Details of a primal-dual formulation of a Schubert problem with 128 solutions in the flag variety Fl(2,4,5;8).

The underlying geometry is described in Example 4.11 [here]. There is some description of the geometry in the documented code below as well.
[Maple file to generate a random instance of the Schubert problem.]

# Maple code
# Output: primal-dual square system of equations for the Schubert problem
#
# 48573 78453^2 68574^2 78465^3 47385 = 128 in Fl(2,4,5;8)
#
# Nickolas Hein
# 27 May 2014
#
# This is a proof of concept for computing a non-Grassmannian Schubert problem using a primal-dual formulation in bertini.
# Therefore, precomputations involving linear algebra are omitted.
# This is used as an example in "Certifiable Numerical Computation in Schubert Calculus, and it references remarks therein.
#
with(LinearAlgebra):
k:=[2,4,5]:
n:=8:
# Given an instance
#
# (48573)F1 n (78453)F2 n ... n (47385)F9
#
# of the Schubert problem, we choose a basis of the ambient space so that F9 is the coordinate flag.
# Our Stiefel coordinates for (47385)F9 are given by the 5x8 matrix R with rows r[i]:
r[1]:=Matrix(1,n,[[ x1 , x2 , x3 , 1 , 0 , 0 , 0 , 0 ]]):
r[2]:=Matrix(1,n,[[ x4 , x5 , x6 , 0 , x7 , x8 , 1 , 0 ]]):
r[3]:=Matrix(1,n,[[ x9 , x10 , 1 , 0 , 0 , 0 , 0 , 0 ]]):
r[4]:=Matrix(1,n,[[ x11 , x12 , 0 , 0 , x13 , x14 , 0 , 1 ]]):
r[5]:=Matrix(1,n,[[ x15 , x16 , 0 , 0 , 1 , 0 , 0 , 0 ]]):
# For a different choice of basis for the dual space, we parametrize ((48573)F1)\perp locally.
# To find the intersection of these two Schubert varieties more efficiently, we apply Remark 4.10, and and the resulting parametrization in the dual space is given by the matrix of columns c[j]:
c[1]:=Matrix(n,1,[[ 0 ],[ 0 ],[ 0 ],[ 0 ],[ 0 ],[ 1 ],[ y1 ],[ y2 ]]):
c[2]:=Matrix(n,1,[[ 0 ],[ 1 ],[ y3 ],[ y4 ],[ y5 ],[ 0 ],[ y6 ],[ y7 ]]):
c[3]:=Matrix(n,1,[[ 1 ],[ 0 ],[ y8 ],[ y9 ],[ y10 ],[ 0 ],[ y11 ],[ y12 ]]):
c[4]:=Matrix(n,1,[[ 0 ],[ 0 ],[ 0 ],[ 0 ],[ 0 ],[ 0 ],[ 1 ],[ y13 ]]):
c[5]:=Matrix(n,1,[[ 0 ],[ 0 ],[ 0 ],[ 0 ],[ 1 ],[ 0 ],[ 0 ],[ y14 ]]):
c[6]:=Matrix(n,1,[[ 0 ],[ 0 ],[ 1 ],[ y15 ],[ 0 ],[ 0 ],[ 0 ],[ y16 ]]):
# Since F3 and F4 are in general position with respect F1 and F2, there is a product P1 of transition matrices such that R*P1*C=0 if and only if the x's parametriz a point in the intersection
#
# X1 := (48573)F1 n (47385)F9
#
# and the y's parametrize a point mapped the image of this variety given by duality.
# One uses actions of GL(n) to determine P, but a priori it could be any full-rank 8x8 matrix:
P1:=RandomMatrix(n,n):
# We will need to do this more than once.
P2:=RandomMatrix(n,n):
# 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,x13,x14,x15,x16;\n"):
fprintf(file,"variable_group y1,y2,y3,y4,y5,y6,y7,y8,y9,y10,y11,y12,y13,y14,y15,y16;\n"):
fprintf(file,"variable_group z1,z2,z3,z4,z5,z6,z7,z8,z9;\n\n"):

# bilinear equations which we have described
fprintf(file,"function B1,B2,B3,B4,B5,B6,B7,B8,B9,B10,B11,B12,B13,B14,B15,B16,B17,B18,B19,B20,B21;\n"):

# bilinear equations which we will use later
fprintf(file,"function BB1,BB2,BB3,BB4,BB5,BB6,BB7,BB8,BB9,BB10,BB11,BB12,BB13,BB14,BB15;\n"):

# determinants
fprintf(file,"function D1,D2,D3,D4,D5;\n\n"):

for i from 1 to 5 do
  for j from 1 to 3 do
    # The following line prints the ij-th equation from RP1C=0
    fprintf(file,"B%a = %a;\n",3*i+j-3,simplify(Multiply(Multiply(r[i],P1),c[j])[1,1])):
  od:
od:
for i from 1 to 2 do
  for j from 4 to 6 do
    # The following line prints the ij-th equation from RP1C=0
    fprintf(file,"B%a = %a;\n\n",3*i+j+9,simplify(Multiply(Multiply(r[i],P1),c[j])[1,1])):
  od:
od:
# The solutions to these 21 bilinear equations are in X1.
# We use Remark 4.8 to obtain bilinear equations that restrict to points in
#
# X2:=X1 n (78453)F2 n (78453)F3.
#
c[1]:=Matrix(n,1,[[ z1 ],[ z2 ],[ 1 ],[ 0 ],[ 0 ],[ 0 ],[ 0 ],[ 0 ]]):
c[2]:=Matrix(n,1,[[ 0 ],[ z3 ],[ z4 ],[ z5 ],[ z6 ],[ z7 ],[ 1 ],[ 0 ]]):
c[3]:=Matrix(n,1,[[ 0 ],[ 0 ],[ 0 ],[ 0 ],[ 0 ],[ z8 ],[ z9 ],[ 1 ]]):
for i from 1 to 5 do
  for j from 1 to 3 do
    # The following line prints the ij-th equation from RP2C=0
    fprintf(file,"BB%a = %a;\n\n",3*i+j-3,simplify(Multiply(Multiply(r[i],P2),c[j])[1,1])):
  od:
od:
# Now we generate the determinantal equations for []F4 and []F5.
# We randomly generate (6-dimensional subspaces of) flags for these Schubert varieties, and we use classical determinantal conditions.
# We place R_2 in a bigger matrix M. (M will be R_2 stacked on top rows spanning Fi_6.)
M:=Matrix(n,n):
for i from 1 to 2 do
  for j from 1 to n do
    M[i,j]:=r[i][1,j]:
  od:
od:
for ii from 1 to 2 do # for each [] condition
  for i from 3 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\n",ii,simplify(Determinant(M))):
od:
# Now we generate the determinantal equations for []F6...[]F8.
# We randomly generate (4-dimensional subspaces of) flags for these Schubert varieties, and we use classical determinantal conditions.
# We place R_4 in a bigger matrix M. (M will be R_4 stacked on top rows spanning Fi_4.)
M:=Matrix(n,n):
for i from 1 to 4 do
  for j from 1 to n do
    M[i,j]:=r[i][1,j]:
  od:
od:
for ii from 3 to 5 do # for each [] condition
  for i from 5 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\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,x13,x14,x15,x16;
variable_group y1,y2,y3,y4,y5,y6,y7,y8,y9,y10,y11,y12,y13,y14,y15,y16;
variable_group z1,z2,z3,z4,z5,z6,z7,z8,z9;

function B1,B2,B3,B4,B5,B6,B7,B8,B9,B10,B11,B12,B13,B14,B15,B16,B17,B18,B19,B20,B21;
function BB1,BB2,BB3,BB4,BB5,BB6,BB7,BB8,BB9,BB10,BB11,BB12,BB13,BB14,BB15;
function D1,D2,D3,D4,D5;

B1 = 1e-2*(33-38*x1-18*x2+87*x3-93*y1*x1-76*y1*x2-72*y1*x3-2*y1+8*y2*x1+69*y2*x2+99*y2*x3+29*y2);

B2 = 1e-2*(-20+99*x1+60*x2-95*x3+24*y3*x1+65*y3*x2+86*y3*x3+20*y3+31*y4*x1-50*y4*x2-80*y4*x3+43*y4+50*y5*x1+10*y5*x2-16*y5*x3-9*y5-93*y6*x1-76*y6*x2-72*y6*x3-2*y6+8*y7*x1+69*y7*x2+99*y7*x3+29*y7);

B3 = 1e-2*(-62*x1-33*x2-68*x3-67+24*y8*x1+65*y8*x2+86*y8*x3+20*y8+31*y9*x1-50*y9*x2-80*y9*x3+43*y9+50*y10*x1+10*y10*x2-16*y10*x3-9*y10-93*y11*x1-76*y11*x2-72*y11*x3-2*y11+8*y12*x1+69*y12*x2+99*y12*x3+29*y12);

B4 = 1e-2*(57-38*x4-18*x5+87*x6-98*x7-77*x8-93*y1*x4-76*y1*x5-72*y1*x6-4*y1-32*y1*x7-74*y1*x8+8*y2*x4+69*y2*x5+99*y2*x6-31*y2+44*y2*x7+92*y2*x8);

B5 = 1e-2*(76+99*x4+60*x5-95*x6-25*x7+51*x8+77*y3+12*y4+45*y5-4*y6-31*y7+24*y3*x4+65*y3*x5+86*y3*x6-61*y3*x7-48*y3*x8+31*y4*x4-50*y4*x5-80*y4*x6+25*y4*x7+94*y4*x8+50*y5*x4+10*y5*x5-16*y5*x6-50*y5*x7-22*y5*x8-93*y6*x4-76*y6*x5-72*y6*x6-32*y6*x7-74*y6*x8+8*y7*x4+69*y7*x5+99*y7*x6+44*y7*x7+92*y7*x8);

B6 = 1e-2*(16-62*x4-33*x5-68*x6+22*x7+14*x8+77*y8+12*y9+45*y10-4*y11-31*y12+24*y8*x4+65*y8*x5+86*y8*x6-61*y8*x7-48*y8*x8+31*y9*x4-50*y9*x5-80*y9*x6+25*y9*x7+94*y9*x8+50*y10*x4+10*y10*x5-16*y10*x6-50*y10*x7-22*y10*x8-93*y11*x4-76*y11*x5-72*y11*x6-32*y11*x7-74*y11*x8+8*y12*x4+69*y12*x5+99*y12*x6+44*y12*x7+92*y12*x8);

B7 = 1e-2*(87-38*x9-18*x10-93*y1*x9-76*y1*x10-72*y1+8*y2*x9+69*y2*x10+99*y2);

B8 = 1e-2*(-95+99*x9+60*x10+24*y3*x9+65*y3*x10+86*y3+31*y4*x9-50*y4*x10-80*y4+50*y5*x9+10*y5*x10-16*y5-93*y6*x9-76*y6*x10-72*y6+8*y7*x9+69*y7*x10+99*y7);

B9 = 1e-2*(-62*x9-33*x10-68+24*y8*x9+65*y8*x10+86*y8+31*y9*x9-50*y9*x10-80*y9+50*y10*x9+10*y10*x10-16*y10-93*y11*x9-76*y11*x10-72*y11+8*y12*x9+69*y12*x10+99*y12);

B10 = 1e-2*(27-38*x11-18*x12-98*x13-77*x14-93*y1*x11-76*y1*x12+27*y1-32*y1*x13-74*y1*x14+8*y2*x11+69*y2*x12+67*y2+44*y2*x13+92*y2*x14);

B11 = 1e-2*(-44+99*x11+60*x12-25*x13+51*x14+9*y3-2*y4-81*y5+27*y6+67*y7+24*y3*x11+65*y3*x12-61*y3*x13-48*y3*x14+31*y4*x11-50*y4*x12+25*y4*x13+94*y4*x14+50*y5*x11+10*y5*x12-50*y5*x13-22*y5*x14-93*y6*x11-76*y6*x12-32*y6*x13-74*y6*x14+8*y7*x11+69*y7*x12+44*y7*x13+92*y7*x14);

B12 = 1e-2*(9-62*x11-33*x12+22*x13+14*x14+9*y8-2*y9-81*y10+27*y11+67*y12+24*y8*x11+65*y8*x12-61*y8*x13-48*y8*x14+31*y9*x11-50*y9*x12+25*y9*x13+94*y9*x14+50*y10*x11+10*y10*x12-50*y10*x13-22*y10*x14-93*y11*x11-76*y11*x12-32*y11*x13-74*y11*x14+8*y12*x11+69*y12*x12+44*y12*x13+92*y12*x14);

B13 = 1e-2*(-98-38*x15-18*x16-93*y1*x15-76*y1*x16-32*y1+8*y2*x15+69*y2*x16+44*y2);

B14 = 1e-2*(-25+99*x15+60*x16+24*y3*x15+65*y3*x16-61*y3+31*y4*x15-50*y4*x16+25*y4+50*y5*x15+10*y5*x16-50*y5-93*y6*x15-76*y6*x16-32*y6+8*y7*x15+69*y7*x16+44*y7);

B15 = 1e-2*(-62*x15-33*x16+22+24*y8*x15+65*y8*x16-61*y8+31*y9*x15-50*y9*x16+25*y9+50*y10*x15+10*y10*x16-50*y10-93*y11*x15-76*y11*x16-32*y11+8*y12*x15+69*y12*x16+44*y12);

B16 = 1e-2*(-2-93*x1-76*x2-72*x3+8*y13*x1+69*y13*x2+99*y13*x3+29*y13);

B17 = 1e-2*(-9+50*x1+10*x2-16*x3+8*y14*x1+69*y14*x2+99*y14*x3+29*y14);

B18 = 1e-2*(20+24*x1+65*x2+86*x3+31*y15*x1-50*y15*x2-80*y15*x3+43*y15+8*y16*x1+69*y16*x2+99*y16*x3+29*y16);

B19 = 1e-2*(-4-93*x4-76*x5-72*x6-32*x7-74*x8+8*y13*x4+69*y13*x5+99*y13*x6-31*y13+44*y13*x7+92*y13*x8);

B20 = 1e-2*(45+50*x4+10*x5-16*x6-50*x7-22*x8+8*y14*x4+69*y14*x5+99*y14*x6-31*y14+44*y14*x7+92*y14*x8);

B21 = 1e-2*(77+24*x4+65*x5+86*x6-61*x7-48*x8+31*y15*x4-50*y15*x5-80*y15*x6+12*y15+25*y15*x7+94*y15*x8+8*y16*x4+69*y16*x5+99*y16*x6-31*y16+44*y16*x7+92*y16*x8);

BB1 = 1e-2*(-36*z1*x1-69*z1*x2+69*z1*x3-15*z1+10*z2*x1-44*z2*x2+26*z2*x3-3*z2+95*x1+63*x2+10*x3-61);

BB2 = 1e-2*(91+10*z3*x1-44*z3*x2+26*z3*x3-3*z3+95*z4*x1+63*z4*x2+10*z4*x3-61*z4+5*z5*x1-91*z5*x2-44*z5*x3-38*z5-23*z6*x1-63*z6*x2-26*z6*x3+30*z6-14*z7*x1+60*z7*x2-35*z7*x3+21*z7-82*x1-70*x2+41*x3);

BB3 = 1e-2*(72-14*z8*x1+60*z8*x2-35*z8*x3+21*z8-82*z9*x1-70*z9*x2+41*z9*x3+91*z9+52*x1-13*x2+82*x3);

BB4 = 1e-2*(-36*z1*x4-69*z1*x5+69*z1*x6+99*z1+2*z1*x7-88*z1*x8+10*z2*x4-44*z2*x5+26*z2*x6+9*z2-62*z2*x7-83*z2*x8+95*x4+63*x5+10*x6-78-26*x7-20*x8);

BB5 = 1e-2*(-32-82*x4-70*x5+41*x6+29*x7+70*x8+9*z3-78*z4-z5+12*z6+19*z7+10*z3*x4-44*z3*x5+26*z3*x6-62*z3*x7-83*z3*x8+95*z4*x4+63*z4*x5+10*z4*x6-26*z4*x7-20*z4*x8+5*z5*x4-91*z5*x5-44*z5*x6-38*z5*x7+91*z5*x8-23*z6*x4-63*z6*x5-26*z6*x6+10*z6*x7+22*z6*x8-14*z7*x4+60*z7*x5-35*z7*x6+90*z7*x7+80*z7*x8);

BB6 = 1e-2*(-59-14*z8*x4+60*z8*x5-35*z8*x6+19*z8+90*z8*x7+80*z8*x8-82*z9*x4-70*z9*x5+41*z9*x6-32*z9+29*z9*x7+70*z9*x8+52*x4-13*x5+82*x6+42*x7+18*x8);

BB7 = 1e-2*(-36*z1*x9-69*z1*x10+69*z1+10*z2*x9-44*z2*x10+26*z2+95*x9+63*x10+10);

BB8 = 1e-2*(41+10*z3*x9-44*z3*x10+26*z3+95*z4*x9+63*z4*x10+10*z4+5*z5*x9-91*z5*x10-44*z5-23*z6*x9-63*z6*x10-26*z6-14*z7*x9+60*z7*x10-35*z7-82*x9-70*x10);

BB9 = 1e-2*(82-14*z8*x9+60*z8*x10-35*z8-82*z9*x9-70*z9*x10+41*z9+52*x9-13*x10);

BB10 = 1e-2*(-36*z1*x11-69*z1*x12-59*z1+2*z1*x13-88*z1*x14+10*z2*x11-44*z2*x12+88*z2-62*z2*x13-83*z2*x14+95*x11+63*x12-4-26*x13-20*x14);

BB11 = 1e-2*(-1-82*x11-70*x12+29*x13+70*x14+88*z3-4*z4+63*z5+45*z6+88*z7+10*z3*x11-44*z3*x12-62*z3*x13-83*z3*x14+95*z4*x11+63*z4*x12-26*z4*x13-20*z4*x14+5*z5*x11-91*z5*x12-38*z5*x13+91*z5*x14-23*z6*x11-63*z6*x12+10*z6*x13+22*z6*x14-14*z7*x11+60*z7*x12+90*z7*x13+80*z7*x14);

BB12 = 1e-2*(12-14*z8*x11+60*z8*x12+88*z8+90*z8*x13+80*z8*x14-82*z9*x11-70*z9*x12-z9+29*z9*x13+70*z9*x14+52*x11-13*x12+42*x13+18*x14);

BB13 = 1e-2*(-36*z1*x15-69*z1*x16+2*z1+10*z2*x15-44*z2*x16-62*z2+95*x15+63*x16-26);

BB14 = 1e-2*(29+10*z3*x15-44*z3*x16-62*z3+95*z4*x15+63*z4*x16-26*z4+5*z5*x15-91*z5*x16-38*z5-23*z6*x15-63*z6*x16+10*z6-14*z7*x15+60*z7*x16+90*z7-82*x15-70*x16);

BB15 = 1e-2*(42-14*z8*x15+60*z8*x16+90*z8-82*z9*x15-70*z9*x16+29*z9+52*x15-13*x16);

D1 = 1e-5*(32646.79694*x2-22915.26710*x3+77637.41890*x5-49439.82126*x6-4816.583600*x7-90784.74972*x8-54316.67536*x1*x5+74945.82999*x1*x6-56623.61690*x1*x7+67445.25397*x1*x8+31872.76862*x1+37714.33146*x2*x6-56065.23413*x2*x7+3673.093905*x2*x8-37714.33146*x5*x3+38042.34393*x3*x7+41761.93752*x3*x8+54316.67536*x4*x2-74945.82999*x4*x3+83077.01468*x4+4375.800220);

D2 = 1e-4*(-8562.777795*x2+7112.335063*x3+9090.717450*x5-6575.380033*x6-5443.822870*x7+1903.330298*x8+3482.244860*x2*x6+5739.054539*x2*x7-7288.632937*x2*x8-3482.244860*x5*x3-2065.814990*x3*x7+4542.839276*x3*x8-14636.14631*x1*x5+12044.27112*x1*x6+11167.26717*x1*x7-6115.770322*x1*x8+277.0511450*x1-2398.657057+14636.14631*x4*x2-12044.27112*x4*x3+3805.832965*x4);

D3 = 1e-4*(-35.48944610*x1-979.8834495*x2-393.1554040*x4-424.6686895*x5+228.8646700*x7+217.1249347*x8-309.9359032*x9-1180.984559*x10-159.7844319*x11-519.7913588*x12+345.5953033*x13-65.79588760*x14-25.03603470*x10*x3*x7*x14+25.03603470*x10*x3*x13*x8-553.3039683*x9*x5*x3*x14+343.2792540*x9*x3*x7*x14-343.2792540*x9*x3*x13*x8-133.5234583*x9*x2*x6*x13+553.3039683*x9*x2*x6*x14+374.2822800*x9*x2*x7*x14-374.2822800*x9*x2*x13*x8-133.5234583*x9*x12*x3*x7+553.3039683*x9*x12*x3*x8+133.5234583*x9*x5*x3*x13+133.5234583*x11*x10*x3*x7-553.3039683*x11*x10*x3*x8-133.5234583*x4*x10*x3*x13+553.3039683*x4*x10*x3*x14+133.5234583*x1*x10*x6*x13-553.3039683*x1*x10*x6*x14-374.2822800*x1*x10*x7*x14+374.2822800*x1*x10*x13*x8-885.0613310*x1*x5+483.0188940*x1*x7+273.8614181*x1*x8+885.0613310*x4*x2+6.520905000*x2*x7-192.9733477*x2*x8-326.3692953*x11*x5+65.99179590*x11*x10+237.4276193*x11*x7-144.7972960*x11*x8+795.4379000*x4*x10+326.3692953*x4*x12-237.4276193*x4*x13+144.7972960*x4*x14-625.9169530*x1*x10-330.2418216*x1*x12+174.8741700*x1*x13+124.3721100*x1*x14+625.9169530*x9*x2+35.48944610*x9*x3-795.4379000*x9*x5+393.1554040*x9*x6+700.1653510*x9*x7+245.6809059*x9*x8-65.99179590*x9*x12+97.38688362*x9*x13+213.9964603*x9*x14+330.2418216*x11*x2+418.2480968-885.0613310*x9*x2*x6-598.9431670*x9*x2*x7+1.009986663*x9*x2*x8+885.0613310*x9*x5*x3-483.0188940*x9*x3*x7-273.8614181*x9*x3*x8+330.2418216*x9*x12*x3-326.3692953*x9*x12*x6-129.0548850*x9*x2*x13-390.9206984*x9*x2*x14-174.8741700*x9*x3*x13-124.3721100*x9*x3*x14+100.8597213*x9*x5*x13+496.9026478*x9*x5*x14-100.8597213*x9*x12*x7-496.9026478*x9*x12*x8+237.4276193*x9*x6*x13-144.7972960*x9*x6*x14-406.2349904*x9*x7*x14+406.2349904*x9*x13*x8-133.5234583*x11*x2*x7+553.3039683*x11*x2*x8-330.2418216*x11*x10*x3+326.3692953*x11*x10*x6+100.8597213*x11*x10*x7+496.9026478*x11*x10*x8+133.5234583*x4*x2*x13-553.3039683*x4*x2*x14-885.0613310*x4*x10*x3-100.8597213*x4*x10*x13-496.9026478*x4*x10*x14-133.5234583*x1*x5*x13+553.3039683*x1*x5*x14+885.0613310*x1*x10*x6+598.9431670*x1*x10*x7+129.0548850*x1*x10*x13-1.009986663*x1*x10*x8+390.9206984*x1*x10*x14+133.5234583*x1*x12*x7-553.3039683*x1*x12*x8-343.2792540*x1*x7*x14+343.2792540*x1*x13*x8+192.9733477*x10*x3*x8-6.520905000*x10*x3*x7+25.03603470*x2*x7*x14-25.03603470*x2*x13*x8+145.3955178*x10*x3*x13-540.5789558*x10*x3*x14-145.3955178*x2*x13+540.5789558*x2*x14+979.8834495*x10*x3-66.47161240*x5*x13+336.6449704*x5*x14+424.6686895*x10*x6+293.2445208*x10*x7+174.0745657*x10*x13-173.9170144*x10*x8+723.3213999*x10*x14+66.47161240*x12*x7-336.6449704*x12*x8-215.4121261*x7*x14+215.4121261*x13*x8+66.47161240*x10*x6*x13-336.6449704*x10*x6*x14-205.2394021*x10*x7*x14+205.2394021*x10*x13*x8);

D4 = 1e-3*(-270.0127009*x1-50.93056200*x2-245.0065939*x4+20.95640680*x5-138.0120666*x7+394.6356430*x8+62.39726970*x9+121.8574988*x10-94.00235930*x11+1.230578800*x12-88.69245970*x13+175.1201258*x14+211.5611808*x10*x3*x7*x14-211.5611808*x10*x3*x13*x8+37.20944889*x9*x5*x3*x14-259.3591997*x9*x3*x7*x14+259.3591997*x9*x3*x13*x8+130.5862793*x9*x2*x6*x13-37.20944889*x9*x2*x6*x14+304.9880256*x9*x2*x7*x14-304.9880256*x9*x2*x13*x8+130.5862793*x9*x12*x3*x7-37.20944889*x9*x12*x3*x8-130.5862793*x9*x5*x3*x13-130.5862793*x11*x10*x3*x7+37.20944889*x11*x10*x3*x8+130.5862793*x4*x10*x3*x13-37.20944889*x4*x10*x3*x14-130.5862793*x1*x10*x6*x13+37.20944889*x1*x10*x6*x14-304.9880256*x1*x10*x7*x14+304.9880256*x1*x10*x13*x8-277.3618983*x1*x5-201.7809294*x1*x7+608.3680265*x1*x8+277.3618983*x4*x2+173.4969305*x2*x7-498.7869793*x2*x8+65.39045070-25.51522092*x11*x5-176.3510880*x11*x10-133.9152376*x11*x7+88.83413702*x11*x8-476.5759056*x4*x10+25.51522092*x4*x12+133.9152376*x4*x13-88.83413702*x4*x14-454.5792844*x1*x10-78.29682790*x1*x12+70.16517890*x1*x13+135.5134323*x1*x14+454.5792844*x9*x2+270.0127009*x9*x3+476.5759056*x9*x5+245.0065939*x9*x6+359.8113251*x9*x7-1621.279086*x9*x8+176.3510880*x9*x12+103.9448285*x9*x13-599.4167306*x9*x14+78.29682790*x11*x2-277.3618983*x9*x2*x6-14.83234928*x9*x2*x7+652.0130723*x9*x2*x8+277.3618983*x9*x5*x3+201.7809294*x9*x3*x7-608.3680265*x9*x3*x8+78.29682790*x9*x12*x3-25.51522092*x9*x12*x6-218.2100128*x9*x2*x13+245.0415876*x9*x2*x14-70.16517890*x9*x3*x13-135.5134323*x9*x3*x14-223.0148555*x9*x5*x13+3.954649820*x9*x5*x14+223.0148555*x9*x12*x7-3.954649820*x9*x12*x8-133.9152376*x9*x6*x13+88.83413702*x9*x6*x14-755.6957645*x9*x7*x14+755.6957645*x9*x13*x8+130.5862793*x11*x2*x7-37.20944889*x11*x2*x8-78.29682790*x11*x10*x3+25.51522092*x11*x10*x6-223.0148555*x11*x10*x7+3.954649820*x11*x10*x8-130.5862793*x4*x2*x13+37.20944889*x4*x2*x14-277.3618983*x4*x10*x3+223.0148555*x4*x10*x13-3.954649820*x4*x10*x14+130.5862793*x1*x5*x13-37.20944889*x1*x5*x14+277.3618983*x1*x10*x6+14.83234928*x1*x10*x7+218.2100128*x1*x10*x13-652.0130723*x1*x10*x8-245.0415876*x1*x10*x14-130.5862793*x1*x12*x7+37.20944889*x1*x12*x8+259.3591997*x1*x7*x14-259.3591997*x1*x13*x8+498.7869793*x10*x3*x8-173.4969305*x10*x3*x7-211.5611808*x2*x7*x14+211.5611808*x2*x13*x8-72.95555760*x10*x3*x13+147.6357663*x10*x3*x14+72.95555760*x2*x13-147.6357663*x2*x14+50.93056200*x10*x3-25.82702160*x5*x13+48.69607684*x5*x14-20.95640680*x10*x6-299.2310399*x10*x7-167.7503021*x10*x13+906.3022315*x10*x8+336.3772410*x10*x14+25.82702160*x12*x7-48.69607684*x12*x8+165.6589824*x7*x14-165.6589824*x13*x8+25.82702160*x10*x6*x13-48.69607684*x10*x6*x14+421.6233027*x10*x7*x14-421.6233027*x10*x13*x8);

D5 = 1e-4*(-2025.686068*x1-4478.485629*x2+3017.415973*x4+4575.114733*x5+658.4099976*x7-1957.472958*x8+1463.243439*x9+3628.474777*x10+1663.576510*x11-4804.868447*x12+2122.731725*x13+3731.789986*x14+693.6572486*x10*x3*x7*x14-693.6572486*x10*x3*x13*x8+2933.177620*x9*x5*x3*x14+3842.547654*x9*x3*x7*x14-3842.547654*x9*x3*x13*x8-6925.158480*x9*x2*x6*x13-2933.177620*x9*x2*x6*x14-1210.057453*x9*x2*x7*x14+1210.057453*x9*x2*x13*x8-6925.158480*x9*x12*x3*x7-2933.177620*x9*x12*x3*x8+6925.158480*x9*x5*x3*x13+6925.158480*x11*x10*x3*x7+2933.177620*x11*x10*x3*x8-6925.158480*x4*x10*x3*x13-2933.177620*x4*x10*x3*x14+6925.158480*x1*x10*x6*x13+2933.177620*x1*x10*x6*x14+1210.057453*x1*x10*x7*x14-1210.057453*x1*x10*x13*x8-1031.007566*x1*x5+4139.650786*x1*x7+2325.438010*x1*x8+1031.007566*x4*x2+6501.656770*x2*x7+2857.072521*x2*x8-4118.018172-5368.927305*x11*x5-3173.113156*x11*x10+1289.418596*x11*x7+3525.183266*x11*x8-1033.048172*x4*x10+5368.927305*x4*x12-1289.418596*x4*x13-3525.183266*x4*x14+193.5484670*x1*x10-4172.749573*x1*x12+3147.919670*x1*x13+3648.638032*x1*x14-193.5484670*x9*x2+2025.686068*x9*x3+1033.048172*x9*x5-3017.415973*x9*x6-692.3101123*x9*x7-339.1920460*x9*x8+3173.113156*x9*x12-1006.968933*x9*x13-1896.481378*x9*x14+4172.749573*x11*x2-1031.007566*x9*x2*x6+1180.706187*x9*x2*x7+680.2442690*x9*x2*x8+1031.007566*x9*x5*x3-4139.650786*x9*x3*x7-2325.438010*x9*x3*x8+4172.749573*x9*x12*x3-5368.927305*x9*x12*x6+3478.575287*x9*x2*x13+2202.483300*x9*x2*x14-3147.919670*x9*x3*x13-3648.638032*x9*x3*x14+790.3885870*x9*x5*x13-603.3595331*x9*x5*x14-790.3885870*x9*x12*x7+603.3595331*x9*x12*x8+1289.418596*x9*x6*x13+3525.183266*x9*x6*x14+663.8658744*x9*x7*x14-663.8658744*x9*x13*x8-6925.158480*x11*x2*x7-2933.177620*x11*x2*x8-4172.749573*x11*x10*x3+5368.927305*x11*x10*x6+790.3885870*x11*x10*x7-603.3595331*x11*x10*x8+6925.158480*x4*x2*x13+2933.177620*x4*x2*x14-1031.007566*x4*x10*x3-790.3885870*x4*x10*x13+603.3595331*x4*x10*x14-6925.158480*x1*x5*x13-2933.177620*x1*x5*x14+1031.007566*x1*x10*x6-1180.706187*x1*x10*x7-3478.575287*x1*x10*x13-680.2442690*x1*x10*x8-2202.483300*x1*x10*x14+6925.158480*x1*x12*x7+2933.177620*x1*x12*x8-3842.547654*x1*x7*x14+3842.547654*x1*x13*x8-2857.072521*x10*x3*x8-6501.656770*x10*x3*x7-693.6572486*x2*x7*x14+693.6572486*x2*x13*x8+3767.612706*x10*x3*x13+1177.824206*x10*x3*x14-3767.612706*x2*x13-1177.824206*x2*x14+4478.485629*x10*x3-3126.580329*x5*x13-1862.052806*x5*x14-4575.114733*x10*x6-1275.119997*x10*x7-1140.503426*x10*x13+155.8689096*x10*x8-1640.470609*x10*x14+3126.580329*x12*x7+1862.052806*x12*x8-1605.684461*x7*x14+1605.684461*x13*x8+3126.580329*x10*x6*x13+1862.052806*x10*x6*x14+625.4875776*x10*x7*x14-625.4875776*x10*x13*x8);

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.
[128 approximate solutions to the sample above, formatted for rational arithmetic in alphaCertified.]
128

-393634392468687/1000000000000000 -95850029906327/156250000000000
1023884783593603/1250000000000000 -1937466220659681/25000000000000000
2568434442768999/10000000000000000 222252310939491/125000000000000
-1331068427861371/500000000000000 7859065085242583/10000000000000000
1520672729432639/500000000000000 -98803021034837/50000000000000
692434949040697/200000000000000 3191606799704911/5000000000000000
576451971134689/500000000000000 3298799283226073/10000000000000000
-71447627858787/50000000000000 -495701619357807/2500000000000000
-6179040695256307/10000000000000000 1432805291775863/10000000000000000
316469897685307/625000000000000 -3488236678762511/10000000000000000
-4924159903895897/50000000000000000 4946907447170657/5000000000000000
-8365240803465013/100000000000000000 -508306087806681/200000000000000
1935147107932693/5000000000000000 2512337824758953/1000000000000000
1908297431774537/1000000000000000 -74594328342251/62500000000000
3770325669093283/10000000000000000 5109405664953677/10000000000000000
-284947244031321/250000000000000 -559335698398011/500000000000000
-890665004496891/500000000000000 -537482972678097/50000000000000
-902909308516383/500000000000000 -4563448661378741/1000000000000000
-853177080450927/500000000000000 3344412902644499/10000000000000000
-8180339838529067/10000000000000000 -8838131383243097/100000000000000000
-338064765852217/1000000000000000 43587981898081/80000000000000
1104744799184179/100000000000000 753276765776573/50000000000000
802869353309901/125000000000000 2985975329770043/500000000000000
-539335040028069/1000000000000000 68502175324979/62500000000000
-9197790694342639/100000000000000000 2020446407037391/10000000000000000
1282253247128317/500000000000000 -3434482817030833/10000000000000000
754457227976951/50000000000000 1264943720279013/1000000000000000
783456098035809/100000000000000 -2946535446727483/5000000000000000
125970629874079/312500000000000 -50585468554519/3125000000000000
3217585027374277/10000000000000000 6083421913611177/100000000000000000
5203524816211601/10000000000000000 2268452937455139/10000000000000000
-2394653604738211/5000000000000000 3256342076464881/10000000000000000
-1302682206888707/20000000000000000 592025479493113/12500000000000000
2216449927350311/10000000000000000 -163195642808613/125000000000000
64425324025737/8000000000000 -8044380996086137/1000000000000000
200642395186249/20000000000000 2115189360529359/500000000000000
-706874409321699/500000000000000 -140545462882499/1000000000000000
3359069840862787/1000000000000000 262548235736691/1250000000000000
1234383827000391/1000000000000000 -78571877635323/62500000000000
1008664657543981/5000000000000000 -169103522509229/400000000000000
-6039063670926283/10000000000000000 3551299711840453/100000000000000000

172146744041239090937/1000000000000000000000 135112345172250358021/500000000000000000000
116676012795719545373/100000000000000000000 -124066786772238878047/200000000000000000000
187918390805692144083/100000000000000000000 -101787570935580615677/125000000000000000000
-138056221146372001471/100000000000000000000 116546952379215951087/250000000000000000000
917931545026064569549/100000000000000000000 20888671743702933779/4000000000000000000
93880905379995940223/5000000000000000000 975548113881263084801/1000000000000000000000
-33038558637179222927/20000000000000000000 965407045560738986717/1000000000000000000000
74085929744829925089/25000000000000000000 -309767817788206572789/100000000000000000000
-826694838188874256079/10000000000000000000000 28445719572646351023/100000000000000000000
57200206622158834797/100000000000000000000 307349422425515890207/1000000000000000000000
163203702685080648759/50000000000000000000 -438669539950406577869/100000000000000000000
63870059089154574447/25000000000000000000 40964568651278378199/10000000000000000000
993508616969858949879/1000000000000000000000 -110533898844174979861/10000000000000000000
-543301514562802598729/1000000000000000000000 768583110218303437039/100000000000000000000
9953206333091485733/6250000000000000000 31060275509218570857/125000000000000000000
-377436545985520396061/500000000000000000000 312950441474939182457/500000000000000000000
-61303626553355844653/50000000000000000000 -83339905757486782079/2500000000000000000000
-77065501730885784513/50000000000000000000 -88033651433736840263/1000000000000000000000
108355797241087093233/100000000000000000000 691190508648474058689/1000000000000000000000
2885956686331549937/12500000000000000000 262458922748259644987/500000000000000000000
-303425758193208785487/100000000000000000000 -52165977193088525297/500000000000000000000
-7292871667023498893/31250000000000000000 77135153000185667239/250000000000000000000
-792321477605062012407/1000000000000000000000 -9454285632664982491/156250000000000000000
550439766174361933253/100000000000000000000 28946676113486431893/25000000000000000000
52088700365390609877/25000000000000000000 3195568873355243179/3125000000000000000
-158332468928796692479/50000000000000000000 -581392141108774811631/1000000000000000000000
-56203721239057597217/25000000000000000000 36963553274782288397/62500000000000000000
-479394973230427627869/100000000000000000000 -80003136768954537933/500000000000000000000
15870957411995736089/20000000000000000000 16396422143339226259/312500000000000000000
156430878627853386413/2000000000000000000000 -182750089270803700551/5000000000000000000000
354369991327787066843/1000000000000000000000 226148538399301659427/1000000000000000000000
-644314723202740245007/1000000000000000000000 420197657187751932189/5000000000000000000000
-104383123955377236881/200000000000000000000 -757030470526595422143/1000000000000000000000
71527055182683831583/20000000000000000000 240851196810812392143/1000000000000000000000
55624697332995430701/25000000000000000000 162027766601791876899/100000000000000000000
372651838291160782883/500000000000000000000 63393871635264661137/20000000000000000000
-85120359538529259799/2000000000000000000000 -25778513218986338703/50000000000000000000
36676682907328270227/200000000000000000000 44347071681862219373/25000000000000000000
-41794079307613534599/25000000000000000000 -477045042207032024377/100000000000000000000
-501553775664642421169/100000000000000000000 29304479413905808237/25000000000000000000
-75743550142947392177/50000000000000000000 -218549621615021766709/250000000000000000000

182600061045740464131/500000000000000000000 -205283793335782431601/5000000000000000000000
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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 173m15.932s
user 1176m39.864s
sys 2m45.176s

3.6 GHz processors used. Total processing power:

~70.76504 GHz-hours


[This is how long it took to certify.]
time output (linux)

real 29m48.417s
user 29m46.804s
sys 0m0.248s

3.6 GHz processors used. Total processing power:

~1.740000777 GHz-hours