Gröbner basis on a particular set of equations

Question

This question is very similar in gist to equation solving with GroebnerBasis, but hopefully when I say that I make the system a little larger I mean little. I have uploaded the code here for those who are trusting enough to download it. There is a lot of code, and my question of interest is related to the particular system of equations in my uploaded notebook, but I will try to give the general gist here.

I have a system of 20 polynomials and 13 variables. If I make one of these variables 0, the GroebnerBasis computes instantly. If I do not, the GroebnerBasis process maxes out 8GB of RAM after a day or two. I'm interested in knowing what characteristic of the more general set of polynomials is causing the GroebnerBasis process to take so long.

As you will see if you download the notebook, the system of equations is generated through an iterative process, and I can generate as many polynomials as desired (although they do get more and more complicated). If you have downloaded the notebook, you can generate more polynomials by adjusting the iterate[22222] command to, say, iterate[222222]. A solution is guaranteed, due to Hilbert's Nullstellensatz (and more specifically in this case because I am able to explicitly show such a solution when I set a particular variable equal to zero).

The question I initially referenced uses a few commands (CoefficientArrays in particular), and I am curious if the output of those commands would be at all useful in my problem, though I don't know what to look for (and that is the more general gist of my question). I guess what I'm hoping is that someone can tell me, "Ah, yes, when that variable is not zero your system is .... < insert property here >, so a Gröbner basis will be very difficult to compute".

---

By request, I added my code inline here:

deg = 3;
d = Table[
   x^i*y^j*z^k, {i, 0, deg}, {j, 0, deg - i}, {k, 0, deg - i - j}];
d = Flatten[d];
v = Table[a[i], {i, 1, Length[d]}];
f = d.v;
cf = CoefficientList[f, {x, y, z}];
Evaluate[cf[[1]][[1]][[1]]] = 0;
Evaluate[cf[[2]][[1]][[1]]] = 0;
Evaluate[cf[[1]][[2]][[1]]] = 0;
Evaluate[cf[[1]][[1]][[2]]] = 1;
Evaluate[cf[[2]][[2]][[1]]] = 0;

cf[[3]][[1]][[1]]
cf[[1]][[3]][[1]]

(* a[17] and a[8] are dependant on a principal curvature assumption. *)

fx = D[f, x];
fy = D[f, y];
fz = D[f, z];
Nf = fx^2 + fy^2 + fz^2;
Lf = D[fx, x] + D[fy, y] + D[fz, z];
NNfNf = fx*D[Nf, x] + fy*D[Nf, y] + fz*D[Nf, z];
g = 2*Nf*Lf - NNfNf - 4*(Nf)^(3/2);
G[0] = g;
Q[0] = Block[{x = 0, y = 0, z = 0}, Simplify[G[0]]]

(* We want Q[0] to be zero, so at the very least we can make the \
following substitution: *)
a[17] = 1 - a[8];

cross1[h_] := D[h, y]*fz - D[h, z]*fy
cross2[h_] := D[h, z]*fx - D[h, x]*fz

nextIndex[index_] := (in = index;
  rem = Mod[in, 10];
  If[rem != 2, Return[in + 1],
   retindex = nextIndex[(in - rem)/10];
   Return[retindex*10 + Mod[retindex, 10]]
   ];
  )

stats[letter_, index_] := 
  Print[letter <> "[" <> 
    ToString[
     NumberForm[index, DigitBlock -> 3, NumberSeparator -> " "]] <> 
    "] (" <> ToString[TimeUsed[]] <> " / " <> 
    ToString[
     NumberForm[MemoryInUse[], DigitBlock -> 3, 
      NumberSeparator -> " "]] <> " / " <> 
    ToString[
     NumberForm[MaxMemoryUsed[], DigitBlock -> 3, 
      NumberSeparator -> " "]] <> ")"];

Gdone = {};
Qdone = {};

iterate[num_] := (i = 0;
  While[i < num,
   i = nextIndex[i];
   stats["G", i];
   If [Mod[i, 10] == 1, G[i] = cross1[G[(i - Mod[i, 10])/10]], 
    G[i] = cross2[G[(i - Mod[i, 10])/10]]];
   AppendTo[Gdone, i];
   stats["Q", i];
   Q[i] = Block[{x = 0, y = 0, z = 0}, Simplify[G[i]]];
   AppendTo[Qdone, i];
   ];
  )

iterate[22222]

q = Table[Q[Qdone[[i]]], {i, Length[Qdone]}];

r = q /. {a[8] -> 0};

GroebnerBasis[r, Variables[r]]

GroebnerBasis[q, Variables[q]]

It was designed to be taken in steps, so running it all in one cell may not be very enlightening, but you have the notebook file above if you want it spaced out as well.

Answer 1

Your question cannot realistically be answered. One almost never knows what specifically comprises such an impediment.

Here is a Groebner basis for your system of polynomials, computed for degree reverse lexicographic order. It takes some time to do this. Not sure if it will run in reasonable time directly; I used a numeric approximation and rationalized (have not validated the result but I'm fairly sure it is correct).

gb2 = {a[15], (-3*a[12])/2 + a[8]*a[12] + a[16]/2 + (3*a[20])/2, a[6]/2 + a[6]*a[8] - (3*a[10])/2 - a[19]/2, 
 (-17*a[6])/18 - (35*a[7])/9 + (23*a[10])/6 + a[8]*a[10] + (29*a[19])/6 - (61*a[8]*a[19])/9, 
 (17*a[12])/18 + (35*a[13])/9 + (35*a[16])/18 - (61*a[8]*a[16])/9 - (29*a[20])/6 + a[8]*a[20], 
 -a[12]/3 + (5*a[13])/6 + a[8]*a[13] + (7*a[16])/6 - (8*a[8]*a[16])/3 - a[20]/2, 
 a[6]/3 - (11*a[7])/6 + a[7]*a[8] + a[10]/2 + (3*a[19])/2 - (8*a[8]*a[19])/3, -(a[6]*a[12]) + a[10]*a[12] + a[12]*a[19], 
 (52*a[12])/21 - a[3]*a[12] + a[9]*a[12] + (125*a[13])/21 + (52*a[16])/21 - (208*a[8]*a[16])/21 + a[12]*a[18] - (52*a[20])/7, 
 -(a[6]*a[12]) + a[6]*a[16] + a[12]*a[19], -(a[12]*a[19]) + a[6]*a[20], (52*a[6])/21 - a[3]*a[6] + (125*a[7])/21 + a[6]*a[9] - 
  (52*a[10])/7 + a[6]*a[18] - (52*a[19])/7 + (208*a[8]*a[19])/21, -(a[7]*a[12]) + a[6]*a[13], 
 (24*a[6])/7 - a[3]*a[6] + (115*a[7])/28 + a[6]*a[9] - (183*a[10])/28 - (211*a[19])/28 + a[3]*a[19] + (101*a[8]*a[19])/14, 
 (-202*a[6])/63 - (1805*a[7])/252 - a[6]*a[9] + (787*a[10])/84 + a[3]*a[10] + (871*a[19])/84 - (1679*a[8]*a[19])/126, 
 (20*a[12])/21 - a[9]*a[12] - (155*a[13])/84 - (235*a[16])/84 + a[3]*a[16] + (113*a[8]*a[16])/42 + (25*a[20])/28, 
 (-46*a[12])/63 - a[3]*a[12] + a[9]*a[12] - (305*a[13])/252 - (121*a[16])/252 + (431*a[8]*a[16])/126 + (163*a[20])/84 + 
  a[3]*a[20], (-47*a[12])/42 - a[4]*a[12] - (199*a[13])/42 + a[3]*a[13] - (34*a[16])/21 + (136*a[8]*a[16])/21 + (34*a[20])/7, 
 (-47*a[6])/42 - a[4]*a[6] - (199*a[7])/42 + a[3]*a[7] + (34*a[10])/7 + (34*a[19])/7 - (136*a[8]*a[19])/21, 
 (22*a[6]^2)/105 + (a[6]*a[7])/3 - (19*a[6]*a[10])/35 - (92*a[6]*a[19])/105 + a[10]*a[19] + a[19]^2/3, 
 (a[6]*a[12])/3 - (a[7]*a[12])/3 - a[12]*a[19] + (a[16]*a[19])/3 + a[19]*a[20], 
 (121*a[6])/84 - (a[3]*a[6])/2 + (a[4]*a[6])/2 + (253*a[7])/168 - (137*a[10])/56 - (137*a[19])/56 + (169*a[8]*a[19])/84 + 
  a[9]*a[19], (377*a[6])/84 - (a[3]*a[6])/2 - (a[4]*a[6])/2 + (1277*a[7])/168 + a[6]*a[9] - (649*a[10])/56 - (649*a[19])/56 + 
  (1193*a[8]*a[19])/84 + a[18]*a[19], (a[6]*a[12])/2 - (a[7]*a[12])/2 - a[12]*a[19] + a[13]*a[19], 
 (23*a[6]^2)/70 - (a[6]*a[7])/2 + (6*a[6]*a[10])/35 - (29*a[6]*a[19])/35 + a[7]*a[19], 
 (26627*a[6])/8064 - (a[3]*a[6])/2 - (a[4]*a[6])/2 + (73307*a[7])/16128 + a[6]*a[9] - (39499*a[10])/5376 - 
  (39499*a[19])/5376 + a[4]*a[19] + (65243*a[8]*a[19])/8064, (227*a[6]^2)/315 - (4*a[6]*a[7])/9 - (134*a[6]*a[10])/105 + 
  a[10]^2 - (52*a[6]*a[19])/315 - a[19]^2/9, (-2*a[6]*a[12])/3 - (a[7]*a[12])/3 + a[10]*a[16] + a[12]*a[19] + (a[16]*a[19])/3, 
 (-4*a[6]*a[12])/9 + (4*a[7]*a[12])/9 - (a[16]*a[19])/9 + a[10]*a[20], (-713*a[6])/252 + (a[3]*a[6])/2 - (a[4]*a[6])/2 - 
  (2789*a[7])/504 - a[6]*a[9] + (1321*a[10])/168 + a[9]*a[10] + (1321*a[19])/168 - (2537*a[8]*a[19])/252, 
 (263*a[6])/252 - (a[3]*a[6])/2 + (a[4]*a[6])/2 + (1619*a[7])/504 - (631*a[10])/168 + a[10]*a[18] - (631*a[19])/168 + 
  (1367*a[8]*a[19])/252, -(a[6]*a[12])/2 - (a[7]*a[12])/2 + a[10]*a[13] + a[12]*a[19], 
 (47*a[6]^2)/70 - (a[6]*a[7])/2 - (41*a[6]*a[10])/35 + a[7]*a[10] - (6*a[6]*a[19])/35, 
 (-86531*a[6])/24192 + (a[3]*a[6])/2 - (a[4]*a[6])/2 - (329051*a[7])/48384 - a[6]*a[9] + (159307*a[10])/16128 + a[4]*a[10] + 
  (159307*a[19])/16128 - (304859*a[8]*a[19])/24192, (22*a[12]^2)/105 + (a[12]*a[13])/3 - (92*a[12]*a[16])/105 + a[16]^2/3 - 
  (19*a[12]*a[20])/35 + a[16]*a[20], (169*a[12])/84 + (a[3]*a[12])/2 - (a[4]*a[12])/2 - a[9]*a[12] + (277*a[13])/168 + 
  (23*a[16])/168 - (361*a[8]*a[16])/84 + a[9]*a[16] - (233*a[20])/56, (121*a[12])/84 - (a[3]*a[12])/2 + (a[4]*a[12])/2 + 
  (253*a[13])/168 - (73*a[16])/168 - (169*a[8]*a[16])/84 + a[16]*a[18] - (137*a[20])/56, 
 (23*a[12]^2)/70 - (a[12]*a[13])/2 - (29*a[12]*a[16])/35 + a[13]*a[16] + (6*a[12]*a[20])/35, 
 -(a[6]*a[12])/2 - (a[7]*a[12])/2 + a[7]*a[16] + a[12]*a[19], (6659*a[12])/8064 + (a[3]*a[12])/2 - (a[4]*a[12])/2 - 
  a[9]*a[12] - (22693*a[13])/16128 - (27947*a[16])/16128 + a[4]*a[16] + (14629*a[8]*a[16])/8064 + (437*a[20])/5376, 
 (227*a[12]^2)/315 - (4*a[12]*a[13])/9 - (52*a[12]*a[16])/315 - a[16]^2/9 - (134*a[12]*a[20])/105 + a[20]^2, 
 (263*a[12])/252 - (a[3]*a[12])/2 + (a[4]*a[12])/2 + (1619*a[13])/504 + (841*a[16])/504 - (1367*a[8]*a[16])/252 - 
  (631*a[20])/168 + a[9]*a[20], (-89*a[12])/252 - (a[3]*a[12])/2 - (a[4]*a[12])/2 + a[9]*a[12] + (211*a[13])/504 + 
  (137*a[16])/504 + (41*a[8]*a[16])/252 + (73*a[20])/168 + a[18]*a[20], 
 (47*a[12]^2)/70 - (a[12]*a[13])/2 - (6*a[12]*a[16])/35 - (41*a[12]*a[20])/35 + a[13]*a[20], 
 (a[6]*a[12])/2 - (a[7]*a[12])/2 - a[12]*a[19] + a[7]*a[20], (-26627*a[12])/24192 - (a[3]*a[12])/2 - (a[4]*a[12])/2 + 
  a[9]*a[12] - (41051*a[13])/48384 - (11989*a[16])/48384 + (65243*a[8]*a[16])/24192 + (39499*a[20])/16128 + a[4]*a[20], 
 (2*a[12])/7 + (a[3]*a[12])/2 - (a[4]*a[12])/2 - a[9]*a[12] + a[13]/14 + a[9]*a[13] + (2*a[16])/7 - (8*a[8]*a[16])/7 - 
  (6*a[20])/7, (-26*a[6])/21 + (a[3]*a[6])/2 - (a[4]*a[6])/2 - (125*a[7])/42 - a[6]*a[9] + a[7]*a[9] + (26*a[10])/7 + 
  (26*a[19])/7 - (104*a[8]*a[19])/21, 1733/1602 - (641*a[3])/534 + (95*a[3]^2)/801 + (3775*a[4])/801 - (440*a[3]*a[4])/267 + 
  (94481*a[6]^2)/100926 - (7325*a[6]*a[7])/14418 - (182*a[8])/89 + (182*a[8]^2)/89 + (67*a[9])/178 + (424*a[3]*a[9])/801 - 
  (803*a[8]*a[9])/267 + a[9]^2 - (7201*a[6]*a[10])/16821 + (94481*a[12]^2)/100926 - (7325*a[12]*a[13])/14418 - 
  (72878*a[12]*a[16])/50463 + (7325*a[16]^2)/7209 - (1405*a[18])/534 + (424*a[3]*a[18])/801 + (803*a[8]*a[18])/267 - 
  (346*a[9]*a[18])/267 + a[18]^2 - (72878*a[6]*a[19])/50463 + (7325*a[19]^2)/7209 - (7201*a[12]*a[20])/16821, 
 (26*a[12])/21 - (a[3]*a[12])/2 - (a[4]*a[12])/2 + a[9]*a[12] + (125*a[13])/42 + (26*a[16])/21 - (104*a[8]*a[16])/21 + 
  a[13]*a[18] - (26*a[20])/7, (58*a[6])/21 - (a[3]*a[6])/2 - (a[4]*a[6])/2 + (253*a[7])/42 + a[6]*a[9] - (58*a[10])/7 + 
  a[7]*a[18] - (58*a[19])/7 + (232*a[8]*a[19])/21, -5455/12816 + (1931*a[3])/4272 - (169*a[3]^2)/6408 + (1627*a[4])/1602 - 
  (572*a[3]*a[4])/267 - (39349*a[6]^2)/807408 + (23765*a[6]*a[7])/57672 + (881*a[8])/1068 - (881*a[8]^2)/1068 - 
  (2169*a[9])/1424 + (7489*a[3]*a[9])/6408 + a[4]*a[9] + (2311*a[8]*a[9])/2136 - (97787*a[6]*a[10])/269136 - 
  (39349*a[12]^2)/807408 + (23765*a[12]*a[13])/57672 + (372059*a[12]*a[16])/807408 - (23765*a[16]^2)/28836 - 
  (1885*a[18])/4272 + (7489*a[3]*a[18])/6408 + a[4]*a[18] - (2311*a[8]*a[18])/2136 - (610*a[9]*a[18])/267 + 
  (372059*a[6]*a[19])/807408 - (23765*a[19]^2)/28836 - (97787*a[12]*a[20])/269136, 
 (47*a[12]^2)/70 - (3*a[12]*a[13])/2 + a[13]^2 - (6*a[12]*a[16])/35 - (6*a[12]*a[20])/35, 
 (a[6]*a[12])/2 - (3*a[7]*a[12])/2 + a[7]*a[13], (-26*a[12])/21 + (a[3]*a[12])/2 - (3*a[4]*a[12])/2 - (125*a[13])/42 + 
  a[4]*a[13] - (26*a[16])/21 + (104*a[8]*a[16])/21 + (26*a[20])/7, (47*a[6]^2)/70 - (3*a[6]*a[7])/2 + a[7]^2 - 
  (6*a[6]*a[10])/35 - (6*a[6]*a[19])/35, (-26*a[6])/21 + (a[3]*a[6])/2 - (3*a[4]*a[6])/2 - (125*a[7])/42 + a[4]*a[7] + 
  (26*a[10])/7 + (26*a[19])/7 - (104*a[8]*a[19])/21, -63037/230688 + (24473*a[3])/76896 - (5191*a[3]^2)/115344 + 
  (77081*a[4])/57672 - (10073*a[3]*a[4])/4806 + a[4]^2 - (20304931*a[6]^2)/581333760 + (7875295*a[6]*a[7])/33219072 + 
  (9641*a[8])/19224 - (9641*a[8]^2)/19224 - (3891*a[9])/2848 + (131599*a[3]*a[9])/115344 + (29083*a[8]*a[9])/38448 - 
  (78341821*a[6]*a[10])/387555840 - (20304931*a[12]^2)/581333760 + (7875295*a[12]*a[13])/33219072 + 
  (316245187*a[12]*a[16])/1162667520 - (7875295*a[16]^2)/16609536 - (46891*a[18])/76896 + (131599*a[3]*a[18])/115344 - 
  (29083*a[8]*a[18])/38448 - (2864*a[9]*a[18])/2403 + (316245187*a[6]*a[19])/1162667520 - (7875295*a[19]^2)/16609536 - 
  (78341821*a[12]*a[20])/387555840, -3/4 + (3*a[3])/4 + (3*a[6]^2)/8 + 2*a[8] - (3*a[3]*a[8])/2 - (3*a[8]^2)/2 + a[8]^3 + 
  (3*a[8]*a[9])/4 - (3*a[6]*a[10])/8 - (3*a[12]^2)/8 + (3*a[12]*a[16])/8 - (3*a[18])/4 + (3*a[8]*a[18])/4 - (3*a[6]*a[19])/8 + 
  (3*a[12]*a[20])/8, -3/4 + (3*a[3])/4 + 2*a[8] - a[3]*a[8] - 2*a[8]^2 + a[3]*a[8]^2 - a[9]/4 - (a[8]*a[9])/2 - (3*a[18])/4 + 
  (a[8]*a[18])/2, (-5*a[6])/4 - (5*a[7])/2 + (15*a[10])/4 + (15*a[19])/4 - (11*a[8]*a[19])/2 + a[8]^2*a[19], 
 (-5*a[12])/4 - (5*a[13])/2 - (3*a[16])/4 + (7*a[8]*a[16])/2 + a[8]^2*a[16] + (15*a[20])/4, 
 11501/4272 - (4217*a[3])/1424 + (575*a[3]^2)/2136 + (122405*a[4])/1068 - (19585*a[3]*a[4])/89 - (1007425*a[6]^2)/38448 + 
  (1106575*a[6]*a[7])/38448 - (1821*a[8])/356 + (93*a[3]*a[8])/2 + (435*a[4]*a[8])/4 - (14733*a[8]^2)/356 - 
  (49223*a[9])/1424 + (12895*a[3]*a[9])/2136 + 180*a[4]*a[9] - (14135*a[8]*a[9])/712 + a[8]^2*a[9] + (135*a[9]^2)/4 - 
  (16525*a[6]*a[10])/6408 + (12989045*a[12]^2)/269136 + (1106575*a[12]*a[13])/38448 - (1310755*a[12]*a[16])/67284 - 
  (1106575*a[16]^2)/19224 - (159373*a[18])/1424 + (469465*a[3]*a[18])/2136 - (80917*a[8]*a[18])/712 - (80585*a[9]*a[18])/356 + 
  (132125*a[6]*a[19])/2403 - (1106575*a[19]^2)/19224 - (3455845*a[12]*a[20])/44856, 
 -15817/4272 + (3829*a[3])/1424 + (2165*a[3]^2)/2136 - (126565*a[4])/1068 + (19685*a[3]*a[4])/89 + (6846695*a[6]^2)/269136 - 
  (1085975*a[6]*a[7])/38448 + (2997*a[8])/356 - (93*a[3]*a[8])/2 - (435*a[4]*a[8])/4 + (13557*a[8]^2)/356 + 
  (51123*a[9])/1424 - (18035*a[3]*a[9])/2136 - 180*a[4]*a[9] + (14527*a[8]*a[9])/712 - (135*a[9]^2)/4 + 
  (125855*a[6]*a[10])/44856 - (13194325*a[12]^2)/269136 - (1085975*a[12]*a[13])/38448 + (1398125*a[12]*a[16])/67284 + 
  (1085975*a[16]^2)/19224 + (163481*a[18])/1424 - (474605*a[3]*a[18])/2136 + (79101*a[8]*a[18])/712 + a[8]^2*a[18] + 
  (81385*a[9]*a[18])/356 - (1806065*a[6]*a[19])/33642 + (1085975*a[19]^2)/19224 + (3466025*a[12]*a[20])/44856, 
 227/534 - (199*a[3])/178 + (185*a[3]^2)/267 - (35*a[4])/1068 + (80*a[3]*a[4])/89 - (665*a[6]^2)/9612 - 
  (3445*a[6]*a[7])/9612 - (14*a[8])/89 - a[4]*a[8] + (14*a[8]^2)/89 + a[4]*a[8]^2 + (647*a[9])/356 - (425*a[3]*a[9])/267 - 
  (121*a[8]*a[9])/89 + (685*a[6]*a[10])/1602 - (665*a[12]^2)/9612 - (3445*a[12]*a[13])/9612 - (695*a[12]*a[16])/2403 + 
  (3445*a[16]^2)/4806 + (163*a[18])/356 - (425*a[3]*a[18])/267 + (121*a[8]*a[18])/89 + (160*a[9]*a[18])/89 - 
  (695*a[6]*a[19])/2403 + (3445*a[19]^2)/4806 + (685*a[12]*a[20])/1602, 70657/4272 - (18737*a[3])/1424 - (7223*a[3]^2)/2136 + 
  (814915*a[4])/1068 - (130405*a[3]*a[4])/89 - (6698009*a[6]^2)/38448 + (7374425*a[6]*a[7])/38448 - (6401*a[8])/178 + 
  (1227*a[3]*a[8])/4 + a[3]^2*a[8] + (2885*a[4]*a[8])/4 - (96757*a[8]^2)/356 - (339915*a[9])/1424 + (105425*a[3]*a[9])/2136 + 
  1200*a[4]*a[9] - (84397*a[8]*a[9])/712 + (873*a[9]^2)/4 - (14092*a[6]*a[10])/801 + (12304915*a[12]^2)/38448 + 
  (7374425*a[12]*a[13])/38448 - (2465245*a[12]*a[16])/19224 - (7374425*a[16]^2)/19224 - (1063001*a[18])/1424 + 
  (3136943*a[3]*a[18])/2136 - (541451*a[8]*a[18])/712 - (538343*a[9]*a[18])/356 + (7036217*a[6]*a[19])/19224 - 
  (7374425*a[19]^2)/19224 - (1639945*a[12]*a[20])/3204, 22159/1424 - (18573*a[3])/1424 - (1793*a[3]^2)/712 + (61120*a[4])/89 - 
  (117345*a[3]*a[4])/89 - (14078993*a[6]^2)/89712 + (2211575*a[6]*a[7])/12816 - (2994*a[8])/89 + (1113*a[3]*a[8])/4 + 
  (2595*a[4]*a[8])/4 - (87081*a[8]^2)/356 - (305731*a[9])/1424 + (31507*a[3]*a[9])/712 + 1080*a[4]*a[9] - 
  (76705*a[8]*a[9])/712 + a[3]*a[8]*a[9] + (783*a[9]^2)/4 - (29209*a[6]*a[10])/1869 + (3685885*a[12]^2)/12816 + 
  (2211575*a[12]*a[13])/12816 - (737155*a[12]*a[16])/6408 - (2211575*a[16]^2)/6408 - (955761*a[18])/1424 + 
  (940553*a[3]*a[18])/712 - (487199*a[8]*a[18])/712 - (484237*a[9]*a[18])/356 + (14780009*a[6]*a[19])/44856 - 
  (2211575*a[19]^2)/6408 - (491455*a[12]*a[20])/1068, 21011/1424 - (17097*a[3])/1424 - (1957*a[3]^2)/712 + (121735*a[4])/178 - 
  (117285*a[3]*a[4])/89 - (14083381*a[6]^2)/89712 + (2217475*a[6]*a[7])/12816 - (2871*a[8])/89 + (1113*a[3]*a[8])/4 + 
  (2595*a[4]*a[8])/4 - (87573*a[8]^2)/356 - (307439*a[9])/1424 + (31903*a[3]*a[9])/712 + 1080*a[4]*a[9] - 
  (74405*a[8]*a[9])/712 + (783*a[9]^2)/4 - (29978*a[6]*a[10])/1869 + (25796807*a[12]^2)/89712 + (2217475*a[12]*a[13])/12816 - 
  (5137241*a[12]*a[16])/44856 - (2217475*a[16]^2)/6408 - (952869*a[18])/1424 + (940237*a[3]*a[18])/712 - 
  (489499*a[8]*a[18])/712 + a[3]*a[8]*a[18] - (484469*a[9]*a[18])/356 + (14802853*a[6]*a[19])/44856 - (2217475*a[19]^2)/6408 - 
  (3443261*a[12]*a[20])/7476, 416159/25632 - (116359*a[3])/8544 - (33541*a[3]^2)/12816 + (560560*a[4])/801 - 
  (717005*a[3]*a[4])/534 - (258204115*a[6]^2)/1614816 + (10125205*a[6]*a[7])/57672 - (37475*a[8])/1068 + (2269*a[3]*a[8])/8 + 
  (2641*a[4]*a[8])/4 + a[3]*a[4]*a[8] - (530873*a[8]^2)/2136 - (622243*a[9])/2848 + (577657*a[3]*a[9])/12816 + 
  1099*a[4]*a[9] - (468767*a[8]*a[9])/4272 + (801*a[9]^2)/4 - (8433875*a[6]*a[10])/538272 + (474432137*a[12]^2)/1614816 + 
  (10125205*a[12]*a[13])/57672 - (190926397*a[12]*a[16])/1614816 - (10125205*a[16]^2)/28836 - (5840587*a[18])/8544 + 
  (17241661*a[3]*a[18])/12816 - (2975533*a[8]*a[18])/4272 - (1479353*a[9]*a[18])/1068 + (541709855*a[6]*a[19])/1614816 - 
  (10125205*a[19]^2)/28836 - (252645959*a[12]*a[20])/538272, -a[6]^2/140 + (a[6]*a[7])/2 - (69*a[6]*a[10])/140 + 
  (a[6]*a[19])/140 - a[19]^2/2 + a[8]*a[19]^2, -(a[16]*a[19])/2 + a[8]*a[16]*a[19], 
 a[12]^2/140 - (a[12]*a[13])/2 - (a[12]*a[16])/140 - a[16]^2/2 + a[8]*a[16]^2 + (69*a[12]*a[20])/140, 
 1360511/68352 - (393735*a[3])/22784 - (89653*a[3]^2)/34176 + (14068805*a[4])/17088 - (2248755*a[3]*a[4])/1424 - 
  (809530465*a[6]^2)/4306176 + (127195375*a[6]*a[7])/615168 - (120847*a[8])/2848 + (21441*a[3]*a[8])/64 + 
  (49875*a[4]*a[8])/64 - (1666555*a[8]^2)/5696 - (5861477*a[9])/22784 + (1813747*a[3]*a[9])/34176 + 1290*a[4]*a[9] - 
  (1500619*a[8]*a[9])/11392 + (15215*a[9]^2)/64 + a[8]*a[9]^2 - (3368215*a[6]*a[10])/179424 + (1487977835*a[12]^2)/4306176 + 
  (127195375*a[12]*a[13])/615168 - (298805105*a[12]*a[16])/2153088 - (127195375*a[16]^2)/307584 - (18304903*a[18])/22784 + 
  (54059773*a[3]*a[18])/34176 - (9329613*a[8]*a[18])/11392 - (9279521*a[9]*a[18])/5696 + (849949045*a[6]*a[19])/2153088 - 
  (127195375*a[19]^2)/307584 - (198195455*a[12]*a[20])/358848, 774529/68352 - (216953*a[3])/22784 - (61835*a[3]^2)/34176 + 
  (8144155*a[4])/17088 - (1303885*a[3]*a[4])/1424 - (469418015*a[6]^2)/4306176 + (73766225*a[6]*a[7])/615168 - 
  (69697*a[8])/2848 + (12447*a[3]*a[8])/64 + (28845*a[4]*a[8])/64 - (968389*a[8]^2)/5696 - (3414171*a[9])/22784 + 
  (1066061*a[3]*a[9])/34176 + 750*a[4]*a[9] - (846901*a[8]*a[9])/11392 + (8721*a[9]^2)/64 - (1956065*a[6]*a[10])/179424 + 
  (861334549*a[12]^2)/4306176 + (73766225*a[12]*a[13])/615168 - (172485487*a[12]*a[16])/2153088 - (73766225*a[16]^2)/307584 - 
  (10600697*a[18])/22784 + (31355075*a[3]*a[18])/34176 - (5422259*a[8]*a[18])/11392 - (5385759*a[9]*a[18])/5696 + 
  a[8]*a[9]*a[18] + (492890795*a[6]*a[19])/2153088 - (73766225*a[19]^2)/307584 - (114808177*a[12]*a[20])/358848, 
 1461919/153792 - (421379*a[3])/51264 - (98891*a[3]^2)/76896 + (7264295*a[4])/19224 - (580030*a[3]*a[4])/801 - 
  (840225875*a[6]^2)/9688896 + (65497975*a[6]*a[7])/692064 - (260135*a[8])/12816 + (1233*a[3]*a[8])/8 + (1425*a[4]*a[8])/4 - 
  (1715131*a[8]^2)/12816 - (2015183*a[9])/17088 + (1871267*a[3]*a[9])/76896 + (1777*a[4]*a[9])/3 - (1531063*a[8]*a[9])/25632 + 
  a[4]*a[8]*a[9] + (431*a[9]^2)/4 - (25581925*a[6]*a[10])/3229632 + (1536148885*a[12]^2)/9688896 + 
  (65497975*a[12]*a[13])/692064 - (619177235*a[12]*a[16])/9688896 - (65497975*a[16]^2)/346032 - (18884147*a[18])/51264 + 
  (55781771*a[3]*a[18])/76896 - (9625265*a[8]*a[18])/25632 - (2392367*a[9]*a[18])/3204 + (1757197525*a[6]*a[19])/9688896 - 
  (65497975*a[19]^2)/346032 - (817706845*a[12]*a[20])/3229632, (320*a[12])/63 - a[12]^3 + (640*a[13])/63 + (320*a[16])/63 - 
  (1280*a[8]*a[16])/63 + a[12]^2*a[16] - (320*a[20])/21 + a[12]^2*a[20], (a[6]^2*a[12])/2 - (a[6]*a[7]*a[12])/2 - 
  a[6]*a[12]*a[19] + a[12]*a[19]^2, -(a[6]*a[12]^2)/2 + (a[7]*a[12]^2)/2 + a[12]*a[16]*a[19], 
 (-608*a[12])/147 + a[12]^3/2 - (1216*a[13])/147 - (a[12]^2*a[13])/2 - (608*a[16])/147 + (2432*a[8]*a[16])/147 - 
  a[12]^2*a[16] + a[12]*a[16]^2 + (608*a[20])/49, (-2141*a[12])/588 - (3*a[3]*a[12])/4 + (a[3]^2*a[12])/2 + (a[4]*a[12])/4 - 
  (a[3]*a[4]*a[12])/2 + a[9]*a[12] - a[3]*a[9]*a[12] + a[9]^2*a[12] - (1144*a[13])/147 - (572*a[16])/147 + 
  (2288*a[8]*a[16])/147 + (572*a[20])/49, (-221*a[12])/588 - (3*a[3]*a[12])/4 + (a[3]^2*a[12])/2 + (a[4]*a[12])/4 - 
  (a[3]*a[4]*a[12])/2 + a[9]*a[12] - a[3]*a[9]*a[12] + a[4]*a[9]*a[12] - (184*a[13])/147 - (92*a[16])/147 + 
  (368*a[8]*a[16])/147 + (92*a[20])/49, (320*a[6])/63 - a[6]^3 + (640*a[7])/63 - (320*a[10])/21 + a[6]^2*a[10] - 
  (320*a[19])/21 + a[6]^2*a[19] + (1280*a[8]*a[19])/63, (-608*a[6])/147 + a[6]^3/2 - (1216*a[7])/147 - (a[6]^2*a[7])/2 + 
  (608*a[10])/49 + (608*a[19])/49 - a[6]^2*a[19] - (2432*a[8]*a[19])/147 + a[6]*a[19]^2, 
 (419*a[6])/588 - (3*a[3]*a[6])/4 + (a[3]^2*a[6])/2 + (a[4]*a[6])/4 - (a[3]*a[4]*a[6])/2 + (136*a[7])/147 + a[6]*a[9] - 
  a[3]*a[6]*a[9] + a[6]*a[9]^2 - (68*a[10])/49 - (68*a[19])/49 + (272*a[8]*a[19])/147, 
 (97*a[6])/196 - (3*a[3]*a[6])/4 + (a[3]^2*a[6])/2 + (a[4]*a[6])/4 - (a[3]*a[4]*a[6])/2 + (24*a[7])/49 + a[6]*a[9] - 
  a[3]*a[6]*a[9] + a[4]*a[6]*a[9] - (36*a[10])/49 - (36*a[19])/49 + (48*a[8]*a[19])/49, 
 8117/5340 - (711*a[3])/356 + (1972*a[3]^2)/1335 - a[3]^3 - (404*a[4])/267 + (121*a[3]*a[4])/89 - (591149*a[6]^2)/1177470 - 
  (4625*a[6]*a[7])/19224 - (1363*a[8])/445 + (1363*a[8]^2)/445 + (597*a[9])/356 - (2452*a[3]*a[9])/1335 + a[3]^2*a[9] + 
  (1207*a[8]*a[9])/890 + (1165907*a[6]*a[10])/1569960 - (591149*a[12]^2)/1177470 - (4625*a[12]*a[13])/19224 + 
  (1231471*a[12]*a[16])/4709880 + (4625*a[16]^2)/9612 + (5399*a[18])/1780 - (2452*a[3]*a[18])/1335 + a[3]^2*a[18] - 
  (1207*a[8]*a[18])/890 + (64*a[9]*a[18])/89 + (1231471*a[6]*a[19])/4709880 + (4625*a[19]^2)/9612 + 
  (1165907*a[12]*a[20])/1569960, -1166467/128160 + (75307*a[3])/8544 - (13609*a[3]^2)/64080 + a[3]^3/2 - 
  (9488027*a[4])/25632 + (3035287*a[3]*a[4])/4272 - (a[3]^2*a[4])/2 + (9522942379*a[6]^2)/113037120 - 
  (85692415*a[6]*a[7])/922752 + (200153*a[8])/10680 - (6003*a[3]*a[8])/40 - 351*a[4]*a[8] + (175331*a[8]^2)/1335 + 
  (101651*a[9])/890 - (34595*a[3]*a[9])/1602 - a[3]^2*a[9] - 579*a[4]*a[9] + (79729*a[8]*a[9])/1335 - (546*a[9]^2)/5 + 
  a[3]*a[9]^2 + (649585639*a[6]*a[10])/75358080 - (17589529901*a[12]^2)/113037120 - (85692415*a[12]*a[13])/922752 + 
  (14184418127*a[12]*a[16])/226074240 + (85692415*a[16]^2)/461376 + (3856139*a[18])/10680 - (5693467*a[3]*a[18])/8010 + 
  (392947*a[8]*a[18])/1068 + (7821119*a[9]*a[18])/10680 - (40040526433*a[6]*a[19])/226074240 + (85692415*a[19]^2)/461376 + 
  (18724567159*a[12]*a[20])/75358080, 95051/64080 - (4277*a[3])/4272 + (143*a[3]^2)/8010 - a[3]^3/2 + (10019*a[4])/25632 + 
  (161*a[3]*a[4])/4272 + (a[3]^2*a[4])/2 - (34854541*a[6]^2)/113037120 - (509585*a[6]*a[7])/922752 - (39503*a[8])/10680 + 
  (39503*a[8]^2)/10680 + (1837*a[9])/2848 + (28481*a[3]*a[9])/64080 + (9557*a[8]*a[9])/21360 + 
  (64852469*a[6]*a[10])/75358080 - (34854541*a[12]^2)/113037120 - (509585*a[12]*a[13])/922752 - 
  (55139243*a[12]*a[16])/226074240 + (509585*a[16]^2)/461376 + (46669*a[18])/42720 + (28481*a[3]*a[18])/64080 - 
  (9557*a[8]*a[18])/21360 - (907*a[9]*a[18])/2136 + a[3]*a[9]*a[18] - (55139243*a[6]*a[19])/226074240 + 
  (509585*a[19]^2)/461376 + (64852469*a[12]*a[20])/75358080, -6752417/4101120 + (669701*a[3])/273408 - 
  (2671829*a[3]^2)/2050560 + a[3]^3/2 - (9574447*a[4])/205056 + (1532833*a[3]*a[4])/17088 - (a[3]^2*a[4])/2 + 
  (3850112213*a[6]^2)/361718784 - (85429445*a[6]*a[7])/7382016 + (468209*a[8])/170880 - (24069*a[3]*a[8])/1280 - 
  (11259*a[4]*a[8])/256 + (1098001*a[8]^2)/68352 + (17558459*a[9])/1367040 - (1087453*a[3]*a[9])/2050560 - a[3]^2*a[9] - 
  (887*a[4]*a[9])/12 + a[3]*a[4]*a[9] + (1735367*a[8]*a[9])/227840 - (17219*a[9]^2)/1280 + (3499277*a[6]*a[10])/3767904 - 
  (7057018279*a[12]^2)/361718784 - (85429445*a[12]*a[13])/7382016 + (1435487737*a[12]*a[16])/180859392 + 
  (85429445*a[16]^2)/3691008 + (61578841*a[18])/1367040 - (182293411*a[3]*a[18])/2050560 + (10473297*a[8]*a[18])/227840 + 
  (31283207*a[9]*a[18])/341760 - (4018077509*a[6]*a[19])/180859392 + (85429445*a[19]^2)/3691008 + 
  (936921757*a[12]*a[20])/30143232, (-6791*a[6])/896 + (3*a[6]^3)/4 - (24959*a[7])/1792 - (3*a[6]^2*a[7])/4 + 
  (38541*a[10])/1792 - (a[6]*a[12]^2)/4 + (a[7]*a[12]^2)/4 + (38541*a[19])/1792 - a[6]^2*a[19] - (24959*a[8]*a[19])/896 + 
  a[16]^2*a[19] + a[19]^3, (-6791*a[12])/896 - (a[6]^2*a[12])/4 + (a[6]*a[7]*a[12])/4 + (3*a[12]^3)/4 - (24959*a[13])/1792 - 
  (3*a[12]^2*a[13])/4 - (11377*a[16])/1792 + (24959*a[8]*a[16])/896 - a[12]^2*a[16] + a[16]^3 + a[16]*a[19]^2 + 
  (38541*a[20])/1792, 183097/16020 - (70685*a[3])/8544 - (500621*a[3]^2)/128160 + (3*a[3]^3)/4 + (52935505*a[4])/102528 - 
  (4229873*a[3]*a[4])/4272 - (3*a[3]^2*a[4])/4 - (7564272227*a[6]^2)/64592640 + (481295435*a[6]*a[7])/3691008 - 
  (1095331*a[8])/42720 + (67419*a[3]*a[8])/320 + (63033*a[4]*a[8])/128 - (15810211*a[8]^2)/85440 - (9377989*a[9])/56960 + 
  (18900251*a[3]*a[9])/512640 - a[3]^2*a[9] + (12879*a[4]*a[9])/16 - (1468459*a[8]*a[9])/17088 + (98233*a[9]^2)/640 + a[9]^3 - 
  (572265257*a[6]*a[10])/43061760 + (98006554411*a[12]^2)/452148480 + (481295435*a[12]*a[13])/3691008 - 
  (78095727247*a[12]*a[16])/904296960 - (481295435*a[16]^2)/1845504 - (86272807*a[18])/170880 + 
  (127300691*a[3]*a[18])/128160 - (21963923*a[8]*a[18])/42720 - (174955669*a[9]*a[18])/170880 + 
  (31973884679*a[6]*a[19])/129185280 - (481295435*a[19]^2)/1845504 - (104643496799*a[12]*a[20])/301432320, 
 -840029/256320 + (65447*a[3])/17088 - (19399*a[3]^2)/64080 - a[3]^3/4 - (18158765*a[4])/102528 + (91111*a[3]*a[4])/267 + 
  (a[3]^2*a[4])/4 + (18218466373*a[6]^2)/452148480 - (166368295*a[6]*a[7])/3691008 + (251057*a[8])/42720 - 
  (23043*a[3]*a[8])/320 - (21561*a[4]*a[8])/128 + (5650367*a[8]^2)/85440 + (3127683*a[9])/56960 - (5157367*a[3]*a[9])/512640 - 
  (4463*a[4]*a[9])/16 + (491297*a[8]*a[9])/17088 - (33081*a[9]^2)/640 + (1441099843*a[6]*a[10])/301432320 - 
  (33949317947*a[12]^2)/452148480 - (166368295*a[12]*a[13])/3691008 + (27138403619*a[12]*a[16])/904296960 + 
  (166368295*a[16]^2)/1845504 + (29704589*a[18])/170880 - (21831221*a[3]*a[18])/64080 + (1884709*a[8]*a[18])/10680 + 
  (60124613*a[9]*a[18])/170880 + a[9]^2*a[18] - (77197165021*a[6]*a[19])/904296960 + (166368295*a[19]^2)/1845504 + 
  (36219622723*a[12]*a[20])/301432320, -354851893/36910080 + (24762505*a[3])/2460672 - (22134121*a[3]^2)/18455040 + 
  (3*a[3]^3)/4 - (652538555*a[4])/1845504 + (104303741*a[3]*a[4])/153792 - (3*a[3]^2*a[4])/4 + 
  (1312270017869*a[6]^2)/16277345280 - (5868675745*a[6]*a[7])/66438144 + (29456641*a[8])/1537920 - (183161*a[3]*a[8])/1280 - 
  (85671*a[4]*a[8])/256 + (381222601*a[8]^2)/3075840 + (439571293*a[9])/4101120 - (356475041*a[3]*a[9])/18455040 - 
  a[3]^2*a[9] - (19933*a[4]*a[9])/36 + (70233365*a[8]*a[9])/1230336 - (394573*a[9]^2)/3840 + a[4]*a[9]^2 + 
  (5231480819*a[6]*a[10])/678222720 - (2415218261551*a[12]^2)/16277345280 - (5868675745*a[12]*a[13])/66438144 + 
  (488696352013*a[12]*a[16])/8138672640 + (5868675745*a[16]^2)/33219072 + (4231973069*a[18])/12303360 - 
  (12508156079*a[3]*a[18])/18455040 + (2157110207*a[8]*a[18])/6151680 + (2146869467*a[9]*a[18])/3075840 - 
  (1375047787697*a[6]*a[19])/8138672640 + (5868675745*a[19]^2)/33219072 + 
(321086984923*a[12]*a[20])/1356445440};

You can check, e.g. via NSolve, that the solution set has positive dimension.

In[93]:= Timing[sol = NSolve[gb2];]

During evaluation of In[93]:= NSolve::infsolns: Infinite solution set has dimension at least 1. Returning intersection of solutions with (78848 a[3])/86491-(52050 a[4])/86491+(57827 a[6])/86491+(148851 a[7])/172982+(101463 a[8])/172982+(188769 a[9])/172982-(191343 a[10])/172982-(89087 a[12])/86491+(78339 a[13])/86491+(140033 a[15])/172982-(83945 a[16])/86491-(56554 a[18])/86491+(83206 a[19])/86491-(107131 a[20])/172982 == 1. >>

During evaluation of In[93]:= NSolve::infsolns: Infinite solution set has dimension at least 2. Returning intersection of solutions with (67842 a[3])/95609-(184441 a[4])/191218-(97766 a[6])/95609-(184729 a[7])/191218+(93018 a[8])/95609-(59375 a[9])/95609+(147179 a[10])/191218+(81420 a[12])/95609-(60031 a[13])/95609+(156301 a[15])/191218+(130811 a[16])/191218+(94526 a[18])/95609-(2863 a[19])/2854+(54539 a[20])/95609 == 1. >>

During evaluation of In[93]:= NSolve::infsolns: Infinite solution set has dimension at least 3. Returning intersection of solutions with -((81281 a[3])/77135)+(86849 a[4])/77135-(65291 a[6])/77135+(177769 a[7])/154270-(76583 a[8])/77135-(130181 a[9])/154270-(15303 a[10])/15427+(54742 a[12])/77135+(64317 a[13])/77135+(83022 a[15])/77135+(70919 a[16])/77135-(30099 a[18])/30854-(80873 a[19])/77135-(70654 a[20])/77135 == 1. >>

During evaluation of In[93]:= General::stop: Further output of NSolve::infsolns will be suppressed during this calculation. >>

Out[93]= {3.13, Null}

In[94]:= sol

Out[94]= {{a[15] -> 0., a[10] -> 0., a[20] -> 34.7986, 
  a[13] -> 34.7986, a[7] -> 12.0763, a[18] -> 11.9732, 
  a[4] -> 11.9732, a[16] -> 0., a[19] -> 12.0763, a[9] -> 0., 
  a[8] -> 0., a[3] -> 12.9732, a[6] -> 12.0763, 
  a[12] -> 34.7986}, {a[15] -> 0., a[10] -> -1.32805, 
  a[20] -> -1.52477, a[13] -> -1.52477, a[7] -> -1.32805, 
  a[18] -> -0.727061, a[4] -> -0.727061, a[16] -> -1.52477, 
  a[19] -> -1.32805, a[9] -> -0.727061, a[8] -> 0.5, 
  a[3] -> -0.954122, a[6] -> -2.65611, 
  a[12] -> -3.04955}, {a[15] -> 0., a[10] -> 1.47403, a[20] -> 0., 
  a[13] -> 1.84961, a[7] -> 1.47403, a[18] -> 0., a[4] -> 0.644773, 
  a[16] -> 1.84961, a[19] -> 0., a[9] -> 0.644773, a[8] -> 1., 
  a[3] -> 1.64477, a[6] -> 1.47403, a[12] -> 1.84961}}

If you run NSolve directly on the original set, it might in fact go to completion. But expect it to take considerable time-- I've had it running for a few hours now, with the end only dimly in sight.