Simplifying with Assumptions without giving trivial assumptions

Question

Given a long list of independent rational expressions, I need to decide in a reliable but quick way whether or not it is possible for each to be purely negative.

Consider the following rational expression:

expr = ((-1)^(1/3)*(x^2 - y))/x^2

Then I would like something like:

Assuming[(# ∈ Reals)& /@ Variables[expr]   (* {x ∈ Reals, y ∈ Reals} *)
 , Simplify[expr ∈ Reals && expr < 0]]

to return False since expr can't be real and negative. But it doesn't work. I have to explicitly tell it that no combination of variables can be zero:

Assuming[{x ∈ Reals, y ∈ Reals, y ≠ x^2, y ≠ 0, x ≠ 0}
 , Simplify[expr ∈ Reals && expr < 0]]

(* False *)

But this breaks my ability to automate this computation. How do I decide whether a rational expression is real and negative in a simpler way?

Answer 1

Use FullSimplify (instead of Simplify) and you can remove most of the extra assumptions.

expr = ((-1)^(1/3)*(x^2 - y))/x^2;
Assuming[{x \[Element] Reals, y \[Element] Reals}, 
     FullSimplify[expr \[Element] Reals && expr < 0]]

(* False *)