I am working with algebraic points on the unit sphere S2 in R3. I need to find the minimal polynomials for the algebraic numbers. Can Singular handle this?
Currently I am using algdep() command in GP-Pari to attempt to find the polynomials, but running into trouble, for example with 10,018 digits, I still cannot successfully recover the coefficients of a 72nd degree polynomial.
If you have worked on algebraic points on the unit sphere, please let me know how well Singular works.
Thanks
Thank you for your interest in Singular.
What does the data of the given algebraic points consist of? For example, I could think of algebraic points given as
- rational coordinates
- rational coordinates with some roots occurring
- floating point coordinates
- ...
Regards,
Andreas
Actually algebraic numbers can be coordinates on the unit sphere. While we think of just quadratic numbers, here's an example:
[R(7*x^2 + 66*x + 23), R(49*x^4 + 3936*x^2 - 3456), 0]
R means a root of the polynomial.
These polynomials are VERY interesting because one of their roots are on the unit sphere. Notice that the 2nd polynomial seems to be a polynomial in x^2. This happens because the polynomial has a square root.
I can cite other examples, but I need something to enable me to find the characteristic polynomial of the determinant of a Sylvester matrix.
I hope Singular can handle this.
Randall
Okay, Singular is running faster than the current stable version of GP-Pari, when it comes to factoring large degree polynomials. I have a 864th degree polynomial which Singular factored into 3 ideals, in almost record time. (about 5 seconds or so). This took at least 2 or 3 minutes on GP-Pari.
Now... how do do the PolCompositum command which takes two polynomials and finds the polynomial whose roots are the same as the input two.
You are probably searching for the commands primitive() and primitive_extra() from the library primitiv.lib (see
http://www.singular.uni-kl.de/Manual/3- ... htm#SEC938).
Regards,
Andreas