I have a polynomial ring $R=k[x,y,z...]$ and a given ideal $I$ (defined by given generators) and several polynomials $f_1,f_2,...$ in the ring. I also have several other elements of $R$ given as polynomials in $f_1,f_2,...$ and $x,y,z...$. I wish to determine whether these elements lie in the ideal or not. I do not have exact expressions for $f_1,f_2,...$ but I know certain relations that hold between these and the indeterminates. Now, my problem is computational. I wish to simplify the expressions I have to a more manageable form where I am able to "see" the membership. So I was wondering whether there are any softwares that will allow me to automate this. Essentially I would like to feed the relations to the software, and it should simplify the expression as much as possible using these relations. So far I have tried Maple and Matlab, but both of those require too much manual intervention. For example the expression I have may have a term like $f_1(x+f_1)$ and I have a relation $f_1^2=f_2+f_3$. Then I would like the term simplified to $xf_1+f_2+f_3$. Is there any software that might help me with this?
Any computer algebra system supporting Grobner bases will suffice, e.g. in Macsyma
$\rm (c_1)\ \ grob\_tot\_reduce(\ f_1*(x+f_1),\ [\:f_1^2-f_2-f_3],\ [\:x,f_3,f_2,f_1]);$
$\rm(d_1)\ /R/\quad\quad\quad\quad\quad\quad\quad\quad\quad x\ f_1 + f_2 + f_3$
In Maple you could use simplify with respect to side relations: http://www.maplesoft.com/support/help/Maple/view.aspx?path=simplify%2Fsiderels
The input for your example would be: simplify( f1*(x+f1), [f1^2 = f2+f3] );