- Usage:
`TE = newToricEmbedding(polynomialList)`

- Inputs:
`polynomialList`, a list, with polynomials {f0,f1,f2,f3}

- Outputs:
`degH`, an object of class ToricEmbedding

Given a list of four polynomials {f0,f1,f2,f3} in a polynomial ring in two variables, namely QQ[s,t] we can compute its Newton polytope, namely, theconvexHull of the union of the Newton polytope of all the polynomial in the list, with the methodpolynomialsToPolytope. Take N= polynomialsToPolytope ({f0,f1,f2,f3}). ThenToricEmbedding of {f0,f1,f2,f3} can be defined asToricEmbedding of N.

Given a polynedron P we construc the ambient ring for building the coordinate ring of the toric variety associated to P. Precisely, assume P has p+1 lattice points, this is length latticePoints P = p+1, and denote with T_0 ... T_p the lattice points of P. Consider the ring QQ[T_0..T_p]. Let J be the toric ideal of P, then newToricEmbedding provides the ring 'QQ[s,t,u,v,w,T_0..T_p,MonomialOrder=>Eliminate 5]' which has all the information of the coordinate ring QQ[T_0..T_p]/J.

METHODS: teLattice, teLatticeLen,teRing

teLattice: returns the list of lattice points of the polytope associated to the toric embedding. Precisely, when tEmb = newToricEmbedding(P), tEmb#teLattice = latticePoints P.

teLatticeLen: returns the length list ''of lattice points of the polytope associated to the toric embedding. Precisely, when tEmb = newToricEmbedding(P), tEmb#teLattice = length latticePoints P.

teRing: returns the ring 'QQ[s,t,u,v,w,T_0..T_tEmb#teLatticeLen,MonomialOrder=>Eliminate 5]'.

i1 : S = QQ[s,t]; |

i2 : f0 = s^2+s^3*t; |

i3 : f1 = s^3*t^6+1; |

i4 : f2 = s*t^2+2*s^3*t^5; |

i5 : f3 = s^2+s^3*t^6; |

i6 : l = {f0,f1,f2,f3}; |

i7 : teFromPolyList = newToricEmbedding(l); |

- teToricRing -- Returns the coordinate ring of the toric variety
- teToricRationalMap -- Computes the rational map G defined by {f0,f1,f2,f3} over the toric coordinate ring associated to a polytope P
- polynomialsToPolytope -- Return the convexHull of the union of the Newton polytope of all the polynomial in the list

- newToricEmbedding(List)