This contains various files associated to my doctoral dissertation.

sage

This is a directory containing sage code that was included in the appendices of my doctoral dissertation.

amoeba.sage

This file creates the reduced A-discriminant contour for n-variate, (n+3)-nomials. Code to generate the contour for the cubic family, a+bx+cx^2+dx^3 is:

runfile "amoeba.sage"
A = [[0,1,2,3]]
a = amoeba(A, 1/100)
show(a, xmin=-10, xmax=10, ymin=-10, ymax=10)

For a multivariate family, like ax^6+by^3+cy+dty^6+etx^3+ftx in Q_p[x,y,t] use

A = [[6,0,0,0,3,1],[0,3,1,6,0,0],[0,0,0,1,1,1]]

trop2d.sage

This file creates the reduced p-adic A-discriminant amoeba for n-variate, (n+3)-nomials. Code to generate the amoeba for the cubic family, a+bx+cx^2+dx^3 when p=3 is:

runfile "trop2d.sage"
A = [[0,1,2,3]]
a = amoeba(A, 3)
show(a, xmin=-4, xmax=25, ymin=-25, ymax=4)

For a multivariate family, like ax^6+by^3+cy+dty^6+etx^3+ftx in Q_p[x,y,t] use

A = [[6,0,0,0,3,1],[0,3,1,6,0,0],[0,0,0,1,1,1]]

There are a couple of helper functions:

The function get_point will give the point a polynomial would map to.

def get_point(A, poly, p = None)

A is the same list as before, poly is the coefficient vector of a representative polynomial and p is the prime. If p is None then it is expected that poly is the component-wise valuation of the representative polynomial.

Finding a polynomial that maps to a particular point is more difficult. The function get_polynomial is an inverse of get_point. (There are several polynomials mapping to each point.) It attempts to get a polynomial that maps to a given point.

def get_polynomial(A, point, p = None)

Again,A is the support. Now we search for a polynomial (coefficient vector) that maps to point and p is the prime we are working with. If p is None then the return value is a collection of valuations, rather than the actual coefficients. This function does not always succeed, and will print a failure message and return None if it fails.

tropdd.sage

This file creates the reduced p-adic A-discriminant amoeba for n-variate, (n+k)-nomials for general k. Code to generate the amoeba for the quartic family (k=4) when p=3 in the box [-10,10]^3 is:

runfile "tropdd.sage"
A = [[0,1,2,3,4]]
box = Polyhedron(list(get_verts(10,3)))
a = amoeba(A, 3)
show(sum(lambda b: box.intersection(b).show(), a))