The Newton Polytope of the Implicit Equation

Abstract

We apply tropical geometry to study the image of a map defined by Laurent polynomials with generic coefficients. If this image is a hypersurface then our approach gives a construction of its Newton polytope.2000 Mathematics Subject Classification. 13P10, 14Q99, 52B20, 68W30.

Computing the Newton Polygon of the Implicit Equation

“…According to [20], "a priori knowledge of the Newton polytope would greatly facilitate the subsequent computation of recovering the coefficients of the implicit equation […] This is a problem of numerical linear algebra …". Reducing implicitization to linear algebra is also the premise of [1,8].…”

“…Tropical geometry can also give the implicit polytope of any hypersurface parameterized by Laurent polynomials [3,20]. In [22], they consider this polytope as the mixed fiber polytope of the input polytopes, while software TrIm computes it.…”

“…In [22], they consider this polytope as the mixed fiber polytope of the input polytopes, while software TrIm computes it. For curves, the implicit polygon is described in [20,Example 1.1], without any hypothesis of genericity. The approach handles rational parameterizations with the same denominator by homogenizing the parameter as well as the implicit space.…”

The Newton Polygon of a Rational Plane Curve

“…In this direction, Dickenstein, Feichtner, Sturmfels and Tevelev have determined the abstract tropical variety associated to a hypersurface parameterized by products of linear forms, which includes the case of curves [3,19,23]. Besides, Sturmfels, Tevelev and Yu have computed the abstract tropical variety of a hypersurface parameterized by generic Laurent polynomials in any number of variables [19][20][21].…”

“…As an application of Theorem 1.1 above, we recover the Newton polygon of the generic Laurent polynomial parametrization of dimension 1 obtained in [20] but this time with explicit genericity conditions. Proofs of the following statements can be found in Sect.…”