A tip for your latex: I would use \langle and \rangle to surround the generators of an ideal, which looks better than using < and >. And I think they are called angle brackets, not square brackets (which would be [,]).
Consider a strict inequality with real variables x_i:
f(x_1,...,x_n) > 0
f(x_1,...,x_n) = y^-2
If f is a polynomial, the above can be written as a polynomial equation like so:
f(x_1,...,x_n) * y^2 = 1
f(x_1,...,x_n) ≥ 0
f(x_1,...,x_n) = y^2