added function that computes Newton polygons of L-polynomials

Author: stevehuang235

src/DeRham.jl

@@ -30,6 +30,7 @@ include("ExamplePolynomials.jl")
 
 include("ZetaFunction.jl")
 include("PointCounts.jl")
+include("NewtonPolygon.jl")
 
 # TODO: export Zeta Function functions
 

src/NewtonPolygon.jl

@@ -0,0 +1,64 @@
+"""
+    vp(a, p)
+    Returns the p-adic valuation of a 
+    
+    INPUTS: 
+    * "a" -- integer
+    * "p" -- integer, a prime number 
+"""
+function vp(a, p)
+    @assert is_prime(p)
+    if a == 0 
+        return Inf 
+    end 
+
+    e = 0 
+    while mod(a, p) == 0
+        a = div(a, p)
+        e = e + 1
+    end
+
+    return e
+end 
+
+"""
+    newton_polygon(f, p)
+    Returns the Newton polygon of f 
+
+    INPUTS: 
+    * "f" -- list, coefficients of a polynomial, constant term first
+    * "p" -- integer, a prime number
+"""
+function newton_polygon(f, p)
+    @assert is_prime(p)
+
+    pts = []
+    for (i, a) in enumerate(f)
+        if a != 0
+            push!(pts, (i-1, vp(a, p)))
+        end 
+    end 
+
+    # compute the lower convex hull of points
+    hull = Tuple{Int64, Int64}[]
+    for pt in pts
+        while length(hull) >= 2
+            (x1, y1) = hull[end-1]
+            (x2, y2) = hull[end]
+            (x3, y3) = pt
+            # Compute cross product to decide whether pt is "below" the last segment.
+            cp = (x2 - x1) * (y3 - y1) - (y2 - y1) * (x3 - x1)
+            if cp <= 0
+                pop!(hull)
+            else
+                break
+            end
+        end
+        push!(hull, pt)
+    end
+    return SlopesPolygon(hull)
+   #return hull
+end 
+
+
+

src/SlopesPolygon.jl

@@ -103,18 +103,22 @@ end
 """
 Given an array of vertices
 """
-function SlopesPolygon(vertices::Array{Tuple{Int,Int}})
+#function SlopesPolygon(vertices::Array{Tuple{Int,Int}})
+function SlopesPolygon(vertices::Vector{Tuple{Int64, Int64}})
     #TODO implement
 
     n = length(vertices)-1
     slopelengths = zeros(Int,n)
     slopes = zeros(Rational{Int},n)
-    for i = 2:n
+    for i = 2:n+1
         slopevec = vertices[i] .- vertices[i-1]
-        push!(slopes, slopevec[2] // slopevec[1])
-        push!(slopelengths,slopevec[2])
+        #push!(slopes, slopevec[2] // slopevec[1])
+        #push!(slopelengths,slopevec[2])
+        slopes[i-1] = slopevec[2] // slopevec[1]
+        slopelengths[i-1] = slopevec[2]
     end
-
+    println(slopelengths)
+    println(slopes)
     SlopesPolygon(slopes,slopelengths,values(slopes,slopelengths)...)
 end