The Luroth invariant is a hypersurface in the 15 coefficients of a plane quartic which cut out all quartics which are "Luroth". A quartic is Luroth whenever it goes through the 10 intersection points of a configuration of five lines. Such as

The 15 coordinates of a plane quartic are sorted as follows x^4 x^3y x^3z x^2y^2 x^2yz x^2z^2 xy^3 xy^2z xyz^2 xz^3 y^4 y^3z y^2z^2 yz^3 z^4

• We are interested in determining the Luroth Polytope which is the Newton polytope of the Luroth invariant.
• The oracle files used for these experiments are input, start, startingPointFile, start_parameters, and targets. Using NumericalNP.m2 we discovered that the directions
• v1={4,2,3,0,1,2,-2,-1,0,1,-4,-3,-2,-1,0}
• v2={4,3,2,2,1,0,1,0,-1,-2,0,-1,-2,-3,-4}
• v3={1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
• expose the entire polytope, implying that the 12 dimensional intersection of the orthogonal hyperplanes to these vectors, L, contains the Luroth polytope. The Luroth polytope is, in fact, 12 dimensional and we have naively searched random directions in R^15, projected onto L.
• Querying directions is easy after the oracle files have been created. For example, to query in the direction {-1,-1,3,-1,-1,-1,-1,-1,-1,-1,2,-1,-1,-1,1} one uses the command
• oracleQuery({-1,-1,3,-1,-1,-1,-1,-1,-1,-1,2,-1,-1,-1,1},OracleLocation=>"Luroth",MakeSageFile=>true)
• which returns the vertex {0,0,24,0,0,0,0,0,0,0,18,0,0,0,12,0} and produces the animation below.

• After 8582 random oracle queries, there were 3272 oracle queries which exposed a vertex. For the rest of the queries, we did not observe convergence.
• These 3272 queries revealed 760 distinct vertices.
• Of these 760 distinct vertices, 7 of them are not contained in L and thus must be incorrect answers (this algorithm is a two-sided Monte Carlo algorithm, so incorrect answers do happen with small probability).
• There is an action of S_3 on the coordinates of a plane quartic which extends to an action on the Newton polytope.
• We have found a total of 301 orbits containing 1713 vertices.