[nsajko]

Inequalities can be transformed into equations quite easily, by adding variables:

Consider a strict inequality with real variables x_i:

    f(x_1,...,x_n) > 0

By adding the variable y we obtain an equivalent equation:

    f(x_1,...,x_n) = y^-2

This is correct because y^-2 is always strictly positive.

If f is a polynomial, the above can be written as a polynomial equation like so:

    f(x_1,...,x_n) * y^2 = 1

To transform a non-strict inequality into an equation, on the other hand, the procedure is the same, just use 2 instead of -2 as the exponent. That is, this inequality:

    f(x_1,...,x_n) ≥ 0

... is equivalent to this equation:

    f(x_1,...,x_n) = y^2

[enedil]

Trouble starts when your field doesn't always have a solution for f(x_1,...,x_n) = y^2, like rational numbers (but I'm sure you can find more examples). But maybe that can be mitigated as well?

[JadeNB]

I'm not an expert in this area, but, precisely for the reason you mention, I'd expect it's easier to solve a polynomial inequality over ℚ by solving it over ℝ and intersecting down to ℚ, rather than by working directly over ℚ.

[macrolocal]

As long as you don’t bump into Gal(ℝ/ℚ) somewhere along the way.