Dear Sureyya:
I suggest you put the parameters in DMP and set the main
equation variables in DMP or POLY. That way, you won't
have any ambiguity.
A groebner basis consisting of just 1 means the ideal
generated by the given set of polynomials is the whole
ring, but you need to know which ring your original set of
parametric polynomials are in to interpret the result.
Moreover, a system of parametric equations is DIFFERENT
from a system of equations whose coefficient ring is
another polynomial ring, because you are then solving the
parametric system generically, rather than parametrically.
As an example, if you solve bx=1, where b is a parameter
and x the unknown in DMP([x], FRAC POLY INT) (pseudo
code only, as I don't remember the correct syntax), you
will get x = 1/b, and clearly this is not valid when b = 0
(but of course, b is NOT zero in FRAC POLY INT---it is an
indeterminate.
Solving a parametric algebraic system [1] should produce a
covering of the parametric space where each "regime" in
the cover is defined by a set of equations and inequations
in the parametric variables and a solution in the main
variables under the parametric conditions of the regime.
In the simple example above, the cover consists of two
regimes: b = 0 (no solution) and b \neq 0 (solution x =
1/b).
[1] Not to be confused with "parametric equations", which
is parametrization or parametric representation of
solutions of an algebraic equation, such as using x = cos
theta, y = sin theta for x^2 + y^2 = 1.
For LINEAR systems in the main variables, I have a package
called PLEQN (Parametric linear equations) that you may
want to check out. See my paper on the subject:
http://www.sciencedirect.com/science/article/pii/S0747717108801046
I am not aware of any package to solve general parametric
algebraic equations in Axiom (but I have not kept
up-to-date on Axiom). Mathematica has a built-in function
called Reduce that seems to do that (it is even more
general as it deals with inequalities as well, but the
algorithm appears to be proprietary and it does not
produce a Groebner basis).
William
On Wed, 19 Feb 2014 08:15:41 -0500
Sureyya Sahin <[email protected]> wrote:
Thank you for the help. I tried your suggestion and I am
indeed getting
some results. But if we extend the number of variables
by defining a few
of the parameters (in this case cb,sb) while keeping the
equations
unchanged, would this effect the polynomial equations
and therefore the
solutions? Also, is this a standard way to attack this
kind of a
problem, i.e. equations with parameters, in axiom? I
guess this needs
some experimentation to obtain the solution.
I was initially thinking that if I get [1] as a Groebner
bases, then it
means that there is no solution to the system of
equations. But given
this example, I guess I was wrong in my interpretation.
What does
getting [1] as a Groebner bases in axiom or another
computer algebra
system mean?
Best Regards,
On Tue, 2014-02-18 at 18:39 -0500, Bill Page wrote:
Try a larger set of variables (generators). Other
unlisted symbols
default to being parameters (from FRAC POLY INT). For
example
[ca,cb,sa,sb,x,y]
gives a basis of 12 polynomials. See
http://axiom-wiki.newsynthesis.org/SandBoxGroebnerBasis#[email protected]
On 18 February 2014 11:23, sahin <[email protected]>
wrote:
> Hello,
>
> I am trying to obtain Groebner bases of a system of
equations. Below is my
> code
>
> (1) -> m : List DMP([ca,sa,x,y],FRAC POLY INT)
> (2) -> m :=
>
[x^2+y^2-r1^2,(x+lab*ca)^2+(y+lab*sa)^2-r2^2,(x+lac*(ca*cb-sa*sb))^2+(y+lac*(sa*cb+ca*sb))^2-r3^2,ca^2+sa^2-1]
>
> asking for groebner bases is leading to
> (3) -> groebner(m)
>
> (3) [1]
> Type:
>
List(DistributedMultivariatePolynomial([ca,sa,x,y],Fraction(Polynomial(Integer))))
>
> which does not make sense to me. The equations are
based on a physical
> system and I can't see any reason that would lead to
an inconsistency. Why
> am I getting [1] as the result? Any help or insight
would be
> well-appreciated.
>
> Best Regards,
>
>
>
> --
> View this message in context:
http://nongnu.13855.n7.nabble.com/Groebner-bases-of-a-set-of-equations-tp179213.html
> Sent from the axiom-math mailing list archive at
Nabble.com.
>
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William Sit, Professor Emeritus
Mathematics, City College of New York
Office: R6/291D Tel: 212-650-5179
Home Page: http://scisun.sci.ccny.cuny.edu/~wyscc/
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