The artificial-parameter homotopy has the following form:

with a start system, and the target system.**The homotopy parameter***k*- determines the power of the continuation
parameter
*t*. Taking*k*>1 has as effect that at the beginning and at the end of the continuation, changes in*t*do not change the homotopy as much as with a homotopy that is linear in*t*so that paths are better to follow. The default value*k*=2 is recommended. **The homotopy parameter***a*- ensures the accessibility and regularity of the solution
paths, i.e.: by choosing a random complex number for
*a*, all paths are regular and do not diverge for*t*<1. **The target value**- for the continuation parameter
*t*is by default 1. To create stepping stones in the continuation stage, it is possible to let the continuation stop at*t*<1, for instance at*t*= 0.9 or even at a complex value for*t*. The solutions at*t*<1 will serve at start solutions to take up the continuation in a later stage. In this stage, the same homotopy parameters k and a must be used. **A projective transformation**- of the homotopy and start solutions makes the equations in the polynomials homogeneous and adds a random hyperplane. The vectors of the start solutions are extended with an additional unknown. For solutions at infinity, this additional unknown equals zero.