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PHCpack :: gamma

gamma -- the constant in the gamma trick, optional input for trackPaths


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);
i4 : fsols = trackPaths(f,q,qsols,gamma => exp(ii*pi/3))

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

o4 : List

Reference: A.J. Sommese, J. Verschelde, and C.W. Wampler. Introduction to numerical algebraic geometry. In: Solving Polynomial Equations. Foundations, Algorithms and Applications, volume 14 of Algorithms and Computation in Mathematics, pages 301-337. Springer-Verlag, 2005.

Functions with optional argument named gamma :

For the programmer

The object gamma is a symbol.