Re: Groebner bases of a set of equations

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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> 
> 
> 
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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/