tDegree -- the degree of the continuation parameter

Description

By default, the homotopy is created with tDegree equal to 2. This option allows the user to give another value for tDegree; e.g., a linear homotopy between start system q and target system f is of the form (1-t)*q + t*f.

In general, if tDegree is k, then the homotopy is of the form (1-t)^k*q + t^k*f.

A reason for changing the tDegree would be the following: higher degree homotopies ensure that the system doesn't change as fast in the beginning and at the end of the homotopy; that is, they force smaller step sizes in the beginning and end. The default value, 2, is usually sufficient.

i1 : R = CC[x,y];

i2 : f = { x^3*y^5 + y^2 + x^2*y, x*y + x^2 - 1};

i3 : (m,q,qsols) = mixedVolume(f,StartSystem=>true);

i4 : fsols = trackPaths(f,q,qsols,tDegree => 1)

o4 = {{-.894935-.624334*ii, .143333+1.14868*ii}, {-1.59272, .964857},
     ------------------------------------------------------------------------
     {.742585-.425943*ii, .270685+1.00715*ii}, {1.33076+.335184*ii,
     ------------------------------------------------------------------------
     -.62414-.513163*ii}, {1.33076-.335184*ii, -.62414+.513163*ii},
     ------------------------------------------------------------------------
     {-.764107, -.544612}, {.742585+.425943*ii, .270685-1.00715*ii},
     ------------------------------------------------------------------------
     {-.894935+.624334*ii, .143333-1.14868*ii}}

o4 : List

Functions with optional argument named tDegree :

trackPaths(...,tDegree=>...) -- Option to specify the degree of the continuation parameter

For the programmer

The object tDegree is a symbol.