The ‘gamma trick’ refers to the following idea:
If the solutions of the start system are regular, then we avoid singular solutions along the paths by multiplying the start system in the homotopy with a random complex constant gamma. This option allows the user to give a specific value of this gamma constant.
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); using temporary files /var/folders/40/dy88l5qd361391m_3v2m51bm0000gn/T/M2-44657-0/0PHCinput and /var/folders/40/dy88l5qd361391m_3v2m51bm0000gn/T/M2-44657-0/0PHCoutput |
i4 : fsols = trackPaths(f,q,qsols,gamma => exp(ii*pi/3)) using temporary files /var/folders/40/dy88l5qd361391m_3v2m51bm0000gn/T/M2-44657-0/3PHCoutput and /var/folders/40/dy88l5qd361391m_3v2m51bm0000gn/T/M2-44657-0/5PHCtargetsols o4 = {{1.33076-.335184*ii, -.62414+.513163*ii}, {-.764107, -.544612}, ------------------------------------------------------------------------ {.742585-.425943*ii, .270685+1.00715*ii}, {.742585+.425943*ii, ------------------------------------------------------------------------ .270685-1.00715*ii}, {1.33076+.335184*ii, -.62414-.513163*ii}, ------------------------------------------------------------------------ {-.894935+.624334*ii, .143333-1.14868*ii}, {-.894935-.624334*ii, ------------------------------------------------------------------------ .143333+1.14868*ii}, {-1.59272, .964857}} o4 : List |