alphaCertified v1.2.0 (August 15, 2011)
  Jonathan D. Hauenstein and Frank Sottile
         GMP v4.3.1 & MPFR v2.3.1


Please note that all coefficients must be complex rational numbers.

alphaCertified is using the polynomial certification algorithms.

Analyzing 4 points using exact arithmetic.

Isolating 4 approximate solutions.

Classifying 4 distinct approximate solutions.

Rational certification results:

Number of points tested:           4
Certified approximate solutions:   4
Certified distinct solutions:      4
Certified real distinct solutions: 2

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The following files have been created:

approxSolns:          A list of points that are certified approximate solutions.
constantValues:       A list of the values of alpha, beta, and gamma for the points.
distinctSolns:        A list of points that correspond to distinct solutions.
isApproxSoln:         A list which describes if the ith point is an approximate solution.
isDistinctSoln:       A list which describes if the ith point is listed in 'distinctSolns'.
isRealSoln:           A list which describes if the ith point corresponds to a real solution.
nonrealDistinctSolns: A list of points that correspond to distinct nonreal solutions.
realDistinctSolns:    A list of points that correspond to distinct real solutions.
redundantSolns:       A list of points that correspond to the same solution as one in 'distinctSolns'.
refinedPoints:        A list of points that are the best internally computed approximation of each solution.
summary:              A human-readable summary for each point - main output file.
unknownPoints:        A list of points that which cannot be certified as approximate solutions.
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