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Topic: response to a Math-Forum post by Robert Lewis
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Posts: 1,039
Registered: 4/26/08
response to a Math-Forum post by Robert Lewis
Posted: Sep 11, 2011 5:48 PM
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In a Math-Forum post named "Using Groebner Basis routines in Maple"
(which I never saw at the Aioe.org NNTP server)


Robert Lewis wrote on July 9, 2011:
> All,
> I occasionally use Maple for Groebner basis calculations. I thought I
> knew what to do, but recently have hit a problem I don't understand.
> I am aware of the page
> http://www.maplesoft.com/support/help/Maple/view.aspx?path=Groebner/Basis_algorithms
> I have a system of equations and I want a GB that will produce one
> polynomial that contains only the variable ca. (So I want to solve
> for ca, in essence.) The system of equations is at the end of this
> note.
> I have tried
> out := fgb_gbasis(sys, [b, c, d, e, f, rr, t50], [ca]);
> out := fgb_gbasis(sys, [b, c, d, e, f, t50, rr], [ca])
> out := Basis(sys, prod(tdeg(rr, b, c, d, e, f, t50), tdeg(ca)))
> out := Basis(sys, lexdeg([b, c, d, e, f, t50, rr], [ca]));
> out := Basis(sys, lexdeg([rr, b, c, d, e, f, t50], [ca]))
> out := Basis(sys, plex(rr, b, c, d, e, f, t50, ca))
> out := Basis(sys, prod(plex(rr, b, c, d, e, f, t50), plex(ca)))
> Nothing works. In all cases, there is no polynomial in the GB
> containing ONLY ca.
> I tried rearranging the order of the equations. I also tried
> Triangularize, but that crashes Maple after 10 minutes or so. This is
> Maple12.
> Robert H. Lewis
> - e^2 + d^2 - 2*ca*b*d + b^2 ,
> - t50^2*e^2 - e^2 - 2*t50^2*d*e - 2*d*e - t50^2*d^2 - d^2 + t50^2*c^2 + c^2 - rr*t50^2*c - 2*t50*c
> + rr*c + t50^2 + 1 ,
> - t50^2*f^2 - f^2 + t50^2*e^2 + e^2 - 4*t50*e + t50^2 + 1 ,
> e^2 - d*e + d^2 - b^2 ,
> f^2 + d*f + d^2 - c^2 ,
> f^2 - e*f + e^2 - 1,
> rr^2 - 3,
> t50^6 - 33*t50^4 + 27*t50^2 - 3

I have played around a bit with this system using the GROEBNER_BASIS
function of Derive 6.10, which computes a minimal/reduced basis for a
lexicographical order of monomials, presumably via the Buchberger
algorithm. In the Derive function, the lexicographical order
is defined by the variable vector supplied as a second argument; thus,
to obtain equations for ca, this variable has to be placed last in the

Preliminary experiments suggested that the following variable order
might produce a comparatively compact basis,



but I lost patience and stopped the calculation when the intermediate
result approached 50 MBytes. Instead, I removed the one equation that
depends on t50 alone, t50^6 - 33*t50^4 + 27*t50^2 - 3, and fixed t50 at
small integer values. (I suppose that a solution for the original
irrational roots could be obtained by systematically substituting
nearby rational numbers and interpolating.) On a modern computer, the
calculations for fixed t50 = -1, 0, 1 are just a matter of minutes:








where the Groebner-basis elements have subsequently been factored.
Although no first element does here depend on ca alone, solutions for
the variable pair [ca,b] are easily obtained from the corresponding
subset of two or three elements:


[b=-SQRT(3)/3,b=SQRT(3)/3,ca=1 AND b=1,ca=1 AND b=-1,ca=1 AND b=~
1/2,ca=1 AND b=-1/2,ca=-1 AND b=1,ca=-1 AND b=-1,ca=-1 AND b=1/2~
,ca=-1 AND b=-1/2,ca=5*SQRT(7)/14 AND b=SQRT(21)/3,ca=5*SQRT(7)/~
14 AND b=-SQRT(21)/3,ca=-5*SQRT(7)/14 AND b=SQRT(21)/3,ca=-5*SQR~
T(7)/14 AND b=-SQRT(21)/3]


for t50 = -1, 0, 1 respectively. Thus all three cases admit solutions
where ca is arbitrary. For other values like t50 = 2, however, Derive's
algorithm again requires much more memory, and takes longer than I cared
too wait.


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