Eliminating Variables in Large Polynomials
Blue Blue
I have a list of five innocuous-seeming polynomials in five variables
(H, J, K, M, G) with a few symbolic constants (S, SX, SY, SZ):
polyH = H^2 - H SX - M
polyJ = J^2 - J SY - M
polyK = K^2 - K SZ - M
polyM = S - 3 M - 2 H J K + H SX + J SY + K SZ
polyG = G H J K + 4 ( M + H J K )^3
My goal is to eliminate four of the variables to arrive at a
polynomial that involves only G and the Ss.
Successive application of Mathematica's Resultant[] function gets me
down to two polynomials with one variable left to eliminate. However,
the degrees of the polynomials (in that last variable) are 20 and 12.
This means that Resultant[] is effectively computing the determinant
of a 32x32 Sylvester Matrix. Applying Resultant[] to *generic*
symbolic polynomials (simply a_20 x^20 + a_19 x^19 + ... and b_12 x^12
+ b_11 x^11 + ... for symbolic constants a_i and b_i) of this size
already grind my Mathematica's kernel to a halt; attempting to apply
Resultant[] when the a_i and b_i coefficients are themselves lengthy
expressions involving G and the Ss seems pretty hopeless.
I've been pondering these polynomials (and their successive
resultants) off-and-on for months, hoping I'd find a clever change of
variables, or see a helpful pattern, or *something* that would
simplify everything, but to no avail so far. I finally decided it was
about time I sent out a signal flare.
Any suggestions?
Thanks in advance ...
clicl...@freenet.de
Blue Blue schrieb:
Hello ...
I understand that you want to eliminate the four variables [H, J, K, M]
from the system of five polynomial equations [polyH = 0, polyJ = 0,
polyK = 0, polyM = 0, polyG = 0], to arrive at a single polynomial in
the fifth variable G. Because of the presence of the four symbolic
parameters [SX, SY, SZ, S], the result can be expected to be huge,
probably too huge to be comprehended by looking at it (say, Megabytes,
or hundreds of Megabytes). I am assuming that you know this and that you
want the result nonetheless.
In Mathematica you may try another technique, which is usually said to
require less space and time: Calculate a suitable Groebner Basis for the
system {polyH, polyJ, polyK, polyM, polyG} in the variables {H, J, K, M,
G}. The position in which the variable G appears here will be important,
and it may be necessary to set a term-ordering option to something like
EliminationOrder (this is from memory, I don't have MMA). One of the
polynomials in the basis (probably the first) will be the one you want.
I have tried the corresponding Derive command
GROEBNER_BASIS([polyH, polyJ, polyK, polyM, polyG], [h, j, k, m, g])
and interrupted the calculation when intermediate results exceeded 50
MByte.
If the Groebner-Basis route fails too (and if you keep thinking you
need the result) you may either replace one or more of the parameters by
a numerical value (perhaps repeating this for different values and
deriving interpolating polynomials later), or you may try to interest
Robert H. Lewis at Fordham University, who has a powerful implementation
of the Dixon resultant in his own Computer Algebra system, Fermat. You
could check older sci.math.symbolic posts of his to get some idea.
Martin.
Robert H. Lewis
>
>
> I have a list of five innocuous-seeming polynomials
> in five variables
> (H, J, K, M, G) with a few symbolic constants (S, SX,
> SY, SZ):
>
> polyH = H^2 - H SX - M
> polyJ = J^2 - J SY - M
> polyK = K^2 - K SZ - M
> polyM = S - 3 M - 2 H J K + H SX + J SY + K SZ
> polyG = G H J K + 4 ( M + H J K )^3
>
> My goal is to eliminate four of the variables to
> arrive at a polynomial that involves only G and the Ss.
>
It is New Year's ever and I don't have time to work it out now. I expect it will be rather routine.
The method of choice for this kind of question is the Dixon resultant. See my articles at
http://fordham.academia.edu/RobertLewis
especially
http://fordham.academia.edu/RobertLewis/Papers/82782/The-Dixon-Resultant
http://fordham.academia.edu/RobertLewis/Papers/114172/Heuristics-to-Accelerate-the-Dixon--Resultant
http://fordham.academia.edu/RobertLewis/attachment/298407/full/Dixon-Beats-Groebner---Almost-Linear--Polynomial-Equations-Arising-in-GPS-Systems-and-in-Nash-Equilibria
Robert H. Lewis
Fordham University
Blue Blue
On Dec 31, 4:01 pm, "Robert H. Lewis" <rle...@fordham.edu> wrote:
> It is New Year's ever and I don't have time to work it out now.
> I expect it will be rather routine.
It certainly wasn't my intention to have anyone give up New Year's
plans for this problem. :) Today just seemed an appropriate time
to turn the page on my current approach and seek help. My research
has been off-and-on for years; there's no rush to complete it.
> The method of choice for this kind of question is the Dixon
> resultant.
I just started looking at the Dixon approach. Your articles seem
much more readable than the stuff I'd found so far. Thank you
(and Martin) very much for the pointers.
I located a Mathematica package (from the mid-90s) implementing
the Dixon resultant. I haven't given it a shot at my polynomials
yet. Something to do once the New Year has already arrived ...
By the way: The upshot of the final polynomial is a step towards
a "Heron-like" formula for determining the volume of a (special
type of) hyperbolic tetrahedron from the areas of its faces. I
fully expect that polynomial to be absolutely enormous, but I
still hope to compute it. (Trying to glean insights from it will
give me something to do for the next few years. :)
Thanks again, and regards ...
Robert H. Lewis
> >
> >
> > I have a list of five innocuous-seeming
> polynomials
> > in five variables
> > (H, J, K, M, G) with a few symbolic constants (S,
> SX,
> > SY, SZ):
> >
> > polyH = H^2 - H SX - M
> > polyJ = J^2 - J SY - M
> > polyK = K^2 - K SZ - M
> > polyM = S - 3 M - 2 H J K + H SX + J SY + K SZ
> > polyG = G H J K + 4 ( M + H J K )^3
> >
> > My goal is to eliminate four of the variables to
> > arrive at a polynomial that involves only G and the
> Ss.
> >
Using Dixon-EDF, the answer modulo the prime 44449 was computed in 3.5 hours. It is degree 12 in g and has 78149 terms. Computing the answer over Z will take longer.
Blue Blue
On Jan 1, 9:44 pm, "Robert H. Lewis" <rle...@fordham.edu> wrote:
> Using Dixon-EDF, the answer modulo the prime 44449 was computed
> in 3.5 hours. It is degree 12 in g and has 78149 terms.
Mr. Lewis has since sent me the result over Z. (A 17.8-hour
computation!) My thanks to him, and others who have responded,
for their interest, insight, and assistance.
Now, if you'll excuse me, I have to go try and make some sense
of the monster result(ant).
clicl...@freenet.de
Blue Blue schrieb:
Nice to learn this!
Blue Blue schrieb:
> By the way: The upshot of the final polynomial is a step towards
> a "Heron-like" formula for determining the volume of a (special
> type of) hyperbolic tetrahedron from the areas of its faces. I
> fully expect that polynomial to be absolutely enormous, but I
> still hope to compute it.
Long live the new theory of hyberbolic soft cheese! And who knows - one
day the polynomial may well become a cornerstone of the final theory of
quantum gravitation.
;)
Martin.
Axel Vogt
A naive question. Since Maple was not able to decide whether the solution
is zero dimensional (ok, had not much patients) I was looking at the task
and see the first 3 equations as quadratics in H,J,K while the 4the is a
kind of permutation (degree 3 + linear, symmetric up to M).
I have not tried to diagonalize (seems you talk over Z and not a field):
but could such (+symmetry) help to simplify the result?
At least for the first 4 eqs Maple had no problem to assert 'dim = 0' and
did quickly find solutions on choices for SX,SY, SZ.
clicl...@freenet.de
Axel Vogt schrieb:
>
> A naive question. Since Maple was not able to decide whether the solution
> is zero dimensional (ok, had not much patients) I was looking at the task
> and see the first 3 equations as quadratics in H,J,K while the 4the is a
> kind of permutation (degree 3 + linear, symmetric up to M).
>
> I have not tried to diagonalize (seems you talk over Z and not a field):
> but could such (+symmetry) help to simplify the result?
>
> At least for the first 4 eqs Maple had no problem to assert 'dim = 0' and
> did quickly find solutions on choices for SX,SY, SZ.
... adding some musings:
It seems that the direct Groebner-Basis route (i.e. the elimination of
h,j,k,m from the five polynomials, producing a single polynomial in g in
one step) is unlikely to succeed in acceptable time even with fast
implementations.
But using your observation that g appears in the fifth equation only, it
may be possible to calculate a basis for the other four equations first.
For example, with sx=sy=sz=1 I readily obtain
[sx:=1,sy:=1,sz:=1]
GROEBNER_BASIS([polyH,polyJ,polyK,polyM],[m,h,j,k])
[8*k^9-12*k^8-34*k^7+k^6*(99-4*s)+k^5*(20*s-78)+k^4*(47-9*s)-k^3~
*(2*s^2+36*s+82)-k^2*(7*s^2-6*s-42)+k*(10*s^2+26*s+12)+s^3+3*s^2~
+2*s,j*(k^3*(4*s+4)-k^2*(10*s+10)+k*(8*s+8)+2*s^2+4*s+2)+8*k^8-4~
*k^7-38*k^6+65*k^5+k^4*(12*s-21)+k^3*(21-2*s)-k^2*(2*s^2+12*s+35~
)-k*(3*s^2+15*s+8)-s^2-s,j*(4*k^4-14*k^3+18*k^2+k*(2*s-6)-2*s-2)~
+4*k^6-8*k^5-5*k^4+26*k^3+k^2*(6*s-15)-k*(8*s+4)-s^2-s,j^2-j-k^2~
+k,h*(8*s^3-4*s^2-142*s-205)+j*(300*k^3-750*k^2+600*k+8*s^3-4*s^~
2+8*s-55)+k^8*(32*s-208)+k^7*(32*s+112)+k^6*(888-192*s)+k^5*(104~
*s-1436)+k^4*(48*s^2-90*s+527)+k^3*(64*s^2-116*s-1500)-k^2*(8*s^~
3-586*s-2418)-k*(24*s^3+56*s^2-112*s+496)-26*s^3-77*s^2-126*s,h*~
(k*(20*s+50)+4*s^2+4*s-15)-j*(60*k^3-150*k^2+k*(70-20*s)-4*s^2+2~
6*s+45)+16*k^8+16*k^7-56*k^6-28*k^5+k^4*(24*s+61)+k^3*(32*s+250)~
-k^2*(4*s^2-34*s+166)-k*(12*s^2+166*s+168)-23*s^2-38*s,h*(70*k^2~
-70*k+4*s^2-26*s-55)+j*(100*k^3-180*k^2+130*k+4*s^2+24*s-5)+16*k~
^8+16*k^7-96*k^6+52*k^5+k^4*(24*s+111)+k^3*(32*s-90)-k^2*(4*s^2+~
26*s-114)-k*(12*s^2+66*s+168)-13*s^2-38*s,h*(70*j+70*k+12*s^2-8*~
s-165)+j*(160*k^3-400*k^2+390*k+12*s^2+72*s-85)+48*k^8+48*k^7-28~
8*k^6+156*k^5+k^4*(72*s+333)+k^3*(96*s-270)-k^2*(12*s^2+78*s-622~
)-k*(36*s^2+198*s+854)-39*s^2-184*s,h^2-h-k^2+k,m-k^2+k]
This basis has ten elements. My computer is not the fastest, and
Derive's Buchberger algorithm runs in pseudocode-Lisp and knows
lexicographic term order only; with a faster implementation on a faster
computer, the four-dimensional case for general sx,sy,sz may be doable.
Would a lexicographic basis for the four-dimensional case be of help in
deriving a lexicographic basis for all five equations in the five
variables ordered as [m,h,j,k,g]? And would there finally be an
algorithm that could convert from this variable order to the required
one, [g,m,h,j,k], in acceptable time?
Or can one expect a basis for all five equations to be easier to obtain
for a degree ordering, which would then have to be converted to the
required lexicographic order? But I understand that's what Mathematica
would do anyway, and no results have been announced.
Martin.
clicl...@freenet.de
clicl...@freenet.de schrieb:
>
> Would a lexicographic basis for the four-dimensional case be of help
> in deriving a lexicographic basis for all five equations in the five
> variables ordered as [m,h,j,k,g]? And would there finally be an
> algorithm that could convert from this variable order to the required
> one, [g,m,h,j,k], in acceptable time?
>
This should have been: Would ... as [g,m,h,j,k]? And ... required one,
[m,h,j,k,g], ... ?
Martin.
Axel Vogt
> Axel Vogt schrieb:
>> A naive question. Since Maple was not able to decide whether the solution
>> is zero dimensional (ok, had not much patients) I was looking at the task
>> and see the first 3 equations as quadratics in H,J,K while the 4the is a
>> kind of permutation (degree 3 + linear, symmetric up to M).
>>
>> I have not tried to diagonalize (seems you talk over Z and not a field):
>> but could such (+symmetry) help to simplify the result?
>>
>> At least for the first 4 eqs Maple had no problem to assert 'dim = 0' and
>> did quickly find solutions on choices for SX,SY, SZ.
>
> ... adding some musings:
>
> It seems that the direct Groebner-Basis route (i.e. the elimination of
> h,j,k,m from the five polynomials, producing a single polynomial in g in
> one step) is unlikely to succeed in acceptable time even with fast
> implementations.
>
> But using your observation that g appears in the fifth equation only, it
> may be possible to calculate a basis for the other four equations first.
> For example, with sx=sy=sz=1 I readily obtain
>
More or less what I meant (being aware of correcting your typos).
However I can get results only for concrete values and my question
is towards: if 'centering' the quadrics and using symmetry can one
do it more easy and feed the 5th equation?
clicl...@freenet.de
Axel Vogt schrieb:
My insight into the matter is too superficial to answer this.
Martin.
Robert H. Lewis
> > By the way: The upshot of the final polynomial is a step towards
> > a "Heron-like" formula for determining the volume of a (special
> > type of) hyperbolic tetrahedron from the areas of its faces. I
> > fully expect that polynomial to be absolutely enormous, but I
> > still hope to compute it.
>
> Long live the new theory of hyberbolic soft cheese!
> And who knows - one
> day the polynomial may well become a cornerstone of
> the final theory of quantum gravitation.
>
Finding Heron-like formulas for higher dimensional volumes is one of the problems I discuss in a paper that is about to appear in MATCOM. The paper is the fourth talk on my web site here:
http://fordham.academia.edu/RobertLewis/Talks
titled "Comparing Acceleration Techniques for the Dixon and Macaulay Resultants."
The Heron formula derivation is problem II in section 3. (This is in the classic Euclidean case.) The "v" for volume occurs in only one equation, just like the equations here.
We see here that Dixon is greatly superior to Macaulay, and yet again that Dixon is greatly superior to Groebner Bases. (of course!! Duh!!!)
On that last topic, also have a look at the third talk there, Dixon Beats Groebner: "Almost Linear" Polynomial Equations Arising in GPS Systems and in Nash Equilibria.
Prof. Robert H. Lewis
Mathematics
Fordham University
http://fordham.academia.edu/RobertLewis
Blue Blue
Robert Lewis wrote:
> Finding Heron-like formulas for higher dimensional volumes
> is one of the problems I discuss in a paper that is about
> to appear in MATCOM.
My research is into Heron-like formulas that relate *face
areas* of (Euclidean and hyperbolic) tetrahedra to their
volumes. Of course, as tetrahedra admit more degrees of
freedom than they have faces, this is tricky. :) I explore
two tracks: considering a trio of "psuedo-face areas" among
the defining parameters; and restricting attention to
"perfect" tetrahedra, whose opposing edges are pairwise
orthogonal.
This is an aspect of what I call "Hedronometry", a
trigonometry of tetrahedra that relates face areas and
dihedral angles. Some of my write-ups on this can be seen
here:
http://www.daylateanddollarshort.com/math/research-dron.html
(They're a bit drafty and include a lot of unnecessary steps
in formula derivation. I was using the projects, in part, to
finally teach myself LaTeX. :)
The polynomials from my original post appear in the "Heron-
like Strategies for Non-Euclidean Tetrahedral Volume". (This
already needed a bit of an update due to a mis-statement or
two, but --thanks again to Mr. Lewis-- it will now need
additional notes regarding the polynomial resultant that
relates face areas to the Gram matrix.)
clicl...@freenet.de
clicl...@freenet.de schrieb:
>
> [...]
>
> Would a lexicographic basis for the four-dimensional case be of help
> in deriving a lexicographic basis for all five equations in the five
> variables ordered as [m,h,j,k,g]?
>
Indeed, the lexicographic basis for all five equations in the five
variables ordered as [g,m,h,j,k] is no problem either for fixed
sx,sy,sz. Two examples:
[sx:=1,sy:=1,sz:=1]
GROEBNER_BASIS([polyH,polyJ,polyK,polyM,polyG],[g,m,h,j,k])
[8*k^9-12*k^8-34*k^7+k^6*(99-4*s)+k^5*(20*s-78)+k^4*(47-9*s)-k^3~
*(2*s^2+36*s+82)-k^2*(7*s^2-6*s-42)+k*(10*s^2+26*s+12)+s^3+3*s^2~
+2*s,j*(k^3*(4*s+4)-k^2*(10*s+10)+k*(8*s+8)+2*s^2+4*s+2)+8*k^8-4~
*k^7-38*k^6+65*k^5+k^4*(12*s-21)+k^3*(21-2*s)-k^2*(2*s^2+12*s+35~
)-k*(3*s^2+15*s+8)-s^2-s,j*(4*k^4-14*k^3+18*k^2+k*(2*s-6)-2*s-2)~
+4*k^6-8*k^5-5*k^4+26*k^3+k^2*(6*s-15)-k*(8*s+4)-s^2-s,j^2-j-k^2~
+k,h*(8*s^3-4*s^2-142*s-205)+j*(300*k^3-750*k^2+600*k+8*s^3-4*s^~
2+8*s-55)+k^8*(32*s-208)+k^7*(32*s+112)+k^6*(888-192*s)+k^5*(104~
*s-1436)+k^4*(48*s^2-90*s+527)+k^3*(64*s^2-116*s-1500)-k^2*(8*s^~
3-586*s-2418)-k*(24*s^3+56*s^2-112*s+496)-26*s^3-77*s^2-126*s,h*~
(k*(20*s+50)+4*s^2+4*s-15)-j*(60*k^3-150*k^2+k*(70-20*s)-4*s^2+2~
6*s+45)+16*k^8+16*k^7-56*k^6-28*k^5+k^4*(24*s+61)+k^3*(32*s+250)~
-k^2*(4*s^2-34*s+166)-k*(12*s^2+166*s+168)-23*s^2-38*s,h*(70*k^2~
-70*k+4*s^2-26*s-55)+j*(100*k^3-180*k^2+130*k+4*s^2+24*s-5)+16*k~
^8+16*k^7-96*k^6+52*k^5+k^4*(24*s+111)+k^3*(32*s-90)-k^2*(4*s^2+~
26*s-114)-k*(12*s^2+66*s+168)-13*s^2-38*s,h*(70*j+70*k+12*s^2-8*~
s-165)+j*(160*k^3-400*k^2+390*k+12*s^2+72*s-85)+48*k^8+48*k^7-28~
8*k^6+156*k^5+k^4*(72*s+333)+k^3*(96*s-270)-k^2*(12*s^2+78*s-622~
)-k*(36*s^2+198*s+854)-39*s^2-184*s,h^2-h-k^2+k,m-k^2+k,g*(224*s~
^6+896*s^5+1120*s^4+448*s^3)+h*(2737510*s^2+12416085*s+13930775)~
-j*(17008800*k^3-42522000*k^2+34017600*k-2737510*s^2-3911685*s-5~
426375)-k^8*(1792*s^5-3584*s^4-70336*s^3-392000*s^2-1974000*s-20~
890040)-k^7*(1792*s^5+25536*s^4+167104*s^3+865200*s^2+4356800*s+~
10988660)+k^6*(11648*s^5+11424*s^4-174944*s^3-1054200*s^2-530880~
0*s-92432190)-k^5*(8512*s^5-80304*s^4-852656*s^3-4609500*s^2-232~
12000*s-152471005)-k^4*(2016*s^6+3696*s^5+45416*s^4+399476*s^3+2~
146550*s^2-5615485*s+53545210)-k^3*(4928*s^6+23072*s^5-2240*s^4-~
285544*s^3-1600200*s^2-29958130*s-119665425)+k^2*(448*s^7+3360*s~
^6-44800*s^5-540120*s^4-3229128*s^3-19255060*s^2-88455140*s-1938~
70065)+k*(1792*s^7+13104*s^6+56336*s^5+257740*s^4+1293096*s^3-16~
91855*s^2-14925415*s+26950280)+224*s^8+2576*s^7+18424*s^6+109004~
*s^5+578690*s^4+2940833*s^3+5920780*s^2+7717905*s,g*(k*(2304*s^3~
+9216*s^2+11520*s+4608)-32*s^5+256*s^4+1376*s^3+1856*s^2+768*s)-~
h*(48520*s^2+220070*s+246925)+j*(301500*k^3-753750*k^2+603000*k-~
48520*s^2-69320*s-96175)+k^8*(256*s^4-3584*s^3-49984*s^2-142784*~
s-437680)+k^7*(256*s^4+576*s^3+21568*s^2+111312*s+220384)-k^6*(1~
664*s^4-9120*s^3-203552*s^2-582216*s-1949088)+k^5*(1216*s^4-720*~
s^3-294032*s^2-1113156*s-3179960)+k^4*(288*s^5-2928*s^4-71272*s^~
3-108340*s^2-48934*s+1071515)+k^3*(704*s^5-5152*s^4-79040*s^3-33~
9928*s^2-1041368*s-2388264)-k^2*(64*s^6-288*s^5-14464*s^4-181224~
*s^3-865504*s^2-2360494*s-3823302)-k*(256*s^6-3504*s^5-63952*s^4~
-263084*s^3-527376*s^2-637428*s+380176)-32*s^7+16*s^6+4088*s^5+1~
8316*s^4+4546*s^3-54901*s^2-120546*s,g*(161280*k^2-161280*k+7280~
0*s^5+237440*s^4+189280*s^3-2240*s^2-26880*s)+h*(154066168*s^2+6~
98761058*s+783989095)-j*(957176820*k^3-2392942050*k^2+1914353640~
*k-154066168*s^2-220172648*s-305400685)-k^8*(582400*s^4-1594880*~
s^3-21676480*s^2-111487040*s-1175427472)-k^7*(582400*s^4+7869120~
*s^3+48502720*s^2+245358960*s+618497248)+k^6*(3785600*s^4+917280~
*s^3-57501920*s^2-300668760*s-5200583232)-k^5*(2766400*s^4-28141~
680*s^3-256308080*s^2-1309436940*s-8579261624)-k^4*(655200*s^5+7~
17360*s^4+14236040*s^3+119232260*s^2-314770018*s+3013353953)-k^3~
*(1601600*s^5+6315680*s^4-5378240*s^3-88998280*s^2-1687087784*s-~
6733367880)+k^2*(145600*s^6+984480*s^5-15285760*s^4-164262840*s^~
3-1082330848*s^2-4979213482*s-10908576642)+k*(582400*s^6+3828720~
*s^5+15486800*s^4+72281020*s^3-95695584*s^2-840072012*s+15167484~
64)+72800*s^7+783440*s^6+5409880*s^5+31427900*s^4+164806250*s^3+~
333139759*s^2+434356374*s,g*(j*(2304*s^3+9216*s^2+11520*s+4608)-~
32*s^5+256*s^4+1376*s^3+1856*s^2+768*s)-h*(48520*s^2+220070*s+24~
6925)+j*(301500*k^3-k^2*(576*s^3+2304*s^2+2880*s+754902)+k*(1152~
*s^4+8064*s^3+19584*s^2+19584*s+609912)+2304*s^5+16704*s^4+46656~
*s^3+14264*s^2-28424*s-85807)+k^8*(256*s^4-3584*s^3-43072*s^2-12~
2048*s-423856)+k^7*(256*s^4+576*s^3+18112*s^2+100944*s+213472)-k~
^6*(1664*s^4-11424*s^3-179936*s^2-495240*s-1888032)+k^5*(1216*s^~
4-3024*s^3-247088*s^2-956196*s-3072248)+k^4*(288*s^5-2928*s^4-68~
392*s^3-125332*s^2-120070*s+1020251)+k^3*(704*s^5-5152*s^4-63488~
*s^3-257848*s^2-903992*s-2317416)-k^2*(64*s^6-288*s^5-16192*s^4-~
170856*s^3-783424*s^2-2212750*s-3745542)-k*(256*s^6-1200*s^5-406~
24*s^4-177836*s^3-386544*s^2-533172*s+407824)-32*s^7+16*s^6+3512~
*s^5+14572*s^4-4094*s^3-63253*s^2-123426*s,g*(j*(161280*k-120960~
*s^2-201600*s-80640)-k*(120960*s^2+201600*s+80640)-21280*s^5-582~
40*s^4-79520*s^3-69440*s^2-26880*s)-h*(41986472*s^2+190426822*s+~
213651605)+j*(261853020*k^3+k^2*(30240*s^2+131040*s-654471270)-k~
*(60480*s^3+120960*s^2-181440*s-524028600)-120960*s^4-594720*s^3~
-42994472*s^2-60195832*s-82886375)+k^8*(170240*s^4-555520*s^3-56~
06720*s^2-26904640*s-318354608)+k^7*(170240*s^4+2210880*s^3+1345~
5680*s^2+65871120*s+167487392)-k^6*(1106560*s^4-312480*s^3-14644~
000*s^2-66001320*s-1408227648)+k^5*(808640*s^4-8650320*s^3-70841~
680*s^2-334621140*s-2322543976)+k^4*(191520*s^5+109200*s^4+45967~
60*s^3+39727660*s^2-86403062*s+816529027)+k^3*(468160*s^5+160048~
0*s^4-1207360*s^3-18803960*s^2-444558376*s-1830126360)-k^2*(4256~
0*s^6+265440*s^5-4497920*s^4-44715720*s^3-284171072*s^2-13313394~
38*s-2964373638)-k*(170240*s^6+1029840*s^5+4545520*s^4+24846500*~
s^3-12538416*s^2-219401508*s+415513616)-21280*s^7-217840*s^6-152~
1800*s^5-9122260*s^4-46546990*s^3-92213981*s^2-118675266*s,g*(15~
36*h+1536*j+1536*k+2080*s^5+6784*s^4+5408*s^3-64*s^2+768*s)+h*(4~
402856*s^2+19967542*s+22401773)-j*(27323772*k^3-68309430*k^2+546~
47544*k-4402856*s^2-6305656*s-8739887)-k^8*(16640*s^4-45568*s^3-~
619328*s^2-3185344*s-33569072)-k^7*(16640*s^4+224832*s^3+1385792~
*s^2+7010256*s+17667488)+k^6*(108160*s^4+26208*s^3-1642912*s^2-8~
590536*s-148534464)-k^5*(79040*s^4-804048*s^3-7323088*s^2-374124~
84*s-245069800)-k^4*(18720*s^5+20496*s^4+406744*s^3+3406636*s^2-~
8976182*s+86110507)-k^3*(45760*s^5+180448*s^4-153664*s^3-2542808~
*s^2-48182584*s-192275736)+k^2*(4160*s^6+28128*s^5-436736*s^4-46~
93224*s^3-30924704*s^2-142251086*s-311541654)+k*(16640*s^6+10939~
2*s^5+442480*s^4+2065172*s^3-2718624*s^2-23921220*s+43374992)+20~
80*s^7+22384*s^6+154568*s^5+897940*s^4+4710286*s^3+9528965*s^2+1~
2419442*s]
[sx:=-1,sy:=1,sz:=1]
GROEBNER_BASIS([polyH,polyJ,polyK,polyM,polyG],[g,m,h,j,k])
[8*k^9-36*k^8+46*k^7+k^6*(4*s+17)+k^5*(4*s-158)+k^4*(201-47*s)-k~
^3*(2*s^2-100*s+10)+k^2*(11*s^2-46*s-66)-k*(14*s^2+30*s+12)-s^3-~
3*s^2-2*s,j*(k^3*(4*s-44)+k^2*(2*s-22)+k*(88-8*s)-2*s^2+20*s+22)~
-8*k^8+44*k^7-90*k^6+29*k^5+k^4*(217-12*s)+k^3*(54*s-363)+k^2*(2~
*s^2-132*s+131)-k*(7*s^2-101*s-56)+13*s^2+13*s,j*(4*k^4+6*k^3-6*~
k^2-k*(2*s+10)-2*s-2)+4*k^6-8*k^5-5*k^4+22*k^3+k^2*(6*s-11)-k*(8~
*s+4)-s^2-s,j^2-j-k^2+k,h*(8*s^3+20*s^2+106*s-59)-j*(6300*k^3+31~
50*k^2-12600*k+8*s^3+20*s^2-3044*s-3209)-k^8*(32*s+1008)+k^7*(22~
4*s+5696)-k^6*(640*s+12080)+k^5*(440*s+4240)-k^4*(48*s^2+278*s-2~
8611)+k^3*(288*s^2+4484*s-48512)+k^2*(8*s^3-656*s^2-16222*s+1767~
0)-k*(40*s^3+92*s^2-13772*s-7760)+106*s^3+1791*s^2+1838*s,h*(k*(~
20*s-10)+4*s^2+4*s-3)-j*(140*k^3+70*k^2+k*(20*s-290)+4*s^2-66*s-~
73)-16*k^8+112*k^7-280*k^6+140*k^5+k^4*(567-24*s)+k^3*(144*s-113~
4)+k^2*(4*s^2-394*s+510)-k*(20*s^2-354*s-200)+43*s^2+46*s,h*(110~
*k^2-110*k-4*s^2-34*s-37)+j*(300*k^3+40*k^2-490*k+4*s^2-116*s-11~
3)+16*k^8-112*k^7+320*k^6-220*k^5+k^4*(24*s-617)+k^3*(1274-144*s~
)-k^2*(4*s^2-454*s+550)+k*(20*s^2-414*s-200)-53*s^2-46*s,h*(110*~
j+110*k-12*s^2+8*s-111)+j*(680*k^3+340*k^2-1470*k+12*s^2-348*s-2~
29)+48*k^8-336*k^7+960*k^6-660*k^5+k^4*(72*s-1851)+k^3*(3822-432~
*s)-k^2*(12*s^2-1362*s+2090)+k*(60*s^2-1242*s-50)-159*s^2-28*s,h~
^2+h-k^2+k,m-k^2+k,g*(1760*s^6+7040*s^5+8800*s^4+3520*s^3)+h*(52~
3626*s^2-3841089*s+1782213)-j*(35092351440*k^3+17546175720*k^2-7~
0184702880*k+523626*s^2-17550016809*s-17544393507)-k^8*(14080*s^~
5+394240*s^4+4720320*s^3+52187520*s^2+580728720*s+6384579624)+k^~
7*(98560*s^5+2214080*s^4+25949440*s^3+287604240*s^2+3194204640*s~
+35110596468)-k^6*(260480*s^5+4620000*s^4+53046400*s^3+590400360~
*s^2+6532756560*s+71804167590)+k^5*(130240*s^5+1491600*s^4+17072~
000*s^3+192893580*s^2+2101702680*s+23126399835)-k^4*(15840*s^6+1~
1440*s^5-4050200*s^4-49663020*s^3-548899890*s^2-6178088409*s-173~
143389618)+k^3*(116160*s^6+1666720*s^5+13247520*s^4+139131080*s^~
3+1539366840*s^2+16741515726*s-289616506221)+k^2*(3520*s^7-26928~
0*s^6-5179680*s^5-58368200*s^4-643139640*s^3-7120659564*s^2-9582~
0077796*s+104512330395)-k*(14080*s^7+22000*s^6-1304160*s^5-17045~
820*s^4-188593680*s^3-2107995375*s^2-84653215071*s-44670791640)+~
1760*s^8+51920*s^7+753720*s^6+8489580*s^5+93127210*s^4+102803552~
5*s^3+11317741752*s^2+10368900489*s,g*(k*(126720*s^3+506880*s^2+~
633600*s+253440)-1760*s^5+14080*s^4+75680*s^3+102080*s^2+42240*s~
)-h*(4276800*k-26472*s^2+20838*s+337749)+j*(1604750940*k^3+80237~
5470*k^2-3205225080*k-26472*s^2-802354632*s-802037721)+k^8*(1408~
0*s^4+225280*s^3+2523840*s^2+27223680*s+293348592)-k^7*(98560*s^~
4+1031360*s^3+12305920*s^2+145635600*s+1607340864)+k^6*(260480*s~
^4+2001120*s^3+23330560*s^2+291489000*s+3275978160)-k^5*(130240*~
s^4+562320*s^3+5825600*s^2+87092940*s+1043330640)+k^4*(15840*s^5~
-178640*s^4-1779800*s^3-22349580*s^2-282796242*s-7927906419)-k^3~
*(116160*s^5+272800*s^4+7059360*s^3+84514760*s^2+797920248*s-132~
10347648)-k^2*(3520*s^6-311520*s^5-2328480*s^4-27828680*s^3-3323~
80032*s^2-4409097738*s+4736378790)+k*(14080*s^6-20240*s^5+396000~
*s^4-603900*s^3-75652200*s^2-3843205548*s-2033973840)-1760*s^7-3~
0800*s^6-257400*s^5-3373260*s^4-44633050*s^3-514390891*s^2-47261~
2302*s,g*(253440*k^2-253440*k+114400*s^5+373120*s^4+297440*s^3-3~
520*s^2-42240*s)+h*(19600248*s^2+68708598*s-40267131)-j*(1956328~
16220*k^3+97816408110*k^2-391265632440*k+19600248*s^2-9774769951~
2*s-97856675241)-k^8*(915200*s^4+24949760*s^3+288404160*s^2+3179~
299200*s+35477012592)+k^7*(6406400*s^4+139184320*s^3+1583985920*~
s^2+17524861200*s+195226139904)-k^6*(16931200*s^4+287796960*s^3+~
3235633280*s^2+35986738920*s+399615522000)+k^5*(8465600*s^4+9070~
2480*s^3+1042771840*s^2+11767635660*s+128998927200)-k^4*(1029600~
*s^5-16720*s^4-263241880*s^3-3033845100*s^2-33322715202*s-963171~
416979)+k^3*(7550400*s^5+102761120*s^4+785268000*s^3+8463459400*~
s^2+94509279768*s-1611641107248)+k^2*(228800*s^6-17672160*s^5-32~
3627040*s^4-3555129160*s^3-39159196992*s^2-533339547978*s+581719~
142070)-k*(915200*s^6+754160*s^5-85319520*s^4-1045093500*s^3-113~
89116600*s^2-470736394428*s-248815646160)+114400*s^7+3290320*s^6~
+46563000*s^5+517449900*s^4+5671096970*s^3+62893090091*s^2+57785~
431422*s,g*(j*(11520*s^3+46080*s^2+57600*s+23040)-160*s^5+1280*s~
^4+6880*s^3+9280*s^2+3840*s)+h*(388800*k-40008*s^2+26382*s-42303~
9)-j*(466065900*k^3-k^2*(43200*s^3+172800*s^2+216000*s-232946550~
)-k*(40320*s^4+167040*s^3+224640*s^2+109440*s+931754520)-11520*s~
^5-48960*s^4-60480*s^3-42888*s^2-232969128*s-233438709)+k^8*(128~
0*s^4+20480*s^3-335040*s^2-6960000*s-84122928)-k^7*(8960*s^4+937~
60*s^3-1985920*s^2-38652240*s-462974016)+k^6*(23680*s^4+101280*s~
^3-4690240*s^2-80668680*s-948261360)-k^5*(11840*s^4-29520*s^3-19~
77440*s^2-27309180*s-306214800)+k^4*(1440*s^5-16240*s^4-746440*s~
^3+625020*s^2+67358058*s+2286327591)-k^3*(10560*s^5+24800*s^4-29~
43840*s^3-29179400*s^2-244288632*s+3823093632)-k^2*(320*s^6-2832~
0*s^5-231840*s^4+5194280*s^3+90574368*s^2+1273780482*s-137735451~
0)+k*(1280*s^6-13360*s^5-446400*s^4-918900*s^3+19961640*s^2+1113~
803772*s+591367440)-160*s^7-2800*s^6-3240*s^5+745980*s^4+1256293~
0*s^3+148865119*s^2+137538198*s,g*(j*(50688*k-38016*s^2-63360*s-~
25344)-k*(38016*s^2+63360*s+25344)-6688*s^5-18304*s^4-24992*s^3-~
21824*s^2-8448*s)-h*(1145352*s^2+3465690*s-2655357)+j*(111938823~
72*k^3-k^2*(142560*s^2+60192*s-5596776450)-k*(133056*s^3+190080*~
s^2+107712*s+22387435272)-38016*s^4-72864*s^3+1132680*s^2-559333~
9272*s-5599482495)+k^8*(53504*s^4+1430528*s^3+16247616*s^2+18126~
8736*s+2029608720)-k^7*(374528*s^4+7940416*s^3+89399552*s^2+9998~
73072*s+11169060864)+k^6*(989824*s^4+16305696*s^3+183113216*s^2+~
2055055992*s+22863277104)-k^5*(494912*s^4+5042928*s^3+59475328*s~
^2+673325796*s+7381079904)+k^4*(60192*s^5-32560*s^4-15217576*s^3~
-169085268*s^2-1895429982*s-55105057125)-k^3*(441408*s^5+5775968~
*s^4+44013024*s^3+489310360*s^2+5423340216*s-92206678416)-k^2*(1~
3376*s^6-1040160*s^5-18339552*s^4-200621080*s^3-2236454496*s^2-3~
0519520374*s+33281696250)+k*(53504*s^6+16016*s^5-4896672*s^4-581~
00724*s^3-646452504*s^2-26931846756*s-14236839600)-6688*s^7-1888~
48*s^6-2640264*s^5-29330004*s^4-323731958*s^3-3598380605*s^2-330~
6496242*s,g*(84480*h-84480*j-84480*k-114400*s^5-373120*s^4-29744~
0*s^3+3520*s^2-42240*s)-h*(19523448*s^2+68562678*s-40808571)+j*(~
195637193820*k^3+97818596910*k^2-391274387640*k+19523448*s^2-977~
50034232*s-97859405481)+k^8*(915200*s^4+24949760*s^3+288404160*s~
^2+3179299200*s+35477719152)-k^7*(6406400*s^4+139184320*s^3+1583~
985920*s^2+17524861200*s+195230072064)+k^6*(16931200*s^4+2877969~
60*s^3+3235633280*s^2+35986738920*s+399623401680)-k^5*(8465600*s~
^4+90702480*s^3+1042771840*s^2+11767635660*s+129000785760)+k^4*(~
1029600*s^5-16720*s^4-263241880*s^3-3033845100*s^2-33321908802*s~
-963190236819)-k^3*(7550400*s^5+102761120*s^4+785268000*s^3+8463~
459400*s^2+94513104408*s-1611671473968)-k^2*(228800*s^6-17672160~
*s^5-323627040*s^4-3555129160*s^3-39159273792*s^2-533350092618*s~
+581729026230)+k*(915200*s^6+754160*s^5-85319520*s^4-1045093500*~
s^3-11388993720*s^2-470745426108*s-248822281680)-114400*s^7-3290~
320*s^6-46563000*s^5-517449900*s^4-5671181450*s^3-62894353451*s^~
2-57787082622*s]
Each basis has 16 elements. But the feasibilty for general sx,sy,sz
remains an open problem, as does the feasibility of converting to a
lexicographic basis for the variable order [m,h,j,k,g] required by Blue
Blue.
Martin.