bug in Pisot number test?

[drewitz]

For the [3,5,3] orthoscheme, CoxIter says the growth rate is a Salem number but not a Pisot number. This is not possible and afaik, it is indeed both:

$ ./coxiter < ../graphs/simplices/3-jkrt_347-02-J3.coxiter 
Reading graph: 
    Number of vertices: 4
    Dimension: 3
    Vertices: 1, 2, 3, 4
    Field generated by the entries of the Gram matrix: Q[sqrt(5)]
File read

Finding connected subgraphs......
Finding graphs products......
Computations......
    Computation time: 0.00967261s

Information
    Cocompact: yes
    Finite covolume: yes
    f-vector: (4, 6, 4, 1)
    Number of vertices at infinity: 0
    Alternating sum of the components of the f-vector: 2
    Euler characteristic: 0
    Signature (numerically): 3,1,0

Growth series: 
f(x) = C(2,2,2,3,6,10)/(1 - 2 * x + x^2 - x^4 + 2 * x^5 - 2 * x^6 + x^7 - x^9 + 2 * x^10 - x^11)

Growth rate: 1.3509803377162373102114035730633264361
    Perron number: yes
    Pisot number: no
    Salem number: yes

[rgugliel]

I'll look into this, hopefully later on this week.

Thanks for raising this issue.

[rgugliel]

Hey Simon, So we have the minimal polynomial of the growth rate which is

x^10 - x^9 - x^6 + x^5 - x^4 - x + 1

The repartition of its roots is the following:

• lambda =~ 1.35098
• its inverse =~ 0.74
• 6 complex roots

If I'm not mistaken, all the complex roots lie on the unit circle and not inside (reason why it is a Salem number but not a Pisot number).

What do you think?

[drewitz]

Hi Rafael, yes, you're right. I had learned a wrong "definition" of Pisot numbers which implied that every Salem number is a Pisot number and trusted the source so that I didn't double check the definition. Sorry about that.