Scaling polynomial systems and solutions

By scaling  the coefficients are transformed so that they do not have extreme values. The purpose is to avoid numerical problems.

Equation Scaling 

means that every polynomial is divided by its average coefficient.

Variable Scaling 

uses transformations like z = (2^c) x. The transformation is such that real solutions remain real. The inverse of the condition number of the linear system that is solved to set up this transformation gives an indication on the condition of the original polynomial system.

Solution Scaling 

transforms the solutions of a scaled system back into the original coordinate system. Note that in the original coordinates, the solutions can be ill-conditioned.