|\^/| Maple V Release 5 (University of Toronto) ._|\| |/|_. Copyright (c) 1981-1997 by Waterloo Maple Inc. All rights \ MAPLE / reserved. Maple and Maple V are registered trademarks of <____ ____> Waterloo Maple Inc. | Type ? for help. # The Schubert Calculus tells us that given 3 general 3-planes A,B,C and # 3 general 2-planes a,b,c in 5-dimensional space, there are 4 partial # flags X < Y such that X meets a, B, and C and Y intersects A in # a 2-dimensional subspace and also meets b and c. # # We show that if A,B,C,a,b,c are chosen to osculate the standard rational # normal curve at the (respective) points 41, 58, 29, -97, 82, 21, then not # all the solutions are real. This disproves the conjecture of Shapiro # and Shapiro, in the original form they stated. # > with(linalg): > with(Groebner): > readlib(realroot): > Digits:= 5: > Mat := s -> matrix([ > [1 , s ,s^2, s^3 , s^4 ], # The flag osculating the rational > [0 , 1 ,2*s,3*s^2,4*s^3], # normal curve at the point s. > [0 , 0 , 1 , 3*s ,6*s^2], # # > [0 , 0 , 1 , x14 , x15 ], # Local coordinates for the partial > [1 , 0 ,x23, x24 , x25 ], # flag X < Y > [0 , 1 ,x33, x34 , x35 ]]): # (Last 2 rows are X) > # W.i are the equations imposed by W on the flag X < Y. # For A.i and a.i, the other 2 maximal minors are redundant. > > B:=det(submatrix(Mat(-1),[1,2,3,5,6],[1,2,3,4,5])): > C:=det(submatrix(Mat(3),[1,2,3,5,6],[1,2,3,4,5])): > > A1:=det(submatrix(Mat(0),[1,2,3,4,5],[1,2,3,4,5])): > A2:=det(submatrix(Mat(0),[1,2,3,4,6],[1,2,3,4,5])): > A3:=det(submatrix(Mat(0),[1,2,3,5,6],[1,2,3,4,5])): > > a1:=det(submatrix(Mat(4),[1,2,5,6],[1,2,3,4])): > a2:=det(submatrix(Mat(4),[1,2,5,6],[1,2,3,5])): > a3:=det(submatrix(Mat(4),[1,2,5,6],[1,2,4,5])): > > b:=det(submatrix(Mat(1),[1,2,4,5,6],[1,2,3,4,5])): > c:=det(submatrix(Mat(-5),[1,2,4,5,6],[1,2,3,4,5])): > > G:=gbasis([A1,A2,A3,B,C,a1,a2,a3,b,c], > wdeg([1,5,5,5,5,5,5,5],[x14,x15,x23,x24,x25,x33,x34,x35])): bytes used=2015216, alloc=1376004, time=2.85 bytes used=4017984, alloc=1572576, time=6.77 bytes used=6037720, alloc=1703624, time=9.89 > > lprint(G[1]); -117556*x14+27063-5952*x14^2+32400*x14^4-10416*x14^3 > fsolve(G[1],x14,complex); -.76310 - 1.2983 I, -.76310 + 1.2983 I, .22729, 1.6204 > quit; bytes used=7850512, alloc=1703624, time=12.62