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Dear GAP-Forum,

Dr. Keith M. Briggs asked:

I would be grateful for any hints for a solution to this problem:

given an irreducible cubic polynomial p with integer coefficients

and three real roots, find a symmetric (preferably unimodular)

integer matrix with p as its characteristic polynomial, if such exists.

I presume this will involve transformations of the companion matrix.

Alternatively, it would be sufficient to have such a matrix whose

charpoly generates the same field as p.

This appears to be a hard (number theoretic) problem, and GAP does

not seem to help much in its solution. As the mail indicates, there need

not exist a symmetric integer matrix with the given polynomial p as its

characteristic polynomial (an example of such a p is: x^3 - 4x - 1).

I do not know of any method to decide whether a given p as above

is the characteristic polynomial of a symmetric integer matrix.

Best wishes, Gerhard Hiss -- Gerhard Hiss Lehrstuhl D fuer Mathematik, RWTH Aachen Templergraben 64, 52062 Aachen Tel.: (+49) (0) 241 / 80-94543

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