Details of a primal-dual formulation of a Schubert problem with 437 solutions in the Grassmannian Gr(3,9) of 3-planes in complex 9-space.
There are 437 3-planes which nontrivially meet 4 general fixed 4-planes and 6 general fixed 6-planes in 9-space.
In the documented code below, [][][] represents the Schubert condition (4,8,9) which corresponds to the geometric condition of meeting a 4-plane nontrivially, and [] represents the hypersurface Schubert condition (6,8,9) which corresponds to meeting a 6-plane nontrivially.
[Maple file to generate a random instance of (4,8,9)^4 * (6,8,9)^6 = 437]
# Maple code
# Output: primal-dual square system of equations for the Schubert problem
#
# [][][]^4 []^6 = 437 in Gr(3,9)
#
# Nickolas Hein
# 23 May 2014
#
# This is a proof of concept for computing a Schubert problem using a primal-dual formulation in bertini.
# Therefore, precomputations involving linear algebra are omitted.
# In particular, when exploiting duality we randomly choose dual flags rather than primal flags.
#
with(LinearAlgebra):
k:=3:
n:=9:
# Given an instance
#
# [][][]F1 n ... n [][][]F4 n []F5 n ... n []F10
#
# of the Schubert problem, we choose a basis of the ambient space so that F1 is the coordinate flag and F2 is the opposite coordinate flag.
# So [][][]F1 n [][][]F2 is locally the row space of the 3x9 matrix R with rows r[i]:
r[1]:=Matrix(1,n,[[ x1 , x2 , x3 , 1 , 0 , 0 , 0 , 0 , 0 ]]):
r[2]:=Matrix(1,n,[[ 0 , x4 , x5 , x6 , x7 , x8 , x9 , 1 , 0 ]]):
r[3]:=Matrix(1,n,[[ 0 , 0 , 0 , 0 , 0 , x10 , x11 , x12 , 1 ]]):
# For a different choice of basis for the dual space, ([][][]F3 n [][][]F4)^\perp is locally the column space of the 9x6 matrix C with columns c[j]:
c[1]:=Matrix(n,1,[[ 1 ],[ y1 ],[ y2 ],[ 0 ],[ 0 ],[ 0 ],[ 0 ],[ 0 ],[ 0 ]]):
c[2]:=Matrix(n,1,[[ 0 ],[ 1 ],[ y3 ],[ y4 ],[ 0 ],[ 0 ],[ 0 ],[ 0 ],[ 0 ]]):
c[3]:=Matrix(n,1,[[ 0 ],[ 0 ],[ 1 ],[ y5 ],[ y6 ],[ 0 ],[ 0 ],[ 0 ],[ 0 ]]):
c[4]:=Matrix(n,1,[[ 0 ],[ 0 ],[ 0 ],[ 0 ],[ 1 ],[ y7 ],[ y8 ],[ 0 ],[ 0 ]]):
c[5]:=Matrix(n,1,[[ 0 ],[ 0 ],[ 0 ],[ 0 ],[ 0 ],[ 1 ],[ y9 ],[ y10 ],[ 0 ]]):
c[6]:=Matrix(n,1,[[ 0 ],[ 0 ],[ 0 ],[ 0 ],[ 0 ],[ 0 ],[ 1 ],[ y11 ],[ y12 ]]):
# Since F3 and F4 are in general position with respect F1 and F2, there is a product P of transition matrices such that RPC=0 if and only if the x's parametriz a point in [][][]F3 n [][][]F4 and the y's parametrize a point mapped into [][][]F1 n [][][]F2 by duality.
# One uses actions of GL(n) to determine P, but a priori it could be any full-rank 9x9 matrix:
P:=RandomMatrix(9,9):
# We now add variables and functions to a file.
# This is part of a bertini input file.
fprintf(file,"variable_group x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12;\n"):
fprintf(file,"variable_group y1,y2,y3,y4,y5,y6,y7,y8,y9,y10,y11,y12;\n\n"):
fprintf(file,"function B1,B2,B3,B4,B5,B6,B7,B8,B9,B10,B11,B12,B13,B14,B15,B16,B17,B18;\n"): # bilinear equations
fprintf(file,"function D1,D2,D3,D4,D5,D6;\n\n"): # determinants
for i from 1 to 3 do
for j from 1 to 6 do
# The following line prints the ij-th equation from RPC=0
fprintf(file,"B%a = %a;\n",6*i+j-6,simplify(Multiply(Multiply(r[i],P),c[j])[1,1])):
od:
od:
# Now we generate the determinantal equations for []Fi.
# We randomly generate (6-dimensional subspaces of) flags for these Schubert varieties, and we use classical determinantal conditions.
# We place R in a bigger matrix M. (M will be R stacked on top rows spanning Fi_6.)
M:=Matrix(n,n):
for i from 1 to k do
for j from 1 to n do
M[i,j]:=r[i][1,j]:
od:
od:
for ii from 1 to 6 do # for each [] condition
for i from k+1 to n do
for j from 1 to n do
M[i,j]:=evalf(rand()/10^11):
od:
od:
fprintf(file,"D%a = %a;\n",ii,simplify(Determinant(M))):
od:
fclose(file):
We scaled equations to condition our system, and we solved using regeneration in bertini.
[Sample bertini file for a random instance of this Schubert problem.]
CONFIG
USEREGENERATION: 1;
TRACKTOLBEFOREEG: 1e-7;
TRACKTOLDURINGEG: 1e-7;
FINALTOL: 1e-11;
SLICETOLBEFOREEG: 1e-7;
SLICETOLDURINGEG: 1e-7;
SLICEFINALTOL: 1e-11;
SECURITYMAXNORM: 1e8;
MAXNORM: 1e8;
PRINTPATHMODULUS: 1000;
SAMPLEFACTOR: 0.1;
NUMSAMPLEPOINTS: 3;
ODEPREDICTOR: 2;
RANDOMSEED: 1000;
END;
INPUT
variable_group x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12;
variable_group y1,y2,y3,y4,y5,y6,y7,y8,y9,y10,y11,y12;
function B1,B2,B3,B4,B5,B6,B7,B8,B9,B10,B11,B12,B13,B14,B15,B16,B17,B18;
function D1,D2,D3,D4,D5,D6;
B1 = 1e-2*(88*x1-82*x2-70*x3+41+52*y1*x1-13*y1*x2+82*y1*x3+72*y1-33*y2*x1-68*y2*x2-67*y2*x3+22*y2);
B2 = 1e-2*(72+52*x1-13*x2+82*x3-33*y3*x1-68*y3*x2-67*y3*x3+22*y3-95*y4*x1-20*y4*x2-25*y4*x3+51*y4);
B3 = 1e-2*(22-33*x1-68*x2-67*x3-95*y5*x1-20*y5*x2-25*y5*x3+51*y5+20*y6*x1-61*y6*x2-48*y6*x3+77*y6);
B4 = 1e-2*(77+20*x1-61*x2-48*x3+25*y7*x1+94*y7*x2+12*y7*x3-2*y7-22*y8*x1+45*y8*x2-81*y8*x3-38*y8);
B5 = 1e-2*(-2+25*x1+94*x2+12*x3-22*y9*x1+45*y9*x2-81*y9*x3-38*y9+57*y10*x1+27*y10*x2-93*y10*x3-76*y10);
B6 = 1e-2*(-38-22*x1+45*x2-81*x3+57*y11*x1+27*y11*x2-93*y11*x3-76*y11+27*y12*x1+8*y12*x2+69*y12*x3+99*y12);
B7 = 1e-2*(-32-82*x4-70*x5+41*x6+91*x7+29*x8+70*x9+12*y1-13*y1*x4+82*y1*x5+72*y1*x6+42*y1*x7+18*y1*x8-59*y1*x9+99*y2-68*y2*x4-67*y2*x5+22*y2*x6+14*y2*x7+16*y2*x8+9*y2*x9);
B8 = 1e-2*(12-13*x4+82*x5+72*x6+42*x7+18*x8-59*x9+99*y3-68*y3*x4-67*y3*x5+22*y3*x6+14*y3*x7+16*y3*x8+9*y3*x9+65*y4-20*y4*x4-25*y4*x5+51*y4*x6+76*y4*x7-44*y4*x8+24*y4*x9);
B9 = 1e-2*(99-68*x4-67*x5+22*x6+14*x7+16*x8+9*x9+65*y5-20*y5*x4-25*y5*x5+51*y5*x6+76*y5*x7-44*y5*x8+24*y5*x9-80*y6-61*y6*x4-48*y6*x5+77*y6*x6+9*y6*x7+31*y6*x8-50*y6*x9);
B10 = 1e-2*(-80-61*x4-48*x5+77*x6+9*x7+31*x8-50*x9-9*y7+94*y7*x4+12*y7*x5-2*y7*x6+50*y7*x7+10*y7*x8-16*y7*x9-98*y8+45*y8*x4-81*y8*x5-38*y8*x6-18*y8*x7+87*y8*x8+33*y8*x9);
B11 = 1e-2*(-9+94*x4+12*x5-2*x6+50*x7+10*x8-16*x9-98*y9+45*y9*x4-81*y9*x5-38*y9*x6-18*y9*x7+87*y9*x8+33*y9*x9-74*y10+27*y10*x4-93*y10*x5-76*y10*x6-72*y10*x7-2*y10*x8-32*y10*x9);
B12 = 1e-2*(-98+45*x4-81*x5-38*x6-18*x7+87*x8+33*x9-74*y11+27*y11*x4-93*y11*x5-76*y11*x6-72*y11*x7-2*y11*x8-32*y11*x9-31*y12+8*y12*x4+69*y12*x5+99*y12*x6+29*y12*x7+44*y12*x8+92*y12*x9);
B13 = 1e-2*(-1+29*x10+70*x11-32*x12-62*y1+18*y1*x10-59*y1*x11+12*y1*x12+60*y2+16*y2*x10+9*y2*x11+99*y2*x12);
B14 = 1e-2*(-62+18*x10-59*x11+12*x12+60*y3+16*y3*x10+9*y3*x11+99*y3*x12+86*y4-44*y4*x10+24*y4*x11+65*y4*x12);
B15 = 1e-2*(60+16*x10+9*x11+99*x12+86*y5-44*y5*x10+24*y5*x11+65*y5*x12+43*y6+31*y6*x10-50*y6*x11-80*y6*x12);
B16 = 1e-2*(43+31*x10-50*x11-80*x12-50*y7+10*y7*x10-16*y7*x11-9*y7*x12-77*y8+87*y8*x10+33*y8*x11-98*y8*x12);
B17 = 1e-2*(-50+10*x10-16*x11-9*x12-77*y9+87*y9*x10+33*y9*x11-98*y9*x12-4*y10-2*y10*x10-32*y10*x11-74*y10*x12);
B18 = 1e-2*(-77+87*x10+33*x11-98*x12-4*y11-2*y11*x10-32*y11*x11-74*y11*x12+67*y12+44*y12*x10+92*y12*x11-31*y12*x12);
D1 = 1e-5*(9115.162600*x1+9333.836050*x2-31679.31855*x3-13866.03986*x4+24377.51186*x5-9801.154429*x7-6357.991982*x8+22174.81938*x9+1883.565808*x10-4791.769050*x11-9760.767994*x5*x10+6121.078800*x5*x11+4839.359070*x5*x12+2730.333364*x7*x10+1703.487057*x7*x11-2237.157490*x7*x12+4791.769050*x9*x12+7282.346890*x8*x11-1883.565808*x8*x12-7282.346890*x10*x9+78667.44940*x1*x4-133765.3327*x1*x5+461.1958760*x1*x6+54233.35519*x1*x7+33558.74766*x1*x8-121681.9356*x1*x9+17049.99891*x1*x10-35446.32898*x1*x11-1551.931377-136430.6442*x4*x3+13866.03986*x2*x6-41262.19578*x2*x7-75542.92226*x2*x8+124003.5279*x2*x9-8816.419800*x2*x10+23137.31628*x2*x11+136430.6442*x2*x5-23893.49412*x3*x7+70252.46931*x3*x8+175.0620000*x3*x9-24377.51186*x3*x6-1261.918950*x3*x10-7989.281020*x3*x11+15574.96572*x4*x10-38438.98188*x4*x11-306.1219700*x4*x12-97382.64023*x1*x5*x10-33963.98022*x1*x5*x11+41510.49363*x1*x5*x12+13475.92773*x1*x6*x10-47528.34715*x1*x6*x11+3425.988540*x1*x6*x12+58664.20819*x1*x7*x10-55781.50978*x1*x7*x11-11619.76762*x1*x7*x12+35446.32898*x1*x9*x12+97106.60110*x1*x8*x11-17049.99891*x1*x8*x12-97106.60110*x1*x10*x9+77907.39836*x1*x4*x10-54852.89340*x1*x4*x11-18807.05833*x1*x4*x12-56121.94260*x4*x3*x10-34533.06924*x4*x3*x11+22856.71241*x4*x3*x12+56121.94260*x2*x5*x10+34533.06924*x2*x5*x11-22856.71241*x2*x5*x12-15574.96572*x2*x6*x10+38438.98188*x2*x6*x11+306.1219700*x2*x6*x12-52017.18774*x2*x7*x10+47079.81175*x2*x7*x11+11242.67783*x2*x7*x12-23137.31628*x2*x9*x12-70131.22229*x2*x8*x11+8816.419800*x2*x8*x12+70131.22229*x2*x10*x9+9760.767994*x3*x6*x10-6121.078800*x3*x6*x11-4839.359070*x3*x6*x12+22760.63368*x3*x7*x10-5966.666930*x3*x7*x11-10692.76316*x3*x7*x12+7989.281020*x3*x9*x12+17710.16012*x3*x8*x11+1261.918950*x3*x8*x12-17710.16012*x3*x10*x9);
D2 = 1e-5*(49997.83456*x1+57571.95201*x2+6225.015197*x3-38807.75437*x4-9458.770738*x5-150.5467160*x7-8924.434140*x8+19055.26715*x9+24501.53711*x10+11764.48110*x11-10724.72190*x5*x10+7373.291839*x5*x11+20945.53908*x5*x12-4380.937420*x7*x10+9106.974882*x7*x11+18750.59103*x7*x12-11764.48110*x9*x12-14648.83633*x8*x11-24501.53711*x8*x12+14648.83633*x10*x9+39038.91483-9240.907918*x1*x4-7739.150622*x1*x5+27436.07983*x1*x6-24900.23547*x1*x7-35302.35816*x1*x8+2408.500100*x1*x9+5092.904230*x1*x10-9153.139728*x1*x11+7761.000406*x4*x3+38807.75437*x2*x6-35170.14970*x2*x7-46928.54743*x2*x8-3011.344498*x2*x9+20055.08123*x2*x10+2535.318045*x2*x11-7761.000406*x2*x5-8542.055450*x3*x7-9653.324277*x3*x8-4544.750311*x3*x9+9458.770738*x3*x6-14318.19480*x3*x10-14916.43074*x3*x11-28580.60732*x4*x10-2675.500977*x4*x11+18478.06357*x4*x12-16327.74368*x1*x5*x10+2476.986998*x1*x5*x11+22008.07868*x1*x5*x12+24125.58585*x1*x6*x10-6478.040144*x1*x6*x11-30210.38189*x1*x6*x12-38701.72189*x1*x7*x10+7025.879825*x1*x7*x11+51220.02812*x1*x7*x12+9153.139728*x1*x9*x12+6217.492366*x1*x8*x11-5092.904230*x1*x8*x12-6217.492366*x1*x10*x9+13806.57422*x1*x4*x10+685.5672319*x1*x4*x11-20887.04304*x1*x4*x12+25480.35542*x4*x3*x10+1803.333693*x4*x3*x11-27942.61424*x4*x3*x12-25480.35542*x2*x5*x10-1803.333693*x2*x5*x11+27942.61424*x2*x5*x12+28580.60732*x2*x6*x10+2675.500977*x2*x6*x11-18478.06357*x2*x6*x12-48355.48722*x2*x7*x10-3865.550562*x2*x7*x11+44292.43360*x2*x7*x12-2535.318045*x2*x9*x12-1017.614114*x2*x8*x11-20055.08123*x2*x8*x12+1017.614114*x2*x10*x9+10724.72190*x3*x6*x10-7373.291839*x3*x6*x11-20945.53908*x3*x6*x12-14239.42140*x3*x7*x10+4514.637667*x3*x7*x11+21852.30947*x3*x7*x12+14916.43074*x3*x9*x12+12677.96457*x3*x8*x11+14318.19480*x3*x8*x12-12677.96457*x3*x10*x9);
D3 = 1e-5*(-23721.19639*x1+18601.64966*x2+2991.478352*x3+5446.379930*x4-7318.675871*x5+2867.073067*x7-4439.648544*x8-4456.283816*x9+3827.813896*x10+23158.22422*x11+5693.855222*x5*x10-5173.901452*x5*x11-22960.19990*x5*x12-12722.44291*x7*x10-8504.339200*x7*x11+39675.26068*x7*x12-23158.22422*x9*x12-6605.530829*x8*x11-3827.813896*x8*x12+6605.530829*x10*x9+12982.53740-3968.368306*x1*x4+14334.58634*x1*x5+4175.713388*x1*x6-3057.280661*x1*x7+6639.676304*x1*x8-1546.280797*x1*x9+854.7763330*x1*x10+9336.851427*x1*x11+11741.32349*x4*x3-5446.379930*x2*x6+1262.900346*x2*x7-4440.927130*x2*x8+6251.819899*x2*x9-1503.024535*x2*x10-12801.68648*x2*x11-11741.32349*x2*x5+4483.799301*x3*x7-3603.429065*x3*x8-18007.87215*x3*x9+7318.675871*x3*x6+8977.362503*x3*x10+58609.23027*x3*x11-5171.477322*x4*x10+2912.541275*x4*x11+19818.37665*x4*x12-13694.85845*x1*x5*x10-2573.174192*x1*x5*x11+47081.44637*x1*x5*x12-6381.422449*x1*x6*x10-192.4985930*x1*x6*x11+22260.97597*x1*x6*x12+17146.38766*x1*x7*x10-2764.109685*x1*x7*x11-60864.34343*x1*x7*x12-9336.851427*x1*x9*x12-2669.148446*x1*x8*x11-854.7763330*x1*x8*x12+2669.148446*x1*x10*x9+1350.895911*x1*x4*x10+1280.679968*x1*x4*x11-4315.391270*x1*x4*x12-9892.910960*x4*x3*x10+4511.016768*x4*x3*x11+37464.46659*x4*x3*x12+9892.910960*x2*x5*x10-4511.016768*x2*x5*x11-37464.46659*x2*x5*x12+5171.477322*x2*x6*x10-2912.541275*x2*x6*x11-19818.37665*x2*x6*x12-11202.11975*x2*x7*x10+14757.26450*x2*x7*x11+46494.73640*x2*x7*x12+12801.68648*x2*x9*x12+3561.404422*x2*x8*x11+1503.024535*x2*x8*x12-3561.404422*x2*x10*x9-5693.855222*x3*x6*x10+5173.901452*x3*x6*x11+22960.19990*x3*x6*x12-12004.06446*x3*x7*x10-39386.85774*x3*x7*x11+21136.14268*x3*x7*x12-58609.23027*x3*x9*x12-16557.36729*x3*x8*x11-8977.362503*x3*x8*x12+16557.36729*x3*x10*x9);
D4 = 1e-5*(75676.92534*x1-44219.55073*x2-31732.72255*x3+19036.36088*x4-15066.13445*x5+14274.87584*x7+12288.20979*x8-49139.91778*x9+50529.41881*x10+124965.6850*x11-18146.17601*x5*x10-29578.37048*x5*x11-9279.402590*x5*x12+29091.79793*x7*x10-19557.11040*x7*x11+55499.45385*x7*x12-124965.6850*x9*x12+83310.47530*x8*x11-50529.41881*x8*x12-83310.47530*x10*x9+48236.62364+16776.39966*x1*x4-33029.78811*x1*x5+14278.21796*x1*x6+21916.15695*x1*x7+8226.339316*x1*x8-80375.47594*x1*x9-57041.68578*x1*x10+155582.7513*x1*x11-26334.62210*x4*x3-19036.36088*x2*x6-12447.13108*x2*x7+3470.485910*x2*x8+49422.53066*x2*x9-10245.52110*x2*x10-103847.8399*x2*x11+26334.62210*x2*x5-9896.498188*x3*x7-19746.00971*x3*x8+28864.48060*x3*x9+15066.13445*x3*x6-26058.20336*x3*x10-80076.76880*x3*x11+8068.430310*x4*x10+96795.73254*x4*x11-46605.84595*x4*x12+35907.16622*x1*x5*x10-175487.1004*x1*x5*x11+101292.8235*x1*x5*x12-29905.35312*x1*x6*x10+216392.1161*x1*x6*x11-176104.0341*x1*x6*x12-6411.872205*x1*x7*x10-53698.25455*x1*x7*x11+92982.27726*x1*x7*x12-155582.7513*x1*x9*x12-43670.85650*x1*x8*x11+57041.68578*x1*x8*x12+43670.85650*x1*x10*x9-3044.581690*x1*x4*x10-59313.14200*x1*x4*x11+88319.90774*x1*x4*x12-7840.300480*x4*x3*x10-86605.69009*x4*x3*x11+22153.27887*x4*x3*x12+7840.300480*x2*x5*x10+86605.69009*x2*x5*x11-22153.27887*x2*x5*x12-8068.430310*x2*x6*x10-96795.73254*x2*x6*x11+46605.84595*x2*x6*x12+1231.840075*x2*x7*x10+29380.71713*x2*x7*x11-52441.86693*x2*x7*x12+103847.8399*x2*x9*x12-3300.737220*x2*x8*x11+10245.52110*x2*x8*x12+3300.737220*x2*x10*x9+18146.17601*x3*x6*x10+29578.37048*x3*x6*x11+9279.402590*x3*x6*x12-31039.69582*x3*x7*x10+8520.244030*x3*x7*x11-36822.07736*x3*x7*x12+80076.76880*x3*x9*x12-73531.45190*x3*x8*x11+26058.20336*x3*x8*x12+73531.45190*x3*x10*x9);
D5 = 1e-5*(11565.08095*x1-470.0857711*x2-1730.746051*x3-52095.90433*x4-188984.3324*x5+37331.94623*x7+132255.8386*x8+109102.3458*x9-94230.18049*x10-114513.9304*x11+243980.9852*x5*x10-46827.87716*x5*x11-186332.9830*x5*x12+36900.58077*x7*x10+79449.42513*x7*x11+18781.44070*x7*x12+114513.9304*x9*x12+173623.6946*x8*x11+94230.18049*x8*x12-173623.6946*x10*x9-28016.92589+21859.95055*x1*x4+79318.91141*x1*x5+21177.54726*x1*x6-18839.12037*x1*x7-69712.92317*x1*x8-48837.27873*x1*x9+74155.01344*x1*x10+56134.07048*x1*x11-47.32588000*x4*x3+52095.90433*x2*x6-7808.576875*x2*x7-34973.44064*x2*x8-7519.771450*x2*x9+11840.19462*x2*x10+1452.772867*x2*x11+47.32588000*x2*x5-28292.66412*x3*x7-126750.3317*x3*x8-27179.78731*x3*x9+188984.3324*x3*x6+127206.5054*x3*x10+26369.61288*x3*x11-86472.48839*x4*x10-139724.9066*x4*x11-18799.26963*x4*x12+26559.16542*x1*x5*x10+47307.99444*x1*x5*x11+79967.03888*x1*x5*x12-161902.7909*x1*x6*x10-105757.5261*x1*x6*x11+49385.96776*x1*x6*x12-25863.13990*x1*x7*x10-24935.42789*x1*x7*x11-15748.95815*x1*x7*x12-56134.07048*x1*x9*x12-25226.02978*x1*x8*x11-74155.01344*x1*x8*x12+25226.02978*x1*x10*x9+88393.43270*x1*x4*x10+69834.03143*x1*x4*x11+8588.762370*x1*x4*x12+147390.6273*x4*x3*x10+31580.97384*x4*x3*x11+1965.088069*x4*x3*x12-147390.6273*x2*x5*x10-31580.97384*x2*x5*x11-1965.088069*x2*x5*x12+86472.48839*x2*x6*x10+139724.9066*x2*x6*x11+18799.26963*x2*x6*x12-6332.929329*x2*x7*x10-19517.98029*x2*x7*x11-2728.701869*x2*x7*x12-1452.772867*x2*x9*x12-81319.39369*x2*x8*x11-11840.19462*x2*x8*x12+81319.39369*x2*x10*x9-243980.9852*x3*x6*x10+46827.87716*x3*x6*x11+186332.9830*x3*x6*x12-45028.02532*x3*x7*x10-24498.68014*x3*x7*x11-29009.33648*x3*x7*x12-26369.61288*x3*x9*x12-66496.51904*x3*x8*x11-127206.5054*x3*x8*x12+66496.51904*x3*x10*x9);
D6 = 1e-5*(-19963.86014*x1-15553.14432*x2+7551.742943*x3+73706.95460*x4-43965.46804*x5-27574.52975*x7-63036.76180*x8+146469.9899*x9-25720.56365*x10-4921.677770*x11+58657.86883*x5*x10+60083.64594*x5*x11+4463.247452*x5*x12+54121.47295*x7*x10-2588.567482*x7*x11-1182.494189*x7*x12+4921.677770*x9*x12+281564.0648*x8*x11+25720.56365*x8*x12-281564.0648*x10*x9-192510.7365*x1*x4+104745.5982*x1*x5+84639.79405*x1*x6+72661.59052*x1*x7+165865.7273*x1*x8-202736.8962*x1*x9+17009.61859*x1*x10+6976.385200*x1*x11-8782.459990*x4*x3-73706.95460*x2*x6-558.5023700*x2*x7-1065.742300*x2*x8-156592.1388*x2*x9+30072.78010*x2*x10+5260.361940*x2*x11+8782.459990*x2*x5+3618.749740*x3*x7+8146.771300*x3*x8+75953.21098*x3*x9+43965.46804*x3*x6-9248.351351*x3*x10-2609.736140*x3*x11-142336.1784*x4*x10+1503.073300*x4*x11+2625.309180*x4*x12-123792.4415*x1*x5*x10+5623.261950*x1*x5*x11+4158.136359*x1*x5*x12-236394.9344*x1*x6*x10+171221.3459*x1*x6*x11+19773.01065*x1*x6*x12-142703.3930*x1*x7*x10+73362.75728*x1*x7*x11+9724.520385*x1*x7*x12-6976.385200*x1*x9*x12-230697.8942*x1*x8*x11-17009.61859*x1*x8*x12+230697.8942*x1*x10*x9+370525.9758*x1*x4*x10-185134.4779*x1*x4*x11-24855.01153*x1*x4*x12+17403.59220*x4*x3*x10+65015.29668*x4*x3*x11+6162.463698*x4*x3*x12-17403.59220*x2*x5*x10-65015.29668*x2*x5*x11-6162.463698*x2*x5*x12+142336.1784*x2*x6*x10-1503.073300*x2*x6*x11-2625.309180*x2*x6*x12+1093.272300*x2*x7*x10+2154.891960*x2*x7*x11+195.2678940*x2*x7*x12-5260.361940*x2*x9*x12-302418.2535*x2*x8*x11-30072.78010*x2*x8*x12+302418.2535*x2*x10*x9-58657.86883*x3*x6*x10-60083.64594*x3*x6*x11-4463.247452*x3*x6*x12-7068.034830*x3*x7*x10-25828.89479*x3*x7*x11-2443.730713*x3*x7*x12+2609.736140*x3*x9*x12+90201.83195*x3*x8*x11+9248.351351*x3*x8*x12-90201.83195*x3*x10*x9+14481.73906);
END;
We solve the random instance given in the sample above in bertini.
To prove our approximate solutions are correct, we hard certified the results using alphaCertified.
This requires rational arithmetic.
[437 approximate solutions to the sample above, formatted for rational arithmetic in alphaCertified.]
437
56463442067983/1250000000000000 4311101728562611/50000000000000000
-527656350211437/2000000000000000 -860809538053643/20000000000000000
1139108393722197/2000000000000000 1925218612020399/10000000000000000
457856464408219/100000000000000 -21433440918403/4000000000000
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-290117485366373/25000000000000 35489023270787/1562500000000
-4642097085485529/1000000000000000 -206286250624709/100000000000000
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2541065989775771/1000000000000000 1764979748603769/1000000000000000
-4902658473427997/10000000000000000 1164596079018891/2000000000000000
2244182433937739/5000000000000000 8911146386421887/100000000000000000
92974851087789/3125000000000000 519436124481897/2500000000000000
-274206336874781/1250000000000000 -2749720878519331/500000000000000000
-5928987103466589/10000000000000000 4868332540858877/100000000000000000
1117999136562691/250000000000000 -33657604981329/10000000000000
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376024496167941/1250000000000000 733738143622289/2500000000000000
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8277881743123341/10000000000000000 -433847955430179/1000000000000000
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280754167006497/1250000000000000 2099567201241291/10000000000000000
815457981236309/1250000000000000 -978661943447477/1000000000000000
33539091128407/25000000000000 -222858727969087/250000000000000
4130418321683151/10000000000000000 -2462798238038303/5000000000000000
573903692652289/2500000000000000 893585938134521/1250000000000000
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321061373259291/100000000000000 -1161751927752717/1000000000000000
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1243157055330881/1000000000000000 1980896412106511/5000000000000000
1228835965068837/1000000000000000 574876351721467/500000000000000
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3357276901808509/10000000000000000 -3405618772105013/5000000000000000
26079233974677/2000000000000000 131502092351351/1250000000000000
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2078683402793653/1000000000000000 2317829215890531/10000000000000000
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384058112997587/250000000000000 1378216399521393/5000000000000000
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51098397416017/125000000000000 -2664680602346271/5000000000000000
91639829653797/40000000000000 1564112257307821/5000000000000000
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5487146111696313/10000000000000000 -1562675036409869/2500000000000000
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2881909225922057/10000000000000000 665503788947621/1250000000000000
-340466439021431/250000000000000 1181369997338657/1000000000000000
1098047922054469/1000000000000000 -9640742982624411/10000000000000000
-2319384579261833/1000000000000000 2879457155159197/1000000000000000
1396907025247659/2500000000000000 2575350404645271/10000000000000000
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9374366392799979/100000000000000000 183306541236727/625000000000000
743082783136553/1250000000000000 -2464693083700571/2500000000000000
235410295552883/1250000000000000 4001673523491609/10000000000000000
-1737293154019/7812500000000 1683985492880663/10000000000000000
665319301246907/12500000000000000 -1465317883171589/10000000000000000
3708139542493849/10000000000000000 -47594661785979/200000000000000
2012301782329963/10000000000000000 1778623146926137/10000000000000000
-201104965731629/1250000000000000 1877127570528007/5000000000000000
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654389109591201/500000000000000 2891493244925103/1000000000000000
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49045065017003/250000000000000 524974929469031/1000000000000000
4949223753072173/10000000000000000 1638701489897391/5000000000000000
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455672666846339/250000000000000 -6988336381241541/10000000000000000
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2840983419744471/10000000000000000 1072938146873463/2500000000000000
3527595079932631/10000000000000000 4808351721589561/5000000000000000
4810416803443821/10000000000000000 717252823636269/2000000000000000
4708310182621647/50000000000000000 122734966469053/100000000000000
323885363559113/200000000000000 853872742708779/2000000000000000
-1004572715530701/1000000000000000 -1440298345381557/1000000000000000
7707369116739219/10000000000000000 1509189946486491/5000000000000000
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1143756924005431/1000000000000000 -1282449405549713/2000000000000000
155371866588399/250000000000000 445855063669791/2000000000000000
2197746378575729/5000000000000000 700321389357053/1250000000000000
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4407787656218769/5000000000000000 7076338867611771/10000000000000000
3744702695183807/10000000000000000 81477959740881/625000000000000
154233056616281/500000000000000 -470010168411577/2000000000000000
730861173056817/20000000000000000 65830605509973/40000000000000
1548379879452861/2500000000000000 -1245510062025567/500000000000000
-185355663939027/125000000000000 -310492556350761/200000000000000000
2035544579736341/2500000000000000 -6222996107474693/100000000000000000
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[Corresponding polynomial system formated for alphaCertified.]
24 24
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0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 81467713/10000 0
0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3797660549/50000 0
0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1099136701/25000 0
0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -9248351351/1000000 0
0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 -130486807/50000 0
0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -177920223/1250 0
0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 15030733/10000 0
0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 131265459/50000 0
1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -247584883/2000 0
1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 112465239/20000 0
1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 4158136359/1000000 0
1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -147746834/625 0
1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1712213459/10000 0
1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 395460213/20000 0
1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -142703393/1000 0
1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 458517233/6250 0
1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1944904077/200000 0
1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 -17440963/2500 0
1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 -1153489471/5000 0
1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 -1700961859/100000 0
1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1153489471/5000 0
1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1852629879/5000 0
1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 -1851344779/10000 0
1 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 -2485501153/100000 0
0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 87017961/5000 0
0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1625382417/25000 0
0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 3081231849/500000 0
0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -87017961/5000 0
0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 -1625382417/25000 0
0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 -3081231849/500000 0
0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 177920223/1250 0
0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 -15030733/10000 0
0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 -131265459/50000 0
0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 10932723/10000 0
0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 53872299/25000 0
0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 97633947/500000 0
0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 -263018097/50000 0
0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 -604836507/2000 0
0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 -300727801/10000 0
0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 604836507/2000 0
0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -5865786883/100000 0
0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 -3004182297/50000 0
0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 -1115811863/250000 0
0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -706803483/100000 0
0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 -2582889479/100000 0
0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 -2443730713/1000000 0
0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 130486807/50000 0
0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1804036639/20000 0
0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 9248351351/1000000 0
0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1804036639/20000 0
The points given above were all cerfied as being approximate solutions via alphaCertified.
[This is how long it took to solve.]
time output (linux)
real 42m51.941s
user 338m42.424s
sys 0m45.936s
3.6 GHz processors used. Total processing power:
~20.36836 GHz-hours
[This is how long it took to certify.]
time output (linux)
real 33m22.362s
user 33m19.704s
sys 0m0.332s
3.6 GHz processors used. Total processing power:
~2.000036 GHz-hours