Non-degeneracy conditions I (B-D-R)

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

Sat(
I ,t )

o
r I :t ^ infinity , or I_c (ideal of conditions)

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



•Assume the theorem is
not algebraically true, then √(Sat( I , t )) is the set of polynomials f, such that

H & {f≠0} ===> T

It is, therefore, the set of all
non-degeneracy conditions.

Remark : Of course I is contained in Sat( I , t ), thus the elements of √ I are already non-degeneracy conditions, but useless (we get a contradictory set of hypotheses, and this always gives a true theorem for no matter what thesis):
we shall call them
trival conditions.

•If √
I is a prime ideal and the theorem is not algebraically true, all conditions are trivial.
========================================================

Let Sat(
I , t ) (or √Sat( I , t )) =(g_1,...,g_s) .

Then

{g_1=0, ..., g_s=0}

is the Zariski closure of the set of points in
V ( I ) where t ≠0.



Proposition :
In fact, let P_1,..., P_m be the prime components of √
I such that t =0 over them,

and let Q_1,...Q_n be the remaining prime components of √
I where, at some

point,
t ≠0.


Then {g_1=0, ..., g_s=0} is the union of
V (Q_1), ..., V (Q_n).



Proof
•Since (g_i)
t ^d belongs to I , hence to Q_j, and t does not belong to Q_j,
it follows that g_i is in Q_j.

Thus {g_1=0, ..., g_s=0} contains the union of
V (Q_1), ..., V (Q_n).


•Conversely, assume some g=0 contains
V (Q_1), ..., V (Q_n).
Then

g
t =0 over all V ( I )

since
t =0 over the remaining components.


Therefore, for some d, (g
t )^d is in I , so g belongs to the √Sat( I , t ), and

g=0 contains {g_1=0, ..., g_s=0}.


Corollary
√Sat(
I , t )=√ I iff the theorem is false over all components
(we say that H===>T is an
absolutely false theorem).

Otherwise, there are non-trival conditions.



Corollary
{g_1≠0 or....or g_s≠0} is the largest open set of
V ( I ) where the thesis holds.

It is non empty iff there are non-trivial conditions.



Corollary
-The Z-closure of
V ( I )-{g_1=0, ....,g_s=0} is the union of V (P_1), ..., V (P_m).


-√(
I : I_c ^ infinity ) = √ I : I_c = √ I :√ I_c = \intersection P_i

(see Bazzotti-Dalzotto-Robbiano)

So this is the
optimal hypothesis ideal , the one where our thesis really holds.
It is (1) iff the theorem is absolutely false.

- we can try to find smaller subsets of the set of generators of
I_c , with same kind of properties.


Remark
Over the components where the thesis fails we know

---there is not an open set where
t =0

---there is a proper closed set, perhaps empty, that contains the set of points where the thesis holds.


So this method is good for detecting and correcting almost true theorems (ie. theorems true over an open (=large) set of the hypothese variety).

It does not work for theorems which are true only over a "thin" set of cases.