# Simplifying expressions

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?

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Try Singular or CoCoA or Macaulay2.

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Thanks. I have used Macaulay2Sage before, but I am not sure how to handle these type of problems in that. I will look at Singular and CoCoA. –  user9902 Apr 21 '11 at 17:49
@Kyle, I'm not sure they can handle your problem, but I thought I'd mention those programs just in case. –  lhf Apr 21 '11 at 17:55
$\rm (c_1)\ \ grob\_tot\_reduce(\ f_1*(x+f_1),\ [\:f_1^2-f_2-f_3],\ [\:x,f_3,f_2,f_1]);$
$\rm(d_1)\ /R/\quad\quad\quad\quad\quad\quad\quad\quad\quad x\ f_1 + f_2 + f_3$