Dominant balance for algebraic roots ($t={ϵ}^{p}s$)

For a polynomial depending on a small parameter, substitute $t={ϵ}^{p}s$ and compare the exponents of the parameter. Candidate root scales arise when two or more lowest exponents balance. Solve the leading scaled equation and expand about each of its nonzero roots. Different balances can reveal different families of roots; real-root counts must be checked separately.

A polynomial whose degree decreases when its parameter vanishes may have roots escaping to infinity. Such a singular perturbation root is found by balancing powers after rescaling $\lambda ={\epsilon }^{-p}\mu$. If the root enters an exponential, even a lower-order additive root correction can matter on a fixed interval when its leading exponential grows. This phenomenon is different from a repeated root of a fixed polynomial.

If $P({\lambda }_0,0)=0$ and $P_{\lambda }({\lambda }_0,0)\ne 0$, the implicit function theorem gives a local root branch. For analytic coefficients it has an ordinary power series in the perturbation parameter. Its corrections follow by substituting that series into the polynomial and comparing coefficients. A vanishing leading root may require its first nonzero correction to capture a slow exponential mode.