|\^/| 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. #Theorem24.maple #interface(quiet=true): > kernelopts(gcfreq=2*10^6): bytes used=164816, alloc=196572, time=0.03 # # This file shows that each of the 5 3-planes in C^5 which meet 6 2-planes # osculating the rational normal curve at real points are real. # (This instance of Shapiros' conjecture is Theorem 2.4 in "Real Schubert # Calculus: Polynomial systems and a conjecture of Shapiro and Shapiro".) # # Frank Sottile updated: # November 25, 1997 2 August 1998 # Toronto, Canada Berkeley, CA # > with(linalg): Warning, new definition for norm Warning, new definition for trace > with(Groebner): > readlib(realroot): # # SYMMETRIZE contains a routine to symmetrize a polynomial in Vars = s,t,u,v # ELEMENTARIZE contains a procedure write a symmetric polynomial in # Vars in terms of the elementary symmetric polynomials. > read SYMMETRIZE: > read ELEMENTARIZE: # # We choose coordinates for C^5 so that the rational normal curve is the # (1, s, s^2, s^3, s^4), and assume that one of the 5-tuple 2-planes # osculates the rational normal curve at s=0, and another osculates # at s = infinity. (This can be done with a projective transformation # preserving the form of the rational normal curve.) We work in local # coordinates K for the set of 3-planes in C^5 meeting these 2 2-planes. # Here, the 3-plane is the row space of the given matrix. # > K :=matrix([ > [1,x12, 0 , 0 , 0 ], > [0, 1 ,x23,x24, 0 ], > [0, 0, 0 , 1 ,x35]]): # # In these coordinates, the equation that a 3-plane meets the 2-plane # osculating the rational normal curve at a point s is: (we divide by the # parameter s, as s cannot equal zero.) # > A := s -> simplify(det(stackmatrix(matrix([ > [1,s,s^2, s^3, s^4], > [0,1,2*s,3*s^2,4*s^3]]),K))/s): # # Next, we compute a purely lexicographic Gr\"obner basis for the ideal # generated by the equations for the 3-plane to meet 4 2-planes osculating # the rational normal curve at the points s, t, u, and v. # We do this over the ring Z(s,t,u,v). (In fact,the calculations proceed # over the ring Z[s,t,u,v][D^{-1}], where D is the product of the vanderMonde # of s,t,u,v and the monomial stuv.) # > G:= gbasis([A(s),A(t),A(u),A(v)],plex(x12,x23,x24,x35)): # # The Gr\"obner basis has 4 elements, with its first element a homogeneous # polynomial of degree 5 in the variable x35 and symmetric in the parameters # s,t,u,v. This is the UniversalEliminant for this set of equations. # The first equation is linear in a and polynomial in d, and the same may be # said of the second equation for b and the third for c. Thus, for any # value of the parameters s, t, u, and v, the number of real roots of the # system equals the number of real roots of the UniversalEliminant. # > UniversalEliminant:=sort(collect(op(1,G),x35),[x35]): > print(sort(Elementarize(UniversalEliminant,[s,t,u,v]),[x35])); 5 4 3 2 3 2 2 x35 - 4 E1 x35 + 6 E2 x35 + 4 E1 x35 - 4 E3 x35 - 12 E2 E1 x35 2 - 4 E4 x35 + 8 E3 E1 x35 + 9 E2 x35 - 12 E3 E2 + 8 E4 E1 # # We compute the Discriminant of the UniversalEliminant. # > Discriminant:=discrim(UniversalEliminant,x35): bytes used=8174896, alloc=6159256, time=1.90 bytes used=16181404, alloc=8452596, time=3.22 bytes used=24203836, alloc=8780216, time=4.61 bytes used=32210868, alloc=9566504, time=6.11 > print(simplify(Elementarize(Discriminant,[s,t,u,v])/4096)); bytes used=40257456, alloc=9697552, time=7.93 5 5 2 2 4 3 2 4 2 3 3 -32 E3 E1 + 738 E4 E3 E2 E1 - 2106 E4 E2 E1 + 2052 E4 E3 E2 E1 3 5 2 3 4 4 4 2 5 - 108 E4 E3 E1 - 54 E4 E3 E2 E1 + 81 E3 E2 - 486 E4 E3 E2 2 6 2 4 4 5 2 3 3 + 729 E4 E2 + 81 E4 E2 E1 - 324 E3 E2 E1 + 2052 E4 E3 E2 E1 3 2 3 2 4 6 2 - 1296 E4 E3 E2 E1 - 3240 E4 E3 E2 E1 - 108 E3 E1 3 5 4 2 2 2 2 2 + 204 E4 E3 E2 E1 + 738 E4 E3 E2 E1 - 2592 E4 E3 E2 E1 3 3 2 2 3 3 2 2 5 4 4 + 3834 E4 E2 E1 - 368 E4 E3 E1 - 324 E4 E3 E2 E1 - 27 E4 E1 3 3 6 4 2 2 2 3 + 1800 E4 E3 E2 E1 + 324 E3 E2 - 2106 E4 E3 E2 + 3834 E4 E3 E2 3 4 2 2 6 5 4 4 - 972 E4 E2 - 108 E4 E3 E1 - 108 E4 E3 E1 - 8 E4 E3 E1 2 3 3 2 4 2 + 1800 E4 E3 E2 E1 - 5544 E4 E3 E2 E1 + 984 E4 E2 E1 3 2 2 2 4 3 6 4 - 634 E4 E3 E1 - 27 E4 E3 + 324 E4 E2 E1 - 352 E4 E3 E1 4 2 3 2 5 4 3 2 + 432 E4 E2 + 984 E4 E3 E2 - 64 E4 - 54 E3 E2 E1 2 4 2 2 5 2 5 3 4 2 4 + 324 E4 E3 E2 E1 - 486 E4 E2 E1 + 204 E3 E2 E1 + 9 E3 E2 E1 # # The following symmetric polynomials are all sums of squares # > A1:=SYMMETRIZE((s*t*u*v*(s-t)*(s-u)*(s-v)*(t-u)*(t-v)*(u-v))^2): # > A2:=SYMMETRIZE(s^4*t^2*u^2*v^2*(s-u)^2*(s-v)^2*(t-u)^2*(t-v)^2*(u-v)^2): # > A3:=SYMMETRIZE(s^4*t^2*u^2*(s-t)^2*(s-u)^2*(s-v)^2*(t-u)^2*(t-v)^2*(u-v)^2): bytes used=48272364, alloc=9697552, time=9.79 # > A4:=SYMMETRIZE(s^2*t^2*u^2*(s-t)^2*(s-u)^2*(s-v)^2*(t-u)^2*(t-v)^2*(u-v)^4): # > A5:=SYMMETRIZE(s^4*t^4*u^2*(s-u)^2*(s-v)^2*(t-u)^2*(t-v)^2*(u-v)^2): # > A6:=SYMMETRIZE(s^2*t^2*u^2*v^2*(s-t)^4*(s-u)^4*(t-v)^2*(u-v)^2): # > A7:=SYMMETRIZE(t^2*u^2*v^2*(s-t)^4*(s-u)^4*(s-v)^2*(t-v)^2*(u-v)^2): bytes used=56275356, alloc=9697552, time=11.62 # > A8:=SYMMETRIZE(s^4*t^4*u^2*(s-t)^2*(s-v)^2*(t-u)^2*(t-v)^2*(u-v)^2): # > A9:=SYMMETRIZE(s^4*t^2*u^2*v^2*(s-t)^4*(s-u)^2*(t-v)^2*(u-v)^2): # > A10:=SYMMETRIZE(t^2*u^2*v^2*(s-t)^4*(s-u)^4*(s-v)^2*(t-u)^2*(u-v)^2): # > A11:=SYMMETRIZE(s^4*t^2*u^2*(s-u)^2*(s-v)^2*(t-u)^4*(t-v)^2*(u-v)^2): # > A12:=SYMMETRIZE(s^2*t^2*v^2*(s-u)^2*(s-v)^4*(t-u)^4*(t-v)^2*(u-v)^2): bytes used=64275580, alloc=9697552, time=13.41 # > A13:=SYMMETRIZE(s^2*t^2*u^2*v^2*(s-t)^4*(t-u)^2*(t-v)^2*(u-v)^4): # > A14:=SYMMETRIZE(s^4*t^2*u^2*(s-u)^2*(s-v)^2*(t-u)^2*(t-v)^4*(u-v)^2): # > A15:=SYMMETRIZE(s^2*t^2*u^2*v^2*(s-t)^4*(s-v)^2*(t-u)^2*(u-v)^4): # # These have a positive linear combination which equals the discriminant # > simplify(6144*A1 + 12288*A2 + 14336*A3 + 14336*A4 + 4096*A5 + 4096*A6 > + 4096*A7 + 2048*A8 + 4096*A9 + 2048*A10 + 2048*A11 + 2048*A12 > + 12288*A13 + 20480*A14 + 6144*A15 - Discriminant); 0 # # Since the Discriminant is a sum of squares, its non-zero locus is # connected. Hence the UniversalEliminant has the same number of real roots # for any value of the parameters s,t,u,v. At the point s=1, t=2, u=3, and # v=4, it has five real roots: # > solve(subs(s=1,t=2,u=3,v=4,UniversalEliminant)); 1/2 1/2 1/2 1/2 8, 8 + 19 , 8 - 19 , 8 + 11 , 8 - 11 # # This completes the proof that each of the 5 3-planes meeting 6 2-planes # which osculate the rational normal curve at real points are real. # > quit; bytes used=69814792, alloc=9697552, time=14.60