Simplifying ∃x
∈ ]a, b[, P1 = 0 and .... Pn
= 0 and
Q1 > 0 and ... Qk
> 0
In order to get rid off several polynomials being equal to 0, we
compute the greatest common divisor (GCD) and we impose the GCD to be equal to 0. Formally, P1
= 0 and P2 =
0 and .... Pn = 0 is equivalent to GCD(P1, P2,
... Pn) = 0.
You have now a formula of the
form ∃x ∈ ]a, b[, P =
0 and Q1 > 0
and Q2 > 0 and ... Qk
> 0.
∃x
∈ ]a, b[, x2 + 5 = 0 and
x2 + 3 = 0
is rewritten in ∃x ∈ ]a, b[, 2 =
0 , i.e it is false.
∃x
∈ ]a, b[, x2 + 7x + 5
= 0 and x2 - 5 = 0 is rewritten in ∃x ∈ ]a, b[, x + 5 =
0.