|\^/| Maple V Release 5.1 (WMI MVR5.1 Campus Wide License) ._|\| |/|_. Copyright (c) 1981-1998 by Waterloo Maple Inc. All rights \ MAPLE / reserved. Maple and Maple V are registered trademarks of <____ ____> Waterloo Maple Inc. | Type ? for help. #Theorem39iii.maple # # Frank Sottile # October 31998 # Berkeley, California # # This MAPLE script establishes Theorem 3.9.iii of "Real Schubert Calculus: # Polynomial systems and a conjecture of Shapiro and Shapiro". This is the # case of Shapiro's conjecture concerning the 6 3-planes in C^6 which # satisfy 2 Schubert conditions of type 135, and 3 hypersurface # Schubert conditions (J_1). # #interface(quiet=true): > with(linalg): Warning, new definition for norm Warning, new definition for trace > with(Groebner): > read ELEMENTARIZE: # # We work in local coordinates X_135,135 for the set of 3-planes satisfying # two Schubert conditiopns of type 135. The Polynomial Poly(s) determines # those planes that also meet the 3-plane osculating the rational normal curve # at the point s. # > Poly := s -> det(matrix([ > [1, s ,s^2, s^3 , s^4 , s^5 ], > [0, 1 ,2*s,3*s^2,4*s^3, 5*s^4], > [0, 0 , 1 , 3*s ,6*s^2,10*s^3], > [1,x12, 0 , 0 , 0 , 0 ], > [0, 0 , 1 , x24 , 0 , 0 ], > [0, 0 , 0 , 0 , 1 , x36 ]]))/s^3: # # We compute the universal eliminant by computing a Gr\"obner basis. This # computation proceeds over the function field R(s,t,u) and the # Gr\"obner basis consists of 3 polynomials, with the first the # universal eliminant, a univariate polynomial in x36 of degree 6. # > G:=gbasis([Poly(s),Poly(t),Poly(u)],plex(x12,x24,x36)): bytes used=1005084, alloc=851812, time=0.29 bytes used=2005492, alloc=1310480, time=0.61 # > UniversalEliminant:=G[1]; 4 3 2 2 2 2 UniversalEliminant := 236 t x36 u - 288 t x36 u + 320 t x36 s 2 2 2 6 4 2 4 2 4 + 1125 s u t + 9 x36 + 64 x36 u + 64 x36 s + 236 x36 u s 3 2 3 2 2 2 2 2 2 - 288 x36 u s - 288 x36 u s + 320 x36 s u + 1180 t x36 s u 2 2 2 2 2 2 3 - 1200 t s u x36 + 1180 t x36 s u - 1200 t x36 s u - 1062 t x36 s u 2 2 5 2 2 5 2 4 + 1180 t x36 s u - 48 u x36 - 1200 t s u x36 - 48 s x36 + 64 t x36 5 2 3 2 3 2 2 2 - 48 t x36 - 288 t x36 s - 288 t x36 u + 320 t x36 u 3 2 4 - 288 t x36 s + 236 t x36 s # # We compute the Discriminant of the UniversalEliminant. # > Discriminant:=factor(sort(primpart(discrim(UniversalEliminant,x36)),[s])); bytes used=3012640, alloc=1507052, time=0.91 bytes used=4023752, alloc=1638100, time=1.11 bytes used=5037984, alloc=1703624, time=1.29 bytes used=6054284, alloc=1769148, time=1.47 bytes used=7072404, alloc=1834672, time=1.66 bytes used=8072768, alloc=1834672, time=1.86 bytes used=9072996, alloc=1834672, time=2.21 bytes used=10073216, alloc=1834672, time=2.54 4 4 4 2 4 3 3 4 2 4 3 2 Discriminant := s u t (4 s t - 7 s t + 4 s t - 7 u t s + 5 u t s 3 2 4 2 4 2 3 2 2 2 3 2 + 5 s u t - 7 s u t + 4 u t + 5 u t s - 12 s u t + 5 s u t 4 2 3 3 3 2 3 2 3 3 4 2 4 + 4 s u - 7 u t + 5 u t s + 5 u t s - 7 s u + 4 u t - 7 u t s 2 4 2 4 3 3 4 2 4 3 2 + 4 s u ) (256 s t - 256 s t + 256 s t - 256 u t s + 32 u t s 3 2 4 2 4 2 3 2 2 2 + 32 s u t - 256 s u t + 256 u t + 32 u t s + 213 s u t 3 2 4 2 3 3 3 2 3 2 3 3 + 32 s u t + 256 s u - 256 u t + 32 u t s + 32 u t s - 256 s u 4 2 4 2 4 2 + 256 u t - 256 u t s + 256 s u ) # # Of the factors, only one is not itself a square; we show that factor # is a sum of squares. # > simplify(op(4,Discriminant) - > ( 7/2*(s-t)^2*(s-u)^2*(t-u)^2 + 1/2*s^2*(t-u)^4 > + 1/2*t^2*(s-u)^4 + 1/2*u^2*(s-t)^4 )); 0 # # Interestingly, even the squared factor is a sum of squares: # > simplify(op(1,op(5,Discriminant)) - > ( 16*(s-t)^2*(s-u)^2*(t-u)^2 > + 112*(s^4*(t-u)^2+t^4*(s-u)^2+u^4*(s-t)^2) > + 112*(s^2*t^2*(s-t)^2+ t^2*u^2*(t-u)^2+ s^2*u^2*(s-u)^2) > + 309*s^2*t^2*u^2 > + 16*(s^2*t^4 + s^4*t^2 + t^2*u^4+u^2*t^4 + s^4*u^2+u^4*s^2))); 0 # # It follows that the number of real roots to the system is independent of # the choice of real values for s, t, and u. Thus we need only show that there # are 6 real roots for a single choice of the parameters. # > fsolve(subs(s=1,t=2,u=3,UniversalEliminant)=0); 1.490697830, 1.682893605, 3.209881761, 5.630259037, 9.213043594, 10.77322417 > quit; bytes used=10354552, alloc=1834672, time=2.62