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# 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 assumptions 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 : mixedVol = computeMixedVolume specializeSystem(p0,polys)

    o4 = 6

    i5 : monodromySolve(polys,p0,{x0},SelectEdgeAndDirection=>selectBestEdgeAndDirection, Potential=>potentialE, TargetSolutionCount=>mixedVol)

    o5 = (HomotopyNode{...7...}, 10)

    o5 : Sequence

## See also

- selectBestEdgeAndDirection -- selects edge and direction based on highest potential for obtaining new information
- potentialLowerBound -- the potential which is equal to the minimal number of new points guaranteed to be discovered
- computeMixedVolume -- compute mixed volume via Gfan
- Potential

## For the programmer

The object potentialE is a function closure.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/MonodromySolver/Documentation.m2:677:0.