L M + 8 K L M + 4 K L M + 6 K L M 4 3 3 4 3 3 2 3 2 3 3 4 - 4 K L M + 4 K L M + 24 K L M + 8 K L M - 4 K L M 2 4 2 2 3 3 2 2 4 4 2 3 3 + 2 K L M + 6 K L M + 2 K L M - 6 K L M - 4 K L M 3 2 2 3 3 2 4 2 2 3 3 3 3 2 - 24 K L M - 4 K L M + 2 K L M - 8 K L M - 3 K L - 11 K L M 3 2 3 3 2 3 2 2 2 2 3 3 2 - 5 K L M + 3 K M - 11 K L M - 14 K L M - 3 K L M + 2 K L M 2 3 3 2 3 2 3 2 2 2 2 + 2 K L M + 4 K L + 12 K L M - 2 K L + 8 K L M + 14 K L M 3 2 2 3 2 2 2 2 2 3 - 2 K L M - 8 K L M - 2 K L M + 9 K L + 8 K L M + K M - K L 2 2 3 3 2 2 3 2 2 + 3 K L M + 5 K L M + K M - L M - 2 L M - L M - 8 K L - 6 K M 2 2 2 2 2 + 6 K L + 4 K L M - 2 K M + 4 L M + 4 L M - 10 K L - 2 K M + L 2 2 2 4 3 2 4 2 - 2 L M + M + 4 K - 4 L - 4 M + 4) z /(%1) - (2 K L M + 2 K L M 2 3 3 2 4 2 3 2 2 2 3 4 2 - 4 K L M - 2 K L M - 4 K L M + 2 K L M - 4 K L M 2 2 2 4 3 2 2 3 2 2 2 2 + 4 K L M + 2 K L M + 2 K L M + 2 K L + 2 K L M - 2 K L M 4 3 2 2 4 3 2 2 2 3 2 - K L + 4 K L M + 5 K L M + L M + L M - 2 K L + 2 K L - 6 K L M 2 3 2 2 3 2 2 2 - 4 K L M - 2 L M - 5 K L + K M - L - 2 L M - L M + 4 K L + 2 L 3 4 4 3 4 3 3 3 4 + 2 L M + L + M - 2) M/(%1) + (6 K L M + 4 K L M - 10 K L M 3 4 2 3 3 3 3 2 4 2 4 3 2 3 4 - 5 K L M - 12 K L M + 5 K L M - 3 K L M - 7 K L M 3 3 2 3 2 3 2 4 2 2 3 3 2 2 4 + 2 K L M + 10 K L M - 4 K L M + 4 K L M + 6 K L M 3 3 3 2 2 3 3 2 4 2 3 2 2 2 3 + 5 K L M + 6 K L M - 5 K L M + K L M + 14 K L M + 15 K L M 2 4 4 2 3 3 2 4 3 2 2 3 - 2 K L M + K L M - K L M + 2 K L M - 6 K L M + 6 K L M 2 2 2 2 3 2 3 2 3 2 2 2 2 - 16 K L M - 10 K L M + 6 K L M - 2 K L - 16 K L M - 3 K L M 2 3 3 2 2 3 3 2 2 3 2 2 + 3 K M + 2 K L M - 10 K L M - 12 K L M + L M + L M + 2 K L 2 3 2 2 3 2 2 3 + 12 K L M - 2 K L - 6 K L M + 10 K L M + 2 K M - 4 L M - 2 L M 2 2 2 2 + 5 K L + 11 K L M + 2 K M + 4 L M - 4 K L - 6 K M + 2 L M + 2 M - L 3 3 3 3 3 2 3 2 3 2 3 2 - 5 M + 2) z/(%1), - (K L M - K L M - K L M - 2 K L M 2 2 3 3 2 3 2 2 3 2 2 2 2 3 - 2 K L M + K L M + K L M + 2 K L M + 6 K L M + 2 K L M 3 2 2 2 2 3 2 2 3 2 2 - K L M - 3 K L M - 3 K L M + K L M + 3 K L M + K L M - K L 2 2 3 2 2 3 2 2 2 - K M - K L - 6 K L M - 6 K L M - K M + K L + K M + 3 K L 2 2 2 2 2 + 6 K L M + 3 K M - L M - L M - K L - K M + L + 4 L M + M - K - 3 L 2 3 3 3 3 3 3 2 3 2 3 - 3 M + 2) K z /((L M - 1) %2) + (5 K L M - 4 K L M - 4 K L M 3 2 2 2 3 2 2 2 3 3 2 3 2 2 3 - 2 K L M - 6 K L M - 6 K L M + 4 K L M + 4 K L M + 4 K L M 2 2 2 2 3 3 3 2 2 3 + 12 K L M + 4 K L M - 3 K L M + K L M + 5 K L M + K L M 2 2 2 2 2 2 2 2 2 - 3 K L - 6 K L M - 3 K M - 4 K L M - 4 K L M + 2 K L + 2 K M 2 2 2 2 - K L - 4 K L M - K M + 4 K L + 4 K M - L - 2 L M - M - K + 4 L + 4 M 2 3 3 3 3 3 2 3 2 3 - 4) K z /((L M - 1) %2) - (7 K L M - 5 K L M - 5 K L M 3 2 2 2 3 2 2 2 3 3 2 3 2 2 3 - 4 K L M - 3 K L M - 3 K L M + 5 K L M + 5 K L M + K L M 2 2 2 2 3 3 2 2 2 2 2 2 + 2 K L M + K L M - 3 K L M + 6 K L M + 6 K L M - 4 K L M 2 2 2 2 2 2 2 2 2 - 2 K L - 8 K L M - 2 K M + 6 K L M + 6 K L M + K L + K M 2 2 2 2 - 2 K L - 8 K L M - 2 K M + L M + L M + 2 K L + 2 K M - 2 L M - L - M 2 3 3 2 3 2 2 2 3 2 2 2 + 2) z/((L M - 1) %2) + (3 K L M - 2 K L M - 2 K L M - 2 K L M 3 2 2 3 2 2 2 2 2 2 2 - K L M - K L M + 2 K L M + 2 K L M - 2 K L M - K L M 2 2 2 2 2 2 + 5 K L M + 5 K L M - K L - 4 K L M - K M + L M + L M - 2 L M - L 4 3 3 4 3 2 4 2 3 4 2 2 - M + 2)/((L M - 1) %2)], [(2 K L M - K L M - K L M - 2 K L M 3 3 2 3 2 3 4 2 4 2 3 3 3 2 2 - 5 K L M - 5 K L M + K L M + K L M + 2 K L M + 10 K L M 3 3 3 2 3 2 2 3 2 2 2 2 3 + 2 K L M + K L M + K L M + 4 K L M + 10 K L M + 4 K L M 3 2 3 3 2 2 3 2 2 2 2 2 3 - K L - 4 K L M - K M - K L - 14 K L M - 14 K L M - K M 2 2 2 2 2 3 2 2 3 2 + K L + 4 K L M + K M - K L - 6 K L M - 6 K L M - K M + 3 K L 2 2 2 2 2 + 3 K M + 5 K L + 16 K L M + 5 K M - 5 K L - 5 K M + L + 2 L M + M 3 4 3 3 4 3 2 4 2 3 - 2 K - 4 L - 4 M + 4) K z /(%3) - (6 K L M - 3 K L M - 3 K L M 4 2 2 3 3 2 3 2 3 4 2 4 2 3 3 - 6 K L M - 8 K L M - 8 K L M + 3 K L M + 3 K L M + 2 K L M 3 2 2 3 3 3 2 3 2 2 3 2 2 2 + 12 K L M + 2 K L M + 8 K L M + 8 K L M + 2 K L M + 6 K L M 2 3 3 2 3 3 2 2 3 2 2 + 2 K L M - 2 K L - 12 K L M - 2 K M + K L - 4 K L M 2 2 2 3 2 2 2 2 2 2 2 - 4 K L M + K M - 5 K L - 8 K L M - 5 K M + 2 K L M + 2 K L M 2 2 2 2 2 + 7 K L + 7 K M - 2 K L - 8 K L M - 2 K M + 6 K L + 6 K M - L - 2 L M 2 2 2 2 3 2 2 2 2 3 - M - 4 K + 4 L + 4 M - 4) z /(%3) - (2 K L M - K L M - K L M 2 2 2 2 2 2 2 2 2 3 - 2 K L M - 4 K L M + K L M + K M + 3 K L M + 2 K L M - K M 2 2 2 2 - K L + 2 K L M + 3 K M + L M + L M - 4 K M - 2 L M - L - M + 2) L/(%3 3 3 3 3 3 2 3 2 3 3 2 2 2 3 2 ) + (6 K L M - 3 K L M - 3 K L M - 6 K L M - 7 K L M 2 2 3 3 2 3 2 2 3 2 2 2 2 3 - 3 K L M + 3 K L M + 3 K L M + 3 K L M + 4 K L M - K L M 2 3 2 2 2 2 3 2 2 3 2 2 - K L + 9 K L M + 10 K L M + 2 K L M - K L M + K L M - K L 2 2 2 2 2 2 2 - 12 K L M - K M + 2 K L M + 4 K L M - 4 K L - 11 K L M - 3 K M 2 2 2 2 + L M + L M + 6 K L + 4 K M - L - 4 L M - M + 3 L + 3 M - 2) z/(%3), - 3 3 3 3 3 2 3 2 3 2 3 2 2 2 3 3 2 (K L M - K L M - K L M - 2 K L M - 2 K L M + K L M 3 2 2 3 2 2 2 2 3 3 2 2 + K L M + 2 K L M + 6 K L M + 2 K L M - K L M - 3 K L M 2 2 3 2 2 3 2 2 2 2 3 - 3 K L M + K L M + 3 K L M + K L M - K L - K M - K L 2 2 3 2 2 2 2 - 6 K L M - 6 K L M - K M + K L + K M + 3 K L + 6 K L M + 3 K M 2 2 2 2 2 3 - L M - L M - K L - K M + L + 4 L M + M - K - 3 L - 3 M + 2) K z /( 3 3 3 3 3 2 3 2 3 3 2 2 (L M - 1) %2) + (5 K L M - 4 K L M - 4 K L M - 2 K L M 2 3 2 2 2 3 3 2 3 2 2 3 - 6 K L M - 6 K L M + 4 K L M + 4 K L M + 4 K L M 2 2 2 2 3 3 3 2 2 3 + 12 K L M + 4 K L M - 3 K L M + K L M + 5 K L M + K L M 2 2 2 2 2 2 2 2 2 - 3 K L - 6 K L M - 3 K M - 4 K L M - 4 K L M + 2 K L + 2 K M 2 2 2 2 - K L - 4 K L M - K M + 4 K L + 4 K M - L - 2 L M - M - K + 4 L + 4 M 2 3 3 3 3 3 2 3 2 3 - 4) K z /((L M - 1) %2) - (7 K L M - 5 K L M - 5 K L M 3 2 2 2 3 2 2 2 3 3 2 3 2 2 3 - 4 K L M - 3 K L M - 3 K L M + 5 K L M + 5 K L M + K L M 2 2 2 2 3 3 2 2 2 2 2 2 + 2 K L M + K L M - 3 K L M + 6 K L M + 6 K L M - 4 K L M 2 2 2 2 2 2 2 2 2 - 2 K L - 8 K L M - 2 K M + 6 K L M + 6 K L M + K L + K M 2 2 2 2 - 2 K L - 8 K L M - 2 K M + L M + L M + 2 K L + 2 K M - 2 L M - L - M 2 3 3 2 3 2 2 2 3 2 2 2 + 2) z/((L M - 1) %2) + (3 K L M - 2 K L M - 2 K L M - 2 K L M 3 2 2 3 2 2 2 2 2 2 2 - K L M - K L M + 2 K L M + 2 K L M - 2 K L M - K L M 2 2 2 2 2 2 + 5 K L M + 5 K L M - K L - 4 K L M - K M + L M + L M - 2 L M - L 4 4 4 4 4 3 4 4 2 - M + 2)/((L M - 1) %2), - (2 K L M - 2 K L M + K L M 4 3 3 4 2 4 3 4 3 3 3 4 4 3 2 - 4 K L M - K L M - 5 K L M - 5 K L M + 2 K L M 4 2 3 3 4 2 3 3 3 4 3 4 2 2 4 3 + 2 K L M + 4 K L M + 10 K L M - K L M + 2 K L M + K L M 3 4 3 3 2 3 2 3 3 4 2 4 2 - 2 K L M + 2 K L M + 10 K L M + 2 K L M + 4 K L M 2 3 3 2 2 4 4 2 3 2 2 3 3 + 10 K L M + 4 K L M - 2 K L M - 14 K L M - 6 K L M 2 4 2 3 2 2 2 3 3 3 3 2 3 3 - 2 K L M - 16 K L M - 12 K L M + K L - 2 K L M - K M 2 4 2 3 2 2 2 2 3 2 4 4 3 2 + K L + 2 K L M - 6 K L M - 8 K L M - K M - K L M - 6 K L M 2 3 4 3 3 2 2 3 2 2 - 6 K L M - K L M + 4 K L M + 2 K M - 2 K L + 14 K L M 2 2 2 3 3 2 2 3 2 + 20 K L M + 4 K M + 6 K L M + 16 K L M + 4 K L M - 4 K L M 2 2 2 2 3 3 2 2 3 2 - 2 K M - 2 K L M + 4 K L M + 2 K M + L M + 2 L M + L M - 2 K L 2 2 2 2 2 - 4 K M - 2 K L - 18 K L M - 8 K M - 4 L M - 4 L M + 3 K L + 7 K M 2 2 3 4 4 4 - L + 2 L M - M + 2 K + 4 L + 4 M - 4) K z /(%1) + (6 K L M 4 4 3 4 3 4 4 4 2 4 3 3 4 2 4 - 8 K L M + 2 K L M + 4 K L M - 12 K L M - 4 K L M 3 4 3 3 3 4 4 3 2 4 2 3 3 4 2 - 8 K L M - 8 K L M + 8 K L M + 4 K L M + 8 K L M 3 3 3 3 2 4 4 3 4 2 2 4 3 + 12 K L M - 4 K L M - 4 K L M + 6 K L M + 4 K L M 3 4 3 3 2 3 2 3 3 4 2 4 2 - 4 K L M + 8 K L M + 24 K L M + 4 K L M + 2 K L M 2 3 3 2 2 4 4 2 3 3 3 2 2 + 6 K L M + 2 K L M - 6 K L M - 4 K L M - 24 K L M 3 3 2 3 2 2 4 3 3 3 2 3 2 - 4 K L M - 8 K L M + 2 K L M + 3 K L - 5 K L M - 11 K L M 3 3 2 3 2 2 2 2 3 3 2 2 3 - 3 K M - 3 K L M - 14 K L M - 11 K L M + 2 K L M + 2 K L M 3 3 2 2 2 2 2 2 3 3 + 12 K L M + 4 K M + 14 K L M + 8 K L M - 2 K M - 2 K L M 2 2 3 2 2 2 2 2 3 2 - 8 K L M - 2 K L M + K L + 8 K L M + 9 K M + K L + 5 K L M 2 3 3 2 2 3 2 2 2 + 3 K L M - K M - L M - 2 L M - L M - 6 K L - 8 K M - 2 K L 2 2 2 2 2 + 4 K L M + 6 K M + 4 L M + 4 L M - 2 K L - 10 K M + L - 2 L M + M 2 2 3 4 2 3 3 2 2 4 + 4 K - 4 L - 4 M + 4) z /(%1) + (2 K L M - 4 K L M + 2 K L M 2 3 2 2 2 3 2 4 2 4 2 2 2 2 3 + 2 K L M - 4 K L M - 2 K L M - 4 K L M + 4 K L M + 2 K L M 4 2 2 2 2 2 3 2 2 3 4 + 2 K L M - 2 K L M + 2 K L M + 2 K M + 5 K L M + 4 K L M - K M 2 3 4 2 2 2 2 3 3 2 + L M + L M - 2 K M - 4 K L M - 6 K L M + 2 K M - 2 L M + K L 2 2 2 3 2 - 5 K M - L M - 2 L M - M + 4 K M + 2 L M + 2 M + L + M - 2) L/(%1) 3 4 4 3 4 3 3 3 4 3 4 2 3 3 3 - (6 K L M - 10 K L M + 4 K L M + 5 K L M - 12 K L M 3 2 4 2 4 3 2 3 4 3 3 2 3 2 3 - 5 K L M - 7 K L M - 3 K L M + 10 K L M + 2 K L M 2 4 2 2 3 3 2 2 4 3 3 3 2 2 + 6 K L M + 4 K L M - 4 K L M - 5 K L M + 6 K L M 3 3 2 4 2 3 2 2 2 3 2 4 4 2 + 5 K L M - 2 K L M + 15 K L M + 14 K L M + K L M + 2 K L M 3 3 2 4 3 2 2 3 2 2 2 2 3 - K L M + K L M - 6 K L M - 10 K L M - 16 K L M + 6 K L M 3 2 2 3 2 2 2 2 2 3 3 + 6 K L M + 3 K L - 3 K L M - 16 K L M - 2 K M - 12 K L M 2 2 3 3 2 2 3 2 2 2 3 - 10 K L M + 2 K L M + L M + L M + 12 K L M + 2 K M + 2 K L 2 2 3 3 2 2 2 + 10 K L M - 6 K L M - 2 K M - 2 L M - 4 L M + 2 K L + 11 K L M 2 2 2 + 5 K M + 4 L M - 6 K L - 4 K M + 2 L + 2 L M - 5 L - M + 2) z/(%1)]] 4 3 3 4 4 2 2 4 4 3 2 %1 := 4 K L M - 4 K L M - 4 K L M + 4 K L M + K L M - 3 K L M 2 3 4 4 2 2 4 3 3 3 2 + 3 K L M - K L M - L M + L M + 4 K L M - 4 K L M + 2 L M 2 3 3 2 2 3 3 3 2 - 2 L M - K L - K L M + K L M + K M + 2 L M - 2 L M - 4 L M 2 2 2 + 4 L M - L + M + 2 L - 2 M 2 2 2 2 2 2 2 2 %2 := 4 K L M - 4 K L M - 4 K L M + K L + 2 K L M + K M - L M - L M + 2 L M + L + M - 2 3 2 2 3 3 3 3 2 2 %3 := 4 K L M - 4 K L M - 4 K L M + 4 K L M + K L + K L M - K L M 3 3 3 2 2 2 2 - K M - L M + L M + 2 L M - 2 L M + L - M - 2 L + 2 M and in Maple notation [[K*z, -(2*K^4*L^3*M^3-K^4*L^3*M^2-K^4*L^2*M^3-2*K^4*L^2*M^2-5*K^3*L^3*M^2-5*K^ 3*L^2*M^3+K^4*L^2*M+K^4*L*M^2+2*K^3*L^3*M+10*K^3*L^2*M^2+2*K^3*L*M^3+K^3*L^2*M+ K^3*L*M^2+4*K^2*L^3*M+10*K^2*L^2*M^2+4*K^2*L*M^3-K^3*L^2-4*K^3*L*M-K^3*M^2-K^2* L^3-14*K^2*L^2*M-14*K^2*L*M^2-K^2*M^3+K^2*L^2+4*K^2*L*M+K^2*M^2-K*L^3-6*K*L^2*M -6*K*L*M^2-K*M^3+3*K^2*L+3*K^2*M+5*K*L^2+16*K*L*M+5*K*M^2-5*K*L-5*K*M+L^2+2*L*M +M^2-2*K-4*L-4*M+4)/(4*K*L^3*M^2-4*K*L^2*M^3-4*K*L^3*M+4*K*L*M^3+K*L^3+K*L^2*M- K*L*M^2-K*M^3-L^3*M+L*M^3+2*L^2*M-2*L*M^2+L^2-M^2-2*L+2*M)*K*z^3+(6*K^4*L^3*M^3 -3*K^4*L^3*M^2-3*K^4*L^2*M^3-6*K^4*L^2*M^2-8*K^3*L^3*M^2-8*K^3*L^2*M^3+3*K^4*L^ 2*M+3*K^4*L*M^2+2*K^3*L^3*M+12*K^3*L^2*M^2+2*K^3*L*M^3+8*K^3*L^2*M+8*K^3*L*M^2+ 2*K^2*L^3*M+6*K^2*L^2*M^2+2*K^2*L*M^3-2*K^3*L^2-12*K^3*L*M-2*K^3*M^2+K^2*L^3-4* K^2*L^2*M-4*K^2*L*M^2+K^2*M^3-5*K^2*L^2-8*K^2*L*M-5*K^2*M^2+2*K*L^2*M+2*K*L*M^2 +7*K^2*L+7*K^2*M-2*K*L^2-8*K*L*M-2*K*M^2+6*K*L+6*K*M-L^2-2*L*M-M^2-4*K+4*L+4*M-\ 4)/(4*K*L^3*M^2-4*K*L^2*M^3-4*K*L^3*M+4*K*L*M^3+K*L^3+K*L^2*M-K*L*M^2-K*M^3-L^3 *M+L*M^3+2*L^2*M-2*L*M^2+L^2-M^2-2*L+2*M)*z^2+(2*K^2*L^3*M^2-K^2*L^3*M-K^2*L^2* M^2-2*K^2*L^2*M-4*K*L^2*M^2+K^2*L^2+K^2*L*M-K*L^3+2*K*L^2*M+3*K*L*M^2+3*K*L^2+2 *K*L*M-K*M^2+L^2*M+L*M^2-4*K*L-2*L*M-L-M+2)/(4*K*L^3*M^2-4*K*L^2*M^3-4*K*L^3*M+ 4*K*L*M^3+K*L^3+K*L^2*M-K*L*M^2-K*M^3-L^3*M+L*M^3+2*L^2*M-2*L*M^2+L^2-M^2-2*L+2 *M)*M-(6*K^3*L^3*M^3-3*K^3*L^3*M^2-3*K^3*L^2*M^3-6*K^3*L^2*M^2-3*K^2*L^3*M^2-7* K^2*L^2*M^3+3*K^3*L^2*M+3*K^3*L*M^2-K^2*L^3*M+4*K^2*L^2*M^2+3*K^2*L*M^3+10*K^2* L^2*M+9*K^2*L*M^2-K^2*M^3+K*L^3*M-K*L^2*M^2+2*K*L*M^3-K^2*L^2-12*K^2*L*M-K^2*M^ 2+4*K*L^2*M+2*K*L*M^2-3*K*L^2-11*K*L*M-4*K*M^2+L^2*M+L*M^2+4*K*L+6*K*M-L^2-4*L* M-M^2+3*L+3*M-2)/(4*K*L^3*M^2-4*K*L^2*M^3-4*K*L^3*M+4*K*L*M^3+K*L^3+K*L^2*M-K*L *M^2-K*M^3-L^3*M+L*M^3+2*L^2*M-2*L*M^2+L^2-M^2-2*L+2*M)*z, (2*K^4*L^3*M^3-K^4*L ^3*M^2-K^4*L^2*M^3-2*K^4*L^2*M^2-5*K^3*L^3*M^2-5*K^3*L^2*M^3+K^4*L^2*M+K^4*L*M^ 2+2*K^3*L^3*M+10*K^3*L^2*M^2+2*K^3*L*M^3+K^3*L^2*M+K^3*L*M^2+4*K^2*L^3*M+10*K^2 *L^2*M^2+4*K^2*L*M^3-K^3*L^2-4*K^3*L*M-K^3*M^2-K^2*L^3-14*K^2*L^2*M-14*K^2*L*M^ 2-K^2*M^3+K^2*L^2+4*K^2*L*M+K^2*M^2-K*L^3-6*K*L^2*M-6*K*L*M^2-K*M^3+3*K^2*L+3*K ^2*M+5*K*L^2+16*K*L*M+5*K*M^2-5*K*L-5*K*M+L^2+2*L*M+M^2-2*K-4*L-4*M+4)/(4*K*L^3 *M^2-4*K*L^2*M^3-4*K*L^3*M+4*K*L*M^3+K*L^3+K*L^2*M-K*L*M^2-K*M^3-L^3*M+L*M^3+2* L^2*M-2*L*M^2+L^2-M^2-2*L+2*M)*K*z^3-(6*K^4*L^3*M^3-3*K^4*L^3*M^2-3*K^4*L^2*M^3 -6*K^4*L^2*M^2-8*K^3*L^3*M^2-8*K^3*L^2*M^3+3*K^4*L^2*M+3*K^4*L*M^2+2*K^3*L^3*M+ 12*K^3*L^2*M^2+2*K^3*L*M^3+8*K^3*L^2*M+8*K^3*L*M^2+2*K^2*L^3*M+6*K^2*L^2*M^2+2* K^2*L*M^3-2*K^3*L^2-12*K^3*L*M-2*K^3*M^2+K^2*L^3-4*K^2*L^2*M-4*K^2*L*M^2+K^2*M^ 3-5*K^2*L^2-8*K^2*L*M-5*K^2*M^2+2*K*L^2*M+2*K*L*M^2+7*K^2*L+7*K^2*M-2*K*L^2-8*K *L*M-2*K*M^2+6*K*L+6*K*M-L^2-2*L*M-M^2-4*K+4*L+4*M-4)/(4*K*L^3*M^2-4*K*L^2*M^3-\ 4*K*L^3*M+4*K*L*M^3+K*L^3+K*L^2*M-K*L*M^2-K*M^3-L^3*M+L*M^3+2*L^2*M-2*L*M^2+L^2 -M^2-2*L+2*M)*z^2-(2*K^2*L^2*M^3-K^2*L^2*M^2-K^2*L*M^3-2*K^2*L*M^2-4*K*L^2*M^2+ K^2*L*M+K^2*M^2+3*K*L^2*M+2*K*L*M^2-K*M^3-K*L^2+2*K*L*M+3*K*M^2+L^2*M+L*M^2-4*K *M-2*L*M-L-M+2)/(4*K*L^3*M^2-4*K*L^2*M^3-4*K*L^3*M+4*K*L*M^3+K*L^3+K*L^2*M-K*L* M^2-K*M^3-L^3*M+L*M^3+2*L^2*M-2*L*M^2+L^2-M^2-2*L+2*M)*L+(6*K^3*L^3*M^3-3*K^3*L ^3*M^2-3*K^3*L^2*M^3-6*K^3*L^2*M^2-7*K^2*L^3*M^2-3*K^2*L^2*M^3+3*K^3*L^2*M+3*K^ 3*L*M^2+3*K^2*L^3*M+4*K^2*L^2*M^2-K^2*L*M^3-K^2*L^3+9*K^2*L^2*M+10*K^2*L*M^2+2* K*L^3*M-K*L^2*M^2+K*L*M^3-K^2*L^2-12*K^2*L*M-K^2*M^2+2*K*L^2*M+4*K*L*M^2-4*K*L^ 2-11*K*L*M-3*K*M^2+L^2*M+L*M^2+6*K*L+4*K*M-L^2-4*L*M-M^2+3*L+3*M-2)/(4*K*L^3*M^ 2-4*K*L^2*M^3-4*K*L^3*M+4*K*L*M^3+K*L^3+K*L^2*M-K*L*M^2-K*M^3-L^3*M+L*M^3+2*L^2 *M-2*L*M^2+L^2-M^2-2*L+2*M)*z], [-(2*K^4*L^3*M^3-K^4*L^3*M^2-K^4*L^2*M^3-2*K^4* L^2*M^2-5*K^3*L^3*M^2-5*K^3*L^2*M^3+K^4*L^2*M+K^4*L*M^2+2*K^3*L^3*M+10*K^3*L^2* M^2+2*K^3*L*M^3+K^3*L^2*M+K^3*L*M^2+4*K^2*L^3*M+10*K^2*L^2*M^2+4*K^2*L*M^3-K^3* L^2-4*K^3*L*M-K^3*M^2-K^2*L^3-14*K^2*L^2*M-14*K^2*L*M^2-K^2*M^3+K^2*L^2+4*K^2*L *M+K^2*M^2-K*L^3-6*K*L^2*M-6*K*L*M^2-K*M^3+3*K^2*L+3*K^2*M+5*K*L^2+16*K*L*M+5*K *M^2-5*K*L-5*K*M+L^2+2*L*M+M^2-2*K-4*L-4*M+4)/(4*K*L^3*M^2-4*K*L^2*M^3-4*K*L^3* M+4*K*L*M^3+K*L^3+K*L^2*M-K*L*M^2-K*M^3-L^3*M+L*M^3+2*L^2*M-2*L*M^2+L^2-M^2-2*L +2*M)*K*z^3+(6*K^4*L^3*M^3-3*K^4*L^3*M^2-3*K^4*L^2*M^3-6*K^4*L^2*M^2-8*K^3*L^3* M^2-8*K^3*L^2*M^3+3*K^4*L^2*M+3*K^4*L*M^2+2*K^3*L^3*M+12*K^3*L^2*M^2+2*K^3*L*M^ 3+8*K^3*L^2*M+8*K^3*L*M^2+2*K^2*L^3*M+6*K^2*L^2*M^2+2*K^2*L*M^3-2*K^3*L^2-12*K^ 3*L*M-2*K^3*M^2+K^2*L^3-4*K^2*L^2*M-4*K^2*L*M^2+K^2*M^3-5*K^2*L^2-8*K^2*L*M-5*K ^2*M^2+2*K*L^2*M+2*K*L*M^2+7*K^2*L+7*K^2*M-2*K*L^2-8*K*L*M-2*K*M^2+6*K*L+6*K*M- L^2-2*L*M-M^2-4*K+4*L+4*M-4)/(4*K*L^3*M^2-4*K*L^2*M^3-4*K*L^3*M+4*K*L*M^3+K*L^3 +K*L^2*M-K*L*M^2-K*M^3-L^3*M+L*M^3+2*L^2*M-2*L*M^2+L^2-M^2-2*L+2*M)*z^2+(2*K^2* L^3*M^2-K^2*L^3*M-K^2*L^2*M^2-2*K^2*L^2*M-4*K*L^2*M^2+K^2*L^2+K^2*L*M-K*L^3+2*K *L^2*M+3*K*L*M^2+3*K*L^2+2*K*L*M-K*M^2+L^2*M+L*M^2-4*K*L-2*L*M-L-M+2)/(4*K*L^3* M^2-4*K*L^2*M^3-4*K*L^3*M+4*K*L*M^3+K*L^3+K*L^2*M-K*L*M^2-K*M^3-L^3*M+L*M^3+2*L ^2*M-2*L*M^2+L^2-M^2-2*L+2*M)*M-(6*K^3*L^3*M^3-3*K^3*L^3*M^2-3*K^3*L^2*M^3-6*K^ 3*L^2*M^2-3*K^2*L^3*M^2-7*K^2*L^2*M^3+3*K^3*L^2*M+3*K^3*L*M^2-K^2*L^3*M+4*K^2*L ^2*M^2+3*K^2*L*M^3+10*K^2*L^2*M+9*K^2*L*M^2-K^2*M^3+K*L^3*M-K*L^2*M^2+2*K*L*M^3 -K^2*L^2-12*K^2*L*M-K^2*M^2+4*K*L^2*M+2*K*L*M^2-3*K*L^2-11*K*L*M-4*K*M^2+L^2*M+ L*M^2+4*K*L+6*K*M-L^2-4*L*M-M^2+3*L+3*M-2)/(4*K*L^3*M^2-4*K*L^2*M^3-4*K*L^3*M+4 *K*L*M^3+K*L^3+K*L^2*M-K*L*M^2-K*M^3-L^3*M+L*M^3+2*L^2*M-2*L*M^2+L^2-M^2-2*L+2* M)*z, (2*K^4*L^4*M^4-2*K^4*L^3*M^4-K^4*L^4*M^2-4*K^4*L^3*M^3+K^4*L^2*M^4-5*K^3* L^4*M^3-5*K^3*L^3*M^4+2*K^4*L^3*M^2+2*K^4*L^2*M^3+10*K^3*L^3*M^3+4*K^3*L^2*M^4+ K^4*L^3*M+2*K^4*L^2*M^2-K^4*L*M^3+2*K^3*L^4*M+10*K^3*L^3*M^2+2*K^3*L^2*M^3-2*K^ 3*L*M^4+4*K^2*L^4*M^2+10*K^2*L^3*M^3+4*K^2*L^2*M^4-2*K^4*L^2*M-6*K^3*L^3*M-14*K ^3*L^2*M^2-12*K^2*L^3*M^2-16*K^2*L^2*M^3-2*K^2*L*M^4-K^3*L^3-2*K^3*L*M^2+K^3*M^ 3-K^2*L^4-8*K^2*L^3*M-6*K^2*L^2*M^2+2*K^2*L*M^3+K^2*M^4-K*L^4*M-6*K*L^3*M^2-6*K *L^2*M^3-K*L*M^4+2*K^3*L^2+4*K^3*L*M+4*K^2*L^3+20*K^2*L^2*M+14*K^2*L*M^2-2*K^2* M^3+4*K*L^3*M+16*K*L^2*M^2+6*K*L*M^3-2*K^2*L^2-4*K^2*L*M+2*K*L^3+4*K*L^2*M-2*K* L*M^2+L^3*M+2*L^2*M^2+L*M^3-4*K^2*L-2*K^2*M-8*K*L^2-18*K*L*M-2*K*M^2-4*L^2*M-4* L*M^2+7*K*L+3*K*M-L^2+2*L*M-M^2+2*K+4*L+4*M-4)/(4*K*L^4*M^3-4*K*L^3*M^4-4*K*L^4 *M^2+4*K*L^2*M^4+K*L^4*M-3*K*L^3*M^2+3*K*L^2*M^3-K*L*M^4-L^4*M^2+L^2*M^4+4*K*L^ 3*M-4*K*L*M^3+2*L^3*M^2-2*L^2*M^3-K*L^3-K*L^2*M+K*L*M^2+K*M^3+2*L^3*M-2*L*M^3-4 *L^2*M+4*L*M^2-L^2+M^2+2*L-2*M)*K*z^3-(6*K^4*L^4*M^4+2*K^4*L^4*M^3-8*K^4*L^3*M^ 4-4*K^4*L^4*M^2-12*K^4*L^3*M^3+4*K^4*L^2*M^4-8*K^3*L^4*M^3-8*K^3*L^3*M^4+4*K^4* L^3*M^2+8*K^4*L^2*M^3-4*K^3*L^4*M^2+12*K^3*L^3*M^3+8*K^3*L^2*M^4+4*K^4*L^3*M+6* K^4*L^2*M^2-4*K^4*L*M^3+4*K^3*L^4*M+24*K^3*L^3*M^2+8*K^3*L^2*M^3-4*K^3*L*M^4+2* 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