Upper convex hull ("inverse" Newton polygon)

[Christophe BAL]

Hello,

I would like to draw a kind of "inverse" Newton polygon for points.

How can I achieve this ?

Christophe

[Dima Pasechnik]

I guess you need to specify more details.

[Christophe BAL]

Ok, there is an applet showing on example of the Newton polygon here :

http://www.math.sc.edu/~filaseta/newton/newton.html

Don't focus on the definition of the Newton polygon but only on the graphic.

What I need to obtain is a similar result but from the "upper" point of viex, and the "lower" one. 

Is it more clear now ? If not, I will produce one picture by hand.

[Dima Pasechnik]

On Thursday, 18 October 2012 14:24:45 UTC+8, projetmbc wrote:

Ok, there is an applet showing on example of the Newton polygon here :

http://www.math.sc.edu/~filaseta/newton/newton.html

Don't focus on the definition of the Newton polygon but only on the graphic.

well, we can't focus on graphics, we're (mostly) mathematicians, not  graphics artists :-)

Also, that applet is apparently so old (the page says it's last modified in 1998!) that it doesn't work for me.

So better provide a definition of what you need...

 

What I need to obtain is a similar result but from the "upper" point of viex, and the "lower" one. 

Is it more clear now ? If not, I will produce one picture by hand.

OK, I'm glad that I asked, as this is not what 95% of people mean by a Newton polytope/polygon (and although I even published one paper in number theory --- funnily, with Filaseta, the author of the applet, it was a while ago, and this kind of definition didn't cross my mind).

Anyhow, you need a part of the convex hull of a bunch of points in the plane, right?

Surely, Sage can do convex hulls.

[Christophe BAL]

Hello.

One method could be the following one where L is a list of points. We note xMin and xMax the smallest and biggest abscissa respectively.  

1. Add two extra points (xMin , -inf) and (xMax , -inf) to the list L. We obtain the list LL.
2. Compute the convex hull of the list of points LL.
3. Remove the three lines with vertex (xMin , -inf)or (xMax , -inf). 
4. Draw only the remaining lines.

A good candidiate for (-inf) could simply be (yMin - 10) for example where yMin is the smallest y-coordinate.

Christophe

[Volker Braun]

Just compute the points and take their convex hull together with a ray in the "up" direction:

sage: R.<x> = ZZ[]

sage: p = sum( x^i for i in range(10) )

sage: points = [ (e, randint(0,10)) for e in p.exponents() ]   # replace with whatever y-coordinate you want

sage: q = Polyhedron(vertices=points, rays=[(0,1)])

sage: q

A 2-dimensional polyhedron in ZZ^2 defined as the convex hull of 4 vertices and 1 ray

(A ray in the direction (0, 1), A vertex at (0, 8), A vertex at (9, 6), A vertex at (2, 2), A vertex at (7, 3))

[Christophe BAL]

Tanks.

I don't like the output produced via q.plot(). Is there a way to obtain one basic line like line does ?

Christophe

[Volker Braun]

You can always roll your own if you don't like it. You can also try to take apart the pieces that do the plot, for example,

sage: proj = sage.geometry.polyhedron.plot.Projection(q)

sage: proj.render_outline_2d(color='red')

[Harald Schilly]

On Monday, October 22, 2012 11:09:51 AM UTC+2, Volker Braun wrote:

sage: q = Polyhedron(vertices=points, rays=[(0,1)])

I just opened the documentation here:
 http://www.sagemath.org/doc/reference/sage/geometry/polyhedron/constructor.html
because I wasn't sure what it is doing, and at the top the definition states:

• linear inequalities , and
• linear equations .

But, I think the second one is also a inequality and this is a bug, right?

H

[Volker Braun]

yes, should be equality.

Btw if anybody feels adventurous I have a big pile of trac tickets relating to polyhedra that are sitting there for 1+ years.

[Christophe BAL]

Thanks for this. 

2012/10/22 Volker Braun <vbrau...@gmail.com>

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