Solving Polynomial equations with parameter coefficients

I was wondering whether it is possible to solve a system of polynomial equations with parameter coeffiecients in terms of the parameter coefficients using one of the libraries of Singular.

Dursun.

email: bulutoglu.dursun@afit.edu
Posted in old Singular Forum on: 2005-06-12 21:05:45+02

The question is what you mean by "solving" in this context.

Assume that the system is 0-dimensional where the parameters belong to the ground field (e.g. a,b parameters, x,y,z variables: ring r=(0,a,b),(x,y,z),lp; )
Then the library 'triang.lib' provides routines for computing triangular systems (analog of row-echelon form for polynomial equations). Each system contains one univariate polynomial with coefficient being rational functions in the parameters, a second polynomial depending on two varaible, etc. This does, in principle, allow an analysis of the solution set in terms of the parameters.

Caveats:
- The polynomials (and coefficients) are usually quite big.
- During Groebner bases computations, Singular extracts and forgets the content of the coefficients. Hence the dependence on the parameters is only generically true.
- A further symbolic analysis would require algebraic extensions of coefficient fields containing parameters which is not implemented.

email: greuel@mathematik.uni-kl.de
Posted in old Singular Forum on: 2005-07-28 18:26:54+02