3 ms·
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 nsajko 3y ago
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 3y agoTrouble 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 3y agoI'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 3y agoAs long as you don’t bump into Gal(ℝ/ℚ) somewhere along the way.