# potentialE -- the "expected" potential of an edge

## Description

This is an option for the Potential option for monodromySolve when we use selectBestEdgeAndDirection option to select edge and direction. This option computes the expected number of new points obtained by tracking points (under suitable randomness asumptions about the permutations generated by the underlying graph.) The expected value is computed by the ratio of unmatched points and the difference between the total solution count and the number of the known points.

 i1 : R = CC[a,b,c,d,e,f,g,h][x,y,z]; i2 : polys = polySystem {a*x+b*y+c*z,d*x*y+e*x*z+f*y*z,g*x*y*z+h};

In here, we need the target number of solutions, and we will use the mixed volume for that.

 i3 : (p0,x0) := createSeedPair polys o3 = ({-.395781-.24074*ii, 1.15221-.036791*ii, -.339811+.226642*ii, ------------------------------------------------------------------------ .225376-.140718*ii, -.324066-.135951*ii, .509177+.360148*ii, ------------------------------------------------------------------------ .287657+.283636*ii, .130507+.183897*ii}, {.892712+.673395*ii, ------------------------------------------------------------------------ .29398+.632944*ii, .025888+.714827*ii}) o3 : Sequence

We will comput the mixed volume to find the number of solution counts.

 i4 : mixedVolume = computeMixedVolume specializeSystem(p0,polys) o4 = 6 i5 : monodromySolve(polys,p0,{x0},SelectEdgeAndDirection=>selectBestEdgeAndDirection, Potential=>potentialE, TargetSolutionCount=>mixedVolume) o5 = (HomotopyNode{...7...}, 10) o5 : Sequence