#Example22.maple # # Frank Sottile # 27 June 1998 # Toronto, Canada # # This Maple script computes the lexocigraphic Gr\"obner basis for # the case m=p=2 of Shapiro's conjecture as described in Example 2.2 # of "Real Schubert Calculus: # Polynomial systems and a conjecture of Shapiro and Shapiro" # # The computation is in the ring R(s,t,u,v)[x21,x22,x12,x11], # with term order x21 < x22 < x12 < x11. # # The Gr\"obner basis is G1, G3, G3, and G4. G4 is the eliminant. # We also express G4 in terms of the elementary symmetric polynomials # in s,t,u,v and compute its discriminant. with(linalg): F := s -> det(matrix(4,4, [1,s,s^2,s^3,0,1,2*s,3*s^2,1,0,x11,x12,0,1,x21,x22])): G1:= sort(simplify(-det(matrix(4,4, [1,s,s^2,F(s),1,t,t^2,F(t),1,u,u^2,F(u),1,v,v^2,F(v)]))/ det(matrix(4,4,[1,s,s^2,s^3, 1,t,t^2,t^3, 1,u,u^2,u^3, 1,v,v^2,v^3]))), [x21,x22,x12,x11,s,t,u,v],plex): G2:= sort(simplify(det(matrix(3,3,[1,s,F(s),1,t,F(t),1,u,F(u)]))/ det(matrix(3,3,[1,s,s^2, 1,t,t^2, 1,u,u^2])) +(s+t+u)*G1), [x21,x22,x12,x11,s,t,u,v],plex): G3:=sort(simplify((F(s)-F(t))/(s-t) + (s^2+s*t+t^2)*G1 - (s+t)*G2), [x21,x22,x12,x11,s,t,u,v],plex): G4:=sort(simplify( 4*F(s) + 2*(x12+2*s^3)*G1 - 4*(x11+s^2)*G2 + (-3*s+t+u+v)*G3), [x21,x22,x12,x11,s,t,u,v],plex): # # G1, G2, G3, G4 are the Gr\"obner basis in the purely lexicographic term # order x21 < x22 < x12 < x11 # G1; G2; G3; G4; # # We represent G4 in terms of the elementary symmetric polynomials: # E1:=s+t+u+v: E2:=s*t+s*u+s*v+t*u+t*v+u*v: E3:=s*t*u+s*t*v+s*u*v+t*u*v: E4:=s*t*u*v: simplify(G4 - (12*x11^2 + 4*x11*E2 + E1*E3 - 4*E4)); # # The Discriminant of G4 is then (4*E2)^2 - 4*12*(E1*E3 - 4*E4) , which # has a nice representation in terms of the differences s-t, s-u, etc.: # simplify(16*E2^2 - 4*12*(E1*E3 - 4*E4) - 8*((s-t)^2*(u-v)^2 + (s-u)^2*(t-v)^2 + (s-v)^2*(u-t)^2) ); quit;