**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(

o r

ie. the set of polynomials f such that for some power

¥Assume the theorem is
__not__
algebraically true, then Ã(Sat(
* I*
,

H & {f0} ===> T

It is, therefore, the set of all

we shall call them

¥If Ã

========================================================

Let Sat(

Then

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

is the Zariski closure of the set of points in

In fact, let P_1,..., P_m be the prime components of Ã

and let Q_1,...Q_n be the remaining prime components of Ã

point,

Then {g_1=0, ..., g_s=0} is the union of

¥Since (g_i)

it follows that g_i is in Q_j.

Thus {g_1=0, ..., g_s=0} contains the union of

¥Conversely, assume some g=0 contains

Then

g

since

Therefore, for some d, (g

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

ÃSat(

(we say that H===>T is an

Otherwise, there are non-trival conditions.

{g_10 or....or g_s0} is the largest open set of

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

-The Z-closure of

-Ã(

(see Bazzotti-Dalzotto-Robbiano)

So this is the

It is (1) iff the theorem is absolutely false.

- we can try to find smaller subsets of the set of generators of

Over the components where the thesis fails we know

---there is not an open set where

---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.