Algebraically true theorems

Definition : Given

- hypotheses polynomial (equations) H={h_1,...,h_r} generating a proper ideal
I

- a thesis T given by one polynomial equation
t ,

we say that the statement H ==> T is
algebraically true (or geometrically true)
if
t belongs to the radical ( I ) (i.e. if t vanishes over the variety V ( I )).


Remark : The only coefficient field that matters is the one generated by the coefficients of H, T over Q.

On the other hand, geometrically speaking, we are working over an algebraically closed field.


Procedure : Compute (adding slack variable k)

NF(1, Ideal(h_1,...,h_r, t*k-1)).

If it is 0, theorem is algebraically true, else it is not true.


Remark : Elim(t, Ideal(h_1,...,h_r, t*k-1)) is the Saturation of I by t ,

Sat(
I , t )

ie. the set of polynomials f such that for some power
t ^d, f* t ^d belongs to I .

So a theorem is algebraically true iff (1)=Sat(
I , t ).