TITLE: How to determine a set of polynomials is algebraically indepedent or not?
QUESTION [5 upvotes]: I have a set of polynomials
$$
a a_1, bb_1, c, c_1, ab, da_1 + bd_1, fa_1+cf_1+d_1e, eb_1+ce_1, de-bf, f_1, e_1, d_1. 
$$
Is there some software which can determine that they are algebraically independent or not. If dependent, how to find the relations they satisfy. Thank you very much.

REPLY [3 votes]: Here's a sketch of how you can check this on a computer, e.g. in Macaulay2. I'll do it in the case $f_1=x^2$, $f_2=xy$ and $f_3=y^2$.
R = QQ[t_1..t_3]
S = QQ[x,y]
f1 = x^2;
f2 = x*y;
f3 = y^2;
f = map(S,R,{f1,f2,f3});
ker f

The output is then
            2
o8 = ideal(t  - t t )
            2    1 3

o8 : Ideal of R

So in this case, the polynomials were algebraically dependent, and satisfies the relation showed above.
In fact, what we have just proved, is that $\mathbb P^1$ can be embedded via the $2$-uple embedding into $\mathbb P^2$ as the zero set of $t_2^2-t_1t_3=0$.