# newToricEmbedding -- constructs a ToricEmbedding

## Synopsis

• Usage:
TE = newToricEmbedding(polynomialList)
• Inputs:
• polynomialList, a list, with polynomials {f0,f1,f2,f3}
• Outputs:

## Description

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);`