Arnold's seminar
November 4, 2003

Oleg Karpenkov.
On bi-cubic characterization polynomials for classes
of 3x3 matrices with integer coefficients and distinct irrational
eigenvalues.

Abstract:
We recall the definition of the bi-quadratic characterization
polynomials for classes of 2x2 matrices and give the generalization
to the three dimensional case (Q_A(x,y,z;m,n)).
We show that every bi-cubic characterization polynomial is the product
of two polynomials of degree 3 with integer coefficients
It appears that one of these two polynomials
is uniquely defined by any matrix of the class whereas
the second one depends on the choice of the basis for the class.
There exist some class of Frobenius matrices conjugate to the given class A
iff there exist integer numbers (x,y,z;m,n) such that
Q_A(x,y,z;m,n)=1 or Q_A(x,y,z;m,n)=-1.
As a consequence we present the class of
3x3 matrices with integer coefficients and distinct irrational eigenvectors
that is not conjugate to any class containing Frobenius matrices.