Re: [Axiom-math] Groebner bases of a set of equations

[Sureyya Sahin]

Hi William:
Thank you for your post regarding my problem on solving the Groebner
bases of a set of polynomial equation. The main problem is that not all
of the variables are associated with the set of polynomials, i.e., in
this case I have Q[x,y,ca,sa]. The rest of the variables are independent
parameters which can be interpreted as rational functions. I already
tried simpler polynomial equations and axiom gives me meaningful
results. Just as an example, if you try the following code in
Q(a2,a3,a4,rx,ry)[c3,s3,c4,s4];

n : List DMP([c3,s3,c4,s4], FRAC POLY INT)
n := [a2*c2+a3*c3-a4*c4-rx,a2*s2+a3*s3-a4*s4-ry,c3^2+s3^2-1,c4^2+s3^2-1]
g := groebner(n)

You will obtain meaningful results. This is the solution of the
kinematic equations of a simple four-bar linkage and the other
parameters a2,a3,a4,rx,ry are just symbols, yet axiom is capable of
giving me solution in this case. Here I am not talking about whether the
Groebner bases is defined or not, I am obtaining a set of Groebner base
solutions in terms of the symbolic variables.

On my original question: I am obtaining [1] as the Groebner base
solution which I interpreted as the solution set is the whole
Q[ca,sa,x,y], which would imply no solution as a consequence of
Nullstellensatz. Thus, I interpreted this as I have inconsistent
(meaningless) equations. But the equations describe a physically
meaningful system. And I can't catch an error on the equations. So, I
thought I may misinterpret the [1] that I receive from axiom. To my
understanding you think [1] may be coming from the symbolic variables;
but this does not make sense to me as I already defined Q[ca,sa,x,y] and
the rest is being taken as symbols as in the simple example I
illustrated. 

I may yet try including the rest of the variables in the definition of
the polynomial, i.e., Q[ca,sa,x,y,cb,sb,r1,r2,r3,..]. But I have my
doubts as I have already shown that axiom can solve equations in
Q(cb,sb,r1,r2,r3,..)[ca,sa,x,y]. I may be misinterpreting your
suggestion, can you give more details of your recommendation?

Thank you for including your study on linear systems as well. Generally,
I am dealing with nonlinear polynomials in trigonometric/transcendental
functions. Thus, I will not be able to use a solution in linear
polynomials. Yet I will read to gain a bit more insight into the
symbolic computation.

Best Regards,


On Wed, 2014-02-19 at 13:26 -0500, William Sit wrote:
> 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 <address@hidden> 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/address@hidden
> >> 
> >> On 18 February 2014 11:23, sahin <address@hidden> 
> >>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.
> >> >
> >> > _______________________________________________
> >> > Axiom-math mailing list
> >> > address@hidden
> >> > https://lists.nongnu.org/mailman/listinfo/axiom-math
> > 
> > 
> > 
> > _______________________________________________
> > Axiom-math mailing list
> > address@hidden
> > https://lists.nongnu.org/mailman/listinfo/axiom-math
> 
> 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/