#3e5p3.maple # # Frank Sottile # 2 November 1998 # Berkeley, CA # # This files computes instances of Shapiro's Conjecture for the case # of 3-planes in C^8 satisfying (J_3)^5 = 6. # # We first test the 1140 cases of # choose([1,2,3,4,5,6,7,8,9,10,11,14,15,16,18,20,21,22,24,25],3): # # time output size # Singular input file: 16.1 3616661 # Maple file: 354 621285 # Checking all roots real 72.670 # # Next, the 4960 cases of # trials:=choose([-23,-22,-21,-20,-19,-18,-17,-16,-15,-14,-13,-12,-11, # -10,-9,-8,-7,-6,-5,-4,-3,-2,-1,1,2,12,13,17,19,23,26,27],3): # # Singular input file: 74.840 15878961 # Maple file: 1549 2695105 # Checking all roots real 312.870 # # Next the 4060 instances of # trials:=choose([-30,-29,-28,-27,-26,-25,-24,-23,-22,3,4,28,29,30, # 31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46],3): # # Singular input file: 64 14568794 # Maple file: 1381 2167139 # Checking all roots real 2057 # interface(quiet=true): with(Ore_algebra): with(linalg): with(Groebner): with(combinat): #trials:=choose([1,2,3,4,5,6,7,8,9,10,11,14,15,16,18,20,21,22,24,25],3): #trials:=choose([-23,-22,-21,-20,-19,-18,-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1,1,2,12,13,17,19,23,26,27],3): trials:=choose([-30,-29,-28,-27,-26,-25,-24,-23,-22,3,4,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46],3): #nops(trials); quit; ST_Nreal:="lprint(`realroots = `,NREAL,`Vals = `,": ST_end:=");": SmMat := s -> matrix([ [ 1 , s ,s^2, s^3 , s^4 , s^5 , s^6, s^7], [ 0 , 1 ,2*s,3*s^2,4*s^3, 5*s^4, 6*s^5, 7*s^6], [ 0 , 0 , 1 , 3*s ,6*s^2,10*s^3,15*s^4,21*s^5], [ 1 , a , b , 0 , 0 , 0 , 0 , 0 ], [ 0 , 1 , c , d , e , f , g , 0 ], [ 0 , 0 , 0 , 0 , 0 , 1 , h , x ]]): Cols:=[ [1,2,3,4,5,6],[1,2,3,4,6,7],[1,2,3,4,6,8],[1,2,3,4,7,8], [1,2,3,5,6,7],[1,2,3,5,6,8],[1,2,3,5,7,8],[1,2,3,6,7,8], [1,2,4,5,6,7],[1,2,4,5,6,8],[1,2,4,5,7,8],[1,2,4,6,7,8], [1,2,5,6,7,8],[1,3,4,5,6,7],[1,3,4,5,6,8],[1,3,4,5,7,8], [1,3,4,6,7,8],[1,3,5,6,7,8],[1,4,5,6,7,8],[2,3,4,5,6,7], [2,3,4,5,6,8],[2,3,4,5,7,8],[2,3,4,6,7,8],[2,3,5,6,7,8]]: #GB:=[ #simplify(det(submatrix(SmMat(s),[1,2,3,4,5,6],Cols[1]))/s), #simplify(det(submatrix(SmMat(s),[1,2,3,4,5,6],Cols[2]))/s), #simplify(det(submatrix(SmMat(s),[1,2,3,4,5,6],Cols[3]))/s), #simplify(det(submatrix(SmMat(s),[1,2,3,4,5,6],Cols[4]))/s), #simplify(det(submatrix(SmMat(s),[1,2,3,4,5,6],Cols[5]))/s^2), #simplify(det(submatrix(SmMat(s),[1,2,3,4,5,6],Cols[6]))/s^2), #simplify(det(submatrix(SmMat(s),[1,2,3,4,5,6],Cols[7]))/s^2), #simplify(det(submatrix(SmMat(s),[1,2,3,4,5,6],Cols[8]))/s^3), #simplify(det(submatrix(SmMat(s),[1,2,3,4,5,6],Cols[9]))/s^4), #simplify(det(submatrix(SmMat(s),[1,2,3,4,5,6],Cols[10]))/s^4), #simplify(det(submatrix(SmMat(s),[1,2,3,4,5,6],Cols[11]))/s^4), #simplify(det(submatrix(SmMat(s),[1,2,3,4,5,6],Cols[12]))/s^5), #simplify(det(submatrix(SmMat(s),[1,2,3,4,5,6],Cols[13]))/s^6), #simplify(det(submatrix(SmMat(s),[1,2,3,4,5,6],Cols[14]))/s^4), #simplify(det(submatrix(SmMat(s),[1,2,3,4,5,6],Cols[15]))/s^4), #simplify(det(submatrix(SmMat(s),[1,2,3,4,5,6],Cols[16]))/s^4), #simplify(det(submatrix(SmMat(s),[1,2,3,4,5,6],Cols[17]))/s^5), #simplify(det(submatrix(SmMat(s),[1,2,3,4,5,6],Cols[18]))/s^6), #simplify(det(submatrix(SmMat(s),[1,2,3,4,5,6],Cols[19]))/s^9), #simplify(det(submatrix(SmMat(s),[1,2,3,4,5,6],Cols[20]))/s^5), #simplify(det(submatrix(SmMat(s),[1,2,3,4,5,6],Cols[21]))/s^5), #simplify(det(submatrix(SmMat(s),[1,2,3,4,5,6],Cols[22]))/s^5), #simplify(det(submatrix(SmMat(s),[1,2,3,4,5,6],Cols[23]))/s^6), #simplify(det(submatrix(SmMat(s),[1,2,3,4,5,6],Cols[24]))/s^7) #]: #time(); #GB:=gbasis(GB,wdeg([1,2,1,2,3,4,5,1,2],[a,b,c,d,e,f,g,h,x])): #time(); #for i from 1 to nops(GB) do indets(GB[i]); lprint(GB[i]); od; quit; GB:=[ s^2*e-s^5+3*s^4*c-3*s*a*e-6*s^3*a*c+3*b*e+6*s^3*b-3*d*s^3+8*d*s^2*a-6*d*s*b, 3*s*h*f-h*s^5+6*d*h*s^3-3*s*g+3*s^6-15*d*s^4-6*a*h*f-10*d*s^2*a*h+6*a*g+24*d*s^3*a-8*e*h*s^2+15*e*h*s*a+15*s^3*e-27*s^2*a*e, -3*d*s^5+6*e*s^4+3*d*h*s^4-d*x*s^3-8*e*h*s^3+6*s^2*h*f-6*s^2*g+3*e*x*s^2-3*s*x*f+x*g, -3*s*x*f+x*s^5-6*d*x*s^3-6*s^7+27*d*s^5+6*a*x*f+10*d*x*s^2*a-42*d*s^4*a+8*e*x*s^2-15*e*x*s*a-24*e*s^4+42*e*s^3*a, 6*b*g-6*b*h*f-s^2*g+20*d*h*s^2*b+15*e*h*s^2*a+s^2*h*f-30*d*s^3*b-27*e*s^3*a-40*d*h*s^3*a-8*e*h*s^3+72*d*s^4*a+15*e*s^4-30*s^4*b*h+30*s^4*a*h*c+18*d*h*s^4+54*s^5*b-54*s^5*a*c-35*d*s^5-16*s^5*h*c+30*s^6*c+5*s^6*h-9*s^7, 6*b*x*f-20*d*x*s^2*b-15*e*x*s^2*a-s^2*x*f+40*d*x*s^3*a+8*e*x*s^3+42*d*s^4*b+42*e*s^4*a+30*s^4*b*x-30*s^4*a*x*c-18*d*x*s^4-112*d*s^5*a-24*e*s^5+16*s^5*x*c-84*s^6*b+84*s^6*a*c+57*d*s^6-5*s^6*x-48*s^7*c+14*s^8 ]: lprint(`option(redSB);`); lprint(`int t=timer;`); lprint(`ring R= 0, (x,h,g,f,e,d,c,b,a), dp;`); lprint(`ideal I;`); lprint(`ideal H;`); lprint(`ring S= 0, (a,b,c,d,e,f,g,h,x), (dp(8),dp(1));`); lprint(`ideal G;`); lprint(`print("interface(quiet=true):");`); lprint(`print("readlib(realroot):");`); for ntimes from 1 to nops(trials) do Vals:=trials[ntimes]: lprint(`setring R;`); lprint(`I = `); equations:=[]: for ii in Vals do for P in GB do equations:=[equations[],subs(s=ii,P)]: od:od: for EQS in equations do lprint(EQS,`,`); od: lprint(`0;`); lprint(`H = std(I);`); lprint(`setring S;`); lprint(`G = fglm(R,H);`); lprint(`print("POLY:=");`); lprint(`short=0;`); lprint(`G[1];`); lprint(`print(":");`); lprint(`print("NREAL:= nops(realroot(POLY, 1/100)): ");`); lprint(`print(" if NREAL<6 then");`); lprint(`print("`,convert(ST_Nreal,symbol)); lprint(convert(Vals,symbol)); lprint(convert(ST_end,symbol),` ");`); lprint(`print("fi: ");`); lprint(`print(" ");`); lprint(`print(" ");`); lprint(`print(" ");`); lprint(`print("#######################################################");`); lprint(`print("# #");`); lprint(`print("# Next One #");`); lprint(`print("# #");`); lprint(`print("#######################################################");`); lprint(`print(" ");`); od: lprint(`print("time(); ");`); lprint(`print("quit; ");`); lprint(`timer-t;`); lprint(`quit;`); time(); quit; quit;