Newton Polygons¶
This module implements finite Newton polygons and infinite Newton polygons having a finite number of slopes (and hence a last infinite slope).

sage.geometry.newton_polygon.
NewtonPolygon
= Parent for Newton polygons¶

class
sage.geometry.newton_polygon.
NewtonPolygon_element
(polyhedron, parent)¶ Bases:
sage.structure.element.Element
Class for infinite Newton polygons with last slope.

last_slope
()¶ Returns the last (infinite) slope of this Newton polygon if it is infinite and
+Infinity
otherwise.EXAMPLES:
sage: from sage.geometry.newton_polygon import NewtonPolygon sage: NP1 = NewtonPolygon([ (0,0), (1,1), (2,8), (3,5) ], last_slope=3) sage: NP1.last_slope() 3 sage: NP2 = NewtonPolygon([ (0,0), (1,1), (2,5) ]) sage: NP2.last_slope() +Infinity
We check that the last slope of a sum (resp. a product) is the minimum of the last slopes of the summands (resp. the factors):
sage: (NP1 + NP2).last_slope() 3 sage: (NP1 * NP2).last_slope() 3

plot
(**kwargs)¶ Plot this Newton polygon.
Note
All usual rendering options (color, thickness, etc.) are available.
EXAMPLES:
sage: from sage.geometry.newton_polygon import NewtonPolygon sage: NP = NewtonPolygon([ (0,0), (1,1), (2,6) ]) sage: polygon = NP.plot()

reverse
(degree=None)¶ Returns the symmetric of
self
INPUT:
degree
– an integer (default: the top right abscissa of this Newton polygon)
OUTPUT:
The image this Newton polygon under the symmetry ‘(x,y) mapsto (degreex, y)`
EXAMPLES:
sage: from sage.geometry.newton_polygon import NewtonPolygon sage: NP = NewtonPolygon([ (0,0), (1,1), (2,5) ]) sage: NP2 = NP.reverse(); NP2 Finite Newton polygon with 3 vertices: (0, 5), (1, 1), (2, 0)
We check that the slopes of the symmetric Newton polygon are the opposites of the slopes of the original Newton polygon:
sage: NP.slopes() [1, 4] sage: NP2.slopes() [4, 1]

slopes
(repetition=True)¶ Returns the slopes of this Newton polygon
INPUT:
repetition
– a boolean (default:True
)
OUTPUT:
The consecutive slopes (not including the last slope if the polygon is infinity) of this Newton polygon.
If
repetition
is True, each slope is repeated a number of times equal to its length. Otherwise, it appears only one time.EXAMPLES:
sage: from sage.geometry.newton_polygon import NewtonPolygon sage: NP = NewtonPolygon([ (0,0), (1,1), (3,6) ]); NP Finite Newton polygon with 3 vertices: (0, 0), (1, 1), (3, 6) sage: NP.slopes() [1, 5/2, 5/2] sage: NP.slopes(repetition=False) [1, 5/2]

vertices
(copy=True)¶ Returns the list of vertices of this Newton polygon
INPUT:
copy
– a boolean (default:True
)
OUTPUT:
The list of vertices of this Newton polygon (or a copy of it if
copy
is set to True)EXAMPLES:
sage: from sage.geometry.newton_polygon import NewtonPolygon sage: NP = NewtonPolygon([ (0,0), (1,1), (2,5) ]); NP Finite Newton polygon with 3 vertices: (0, 0), (1, 1), (2, 5) sage: v = NP.vertices(); v [(0, 0), (1, 1), (2, 5)]


class
sage.geometry.newton_polygon.
ParentNewtonPolygon
¶ Bases:
sage.structure.parent.Parent
,sage.structure.unique_representation.UniqueRepresentation
Construct a Newton polygon.
INPUT:
arg
– a list/tuple/iterable of vertices or of slopes. Currently, slopes must be rational numbers.sort_slopes
– boolean (default:True
). Specifying whether slopes must be first sortedlast_slope
– rational or infinity (default:Infinity
). The last slope of the Newton polygon
OUTPUT:
The corresponding Newton polygon.
Note
By convention, a Newton polygon always contains the point at infinity \((0, \infty)\). These polygons are attached to polynomials or series over discrete valuation rings (e.g. padics).
EXAMPLES:
We specify here a Newton polygon by its vertices:
sage: from sage.geometry.newton_polygon import NewtonPolygon sage: NewtonPolygon([ (0,0), (1,1), (3,5) ]) Finite Newton polygon with 3 vertices: (0, 0), (1, 1), (3, 5)
We note that the convex hull of the vertices is automatically computed:
sage: NewtonPolygon([ (0,0), (1,1), (2,8), (3,5) ]) Finite Newton polygon with 3 vertices: (0, 0), (1, 1), (3, 5)
Note that the value
+Infinity
is allowed as the second coordinate of a vertex:sage: NewtonPolygon([ (0,0), (1,Infinity), (2,8), (3,5) ]) Finite Newton polygon with 2 vertices: (0, 0), (3, 5)
If last_slope is set, the returned Newton polygon is infinite and ends with an infinite line having the specified slope:
sage: NewtonPolygon([ (0,0), (1,1), (2,8), (3,5) ], last_slope=3) Infinite Newton polygon with 3 vertices: (0, 0), (1, 1), (3, 5) ending by an infinite line of slope 3
Specifying a last slope may discard some vertices:
sage: NewtonPolygon([ (0,0), (1,1), (2,8), (3,5) ], last_slope=3/2) Infinite Newton polygon with 2 vertices: (0, 0), (1, 1) ending by an infinite line of slope 3/2
Next, we define a Newton polygon by its slopes:
sage: NP = NewtonPolygon([0, 1/2, 1/2, 2/3, 2/3, 2/3, 1, 1]) sage: NP Finite Newton polygon with 5 vertices: (0, 0), (1, 0), (3, 1), (6, 3), (8, 5) sage: NP.slopes() [0, 1/2, 1/2, 2/3, 2/3, 2/3, 1, 1]
By default, slopes are automatically sorted:
sage: NP2 = NewtonPolygon([0, 1, 1/2, 2/3, 1/2, 2/3, 1, 2/3]) sage: NP2 Finite Newton polygon with 5 vertices: (0, 0), (1, 0), (3, 1), (6, 3), (8, 5) sage: NP == NP2 True
except if the contrary is explicitly mentioned:
sage: NewtonPolygon([0, 1, 1/2, 2/3, 1/2, 2/3, 1, 2/3], sort_slopes=False) Finite Newton polygon with 4 vertices: (0, 0), (1, 0), (6, 10/3), (8, 5)
Slopes greater that or equal last_slope (if specified) are discarded:
sage: NP = NewtonPolygon([0, 1/2, 1/2, 2/3, 2/3, 2/3, 1, 1], last_slope=2/3) sage: NP Infinite Newton polygon with 3 vertices: (0, 0), (1, 0), (3, 1) ending by an infinite line of slope 2/3 sage: NP.slopes() [0, 1/2, 1/2]
Be careful, do not confuse Newton polygons provided by this class with Newton polytopes. Compare:
sage: NP = NewtonPolygon([ (0,0), (1,45), (3,6) ]); NP Finite Newton polygon with 2 vertices: (0, 0), (3, 6) sage: x, y = polygen(QQ,'x, y') sage: p = 1 + x*y**45 + x**3*y**6 sage: p.newton_polytope() A 2dimensional polyhedron in ZZ^2 defined as the convex hull of 3 vertices sage: p.newton_polytope().vertices() (A vertex at (0, 0), A vertex at (1, 45), A vertex at (3, 6))

Element
¶ alias of
NewtonPolygon_element