We can use this to write the general solution of this regular perturbation theory problem as,
tilde(x)ᵏ=∑ₙ₌₀[infinity]aₙᵏεlonⁿ
and matching coefficients, after some algebra we can find the 3 roots to any order in εlon, giving
x¹=1+εlon/2+5/8εlon²+dots,
x²=-1+εlon/2-5/8εlon²+dots,
x³=1/εlon-εlon-2εlon³+dots
Problem: Consider now a more sophisticated version of this toy singular perturbation algebraic problem,
εlon²x⁶-εlonx⁴-x³+8=0
The naive scaling in εlon→0 limit, gives x³=8, which has the three solutions x=2,2ω,2ω², with ω-=e²πi/3 a complex root of unity (ω³3=1). But of
course this polynomial has six roots, so three of them have disappeared. Consider balancing two terms of this equation as the dominant ones
in the εlon→0 limit. Since there are four terms, there are six possible pairings to consider. Find the consistent ones.