Suppose I have an affine algebraic variety, and I suspect that a certain element of its coordinate ring can be represented as a product of two elements having some nice form. Are there computer algorithms for such factoring?
I'm not sure I understand the question. If it amounts to factoring modulo side relations in some setting e.g. polynomials over Q, then you might be able to do this with Groebner basis methods. Some idea of what I mean can be found at the link below. It is more geared toward factoring over (numeric) extension fields but I think the same basic method works in the more general setting of factoring mod algebraic relations.
https://www.researchgate.net/publication/224219544_Polynomial_GCD_and_Factorization_via_Approximate_Grbner_Bases
The article shows explicit code for polynomials in two variables but there are fairly well known ways to then extend: basically by specializing to two variables and then lifting.
Conference Paper Polynomial GCD and Factorization via Approximate Gröbner Bases
If you search for software maybe the best (as far as I know) for your problem is Singular , written in C++ with GNU General Public License, and available at http://www.singular.uni-kl.de
There are online free manuals available and an active community of users.
Dear Miguel,
Thank you for your advice, and Happy New 2015 Year!
The point is, however, that I have already tried Singular! Apparently, now I must "switch on my own brain" (as we say here) and help my computer by studying closer the specific beauty of my mathematical problem :)
I have now made some progress; the program the helped me was Singular.
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