Mathematics in Computer Science

, Volume 8, Issue 2, pp 235–251 | Cite as

Newton Polytopes and Witness Sets

  • Jonathan D. Hauenstein
  • Frank SottileEmail author


We present two algorithms that compute the Newton polytope of a polynomial f defining a hypersurface \({\mathcal{H}}\) in \({\mathbb{C}^n}\) using numerical computation. The first algorithm assumes that we may only compute values of f—this may occur if f is given as a straight-line program, as a determinant, or as an oracle. The second algorithm assumes that \({\mathcal{H}}\) is represented numerically via a witness set. That is, it computes the Newton polytope of \({\mathcal{H}}\) using only the ability to compute numerical representatives of its intersections with lines. Such witness set representations are readily obtained when \({\mathcal{H}}\) is the image of a map or is a discriminant. We use the second algorithm to compute a face of the Newton polytope of the Lüroth invariant, as well as its restriction to that face.


Hypersurface Polynomial system Newton polytope Numerical algebraic geometry Witness set 

Mathematics Subject Classification (1991)

14Q15 65H10 


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© Springer Basel 2014

Authors and Affiliations

  1. 1.Department of Applied and Computational Mathematics and StatisticsUniversity of Notre DameNotre DameUSA
  2. 2.Department of MathematicsTexas A&M UniversityCollege StationUSA

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