Algebraic method 



1) Translate the geometric construction and the thesis by means of a collection of equations and inequations:

                                                  H=0, T=0



2) Verify whether  H=0  ===> T=0,   by deciding if its negation
                             
                                             [not(T=0)] &(H=0)

is the empty system (Nullstellensatz).


If empty, then   H=0  ===>  T=0
 

 



3) If the system is not empty, we would like to find other hypotheses  H' such that  either
            
                                (H=0)&(H'=0)  ===>  T=0
                                                           
                                                          or

                                (H=0)&(H'≠0)  ===>  T=0



To find complementary hypotheses H' is equivalent to

                 
eliminate variables in one of the following systems....


•H=0 & notT=0 (
non-degeneracy conditions)

because, obviously
 
                H=0 & not(H=0 & notT=0) ===> T=0


If H'=(h_1, h_2,..., h_r), then we obtain that

H=0 &{h_1≠0 or h_2≠0 or.....or h_r≠0}  ===> T=0

thus theorem will be true for any subset of these non-degeneracy conditions.



•H=0 & T=0  (discovery conditions)

because, obviously

               H=0 & (H=0 & T=0)  ===>T=0


If H'=(h_1, h_2,..., h_r), then we obtain that

H=0 &{h_1=0 and h_2=0 and.....and h_r=0}  ===> T=0

so theorem could be true for any subset of these discovery conditions (but the more we consider, the better...).



•Elimination lead us to rewrite our given data in terms of

                                   some
privileged variables

•Privileged in some
human predetermined sense...