**Non-degeneracy conditions II**

¥If we care for a set of independent privileged variables, say [
**x (privileged), y**
], it makes sense to eliminate
**y**
in Sat(
* I*
,

Let Sat(

Thus non-degeneracy conditions are now expressed in terms of privileged variables.

Of course, the zero set of (g_1, ...,g_s) in

¥There will be components of the hypothesis variety where the privileged variables remain independent, ie.

no polynomial in these variables vanishes over the component <===> its projection onto

¥ It can happen that theorem holds over all components where the privileged variables remain independent.

The we say that we have a

(g_1, ...,g_s)(0) iff

==>

If

<==

Conversely, assume

In the example Bisecting Diagonals, privileging x_1 and x_2, we have

Elim([x[3],x[4]], P1)

Ideal(0)

-------------------------------

Elim([x[3],x[4]], P2)

Ideal(x[2])

-------------------------------

Elim([x[3],x[4]], P3)

Ideal(x[1])

-------------------------------

Elim([x[3],x[4]], P4)

Ideal(x[2], x[1])

-------------------------------

So only P_1 is a prime component where the two variables remain independent.

Then we compute Sat(

Elim([x[3],x[4]],Ideal(x[3]x[4], x[1]x[4], x[2]x[3], x[1]x[2]))

Ideal(x[1]x[2])

-------------------------------

It can happen that Sat(