|\^/| 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. #Theorem39ii.maple # # Frank Sottile # October 1998 # Berkeley, California # # This MAPLE script establishes Theorem 3.9.ii of "Real Schubert Calculus: # Polynomial systems and a conjecture of Shapiro and Shapiro". This is the # case of Shapiro's conjecture concerning the 3 3-planes in C^6 which meet 4 # 2-planes osculating the rational normal curve and one 3-plane. # # Our method is to show that the discriminant of the degree 3 eliminant is # again a sum of squares. What makes ths interesting is that the # discriminant we compute is NOT symmetric in the parameters. # #interface(quiet=true): > with(linalg): Warning, new definition for norm Warning, new definition for trace > with(Groebner): ######################################################### # # We set up the polynomial system. The first 3 rows of the following # are the 3-plane in C^6 osculating the rational normal curve, and the # last 3 rows are local coordinates for the set of 3-planes # satisfying conditions J_2 for s=0 and s=\infty. # > Mat := s -> 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 , 1 ,x23, x24 , x25 , 0 ], > [ 0 , 0 , 0 , 0 , 1 , x36 ]]): > # The polynomial A is the condition J_1, to meet a 3-plane osculating at s # > A:=simplify(-det(Mat(s))/s/s): # # The polynomials B, C, E are the condition J_2, to meet a 2-plane # osculating at t # > B:=simplify(-det(submatrix(Mat(t),[1,2,4,5,6],[1,3,4,5,6]))/t^4): > C:=simplify(det(submatrix(Mat(t),[1,2,4,5,6],[1,2,4,5,6]))/t/t): > E:=simplify(det(submatrix(Mat(t),[1,2,4,5,6],[1,2,3,4,5]))/t): # # We now compute the universal eliminant for the set of 3-planes meeting a # 3-plane osculating at s, and two 2-planes aoculating at t and at u. # > G:=gbasis([A,B,C,E,subs(t=u,B),subs(t=u,C),subs(t=u,E)], > wdeg([4,4,4,4,1],[x12,x23,x24,x25,x36])): bytes used=1000188, alloc=917336, time=0.28 bytes used=2000580, alloc=1376004, time=0.62 bytes used=3001012, alloc=1507052, time=0.96 bytes used=4001212, alloc=1507052, time=1.33 bytes used=5002388, alloc=1572576, time=1.69 bytes used=6003864, alloc=1572576, time=2.01 # # # The computation proceeds over the function field R(s,t,u) and the # Gr\"obner basis consists of 6 polynomials, with the first the # universal eliminant, a univariate polynomial in x36 of degree 3 # # # > UniversalEliminant:=G[1]; 2 2 2 2 UniversalEliminant := -6 t u - 8 t s + 4 t x36 - 6 u t + 11 u t x36 2 2 2 2 - 17 s t u - 4 t x36 + 10 t s x36 - 8 s u + 4 u x36 - 4 u x36 2 3 + 10 s u x36 - 3 x36 s + x36 # # We compute the Discriminant of the UniversalEliminant. # > Discriminant:=sort(primpart(discrim(UniversalEliminant,x36)),[s]); 4 2 4 2 4 3 3 2 3 3 3 Discriminant := -9 u t s + 9 t s + 9 u s - 4 t s - 12 u t s - 4 u s 2 3 3 2 2 2 2 3 2 4 2 4 2 - 12 u t s - 2 u t s + 42 t u s - 2 t u s + 4 u s + 4 t s 2 3 4 3 2 4 4 2 4 2 - 12 t u s - 4 u t s - 12 t u s - 4 t u s + 9 t u + 9 u t 3 3 - 9 t u # # The Discriminant is a sum of squares: # > simplify(Discriminant - > ( 7/2*s^4*(t-u)^2 + s^2*(t-u)^4 + t^4*(s-u)^2 + u^4*(s-t)^2 > + 7/2*t^2*(s-u)^4 + 7/2*u^2*(s-t)^4 + s^2*t^2*(s-t)^2 + s^2*u^2*(s-u)^2 > + 7/2*t^2*u^2*(t-u)^2 + (t-u)^2*(s-u)^2*(s-t)^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 3 real roots for a single choice of the parameters. # > fsolve(subs(s=1,t=2,u=3,UniversalEliminant)=0,x36); 4.736457153, 7.755640076, 10.50790277 > quit; bytes used=6539376, alloc=1638100, time=2.17