Regularize II

The underlying idea of regularization is to recover the roots that are lost in a singular limit. When some coefficients of a polynomial are multiplied by powers of a small parameter , the limiting polynomial obtained by setting may have fewer roots than the original polynomial. In this situation, some roots have escaped to a different scale and are therefore invisible in the limit.

To recover these missing roots, we introduce a rescaling

where the exponent is chosen so that at least two dominant terms of the polynomial balance each other. Geometrically, this choice is determined from the Newton polygon (or Newton diagram) by examining the exponents of . After an additional normalization by a factor , one obtains a new limiting polynomial in the variable . This reduced polynomial captures the asymptotic behavior of the roots on the corresponding scale and typically restores the correct number of roots.

In this sense, the singularity is not an intrinsic property of the problem but rather the consequence of observing the problem at an inappropriate scale.

As a simple example, consider

Setting gives

which has only one root, although the original polynomial is quadratic and therefore has two roots. To recover the missing root, let

Substituting into the polynomial gives

Multiplying by yields

Taking the limit gives the regular polynomial

which has two roots. One corresponds to the finite root seen in the original limit, while the second reveals a root located on the larger scale . The rescaling has therefore recovered the root that was hidden in the singular limit.