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Recognizing Salem numbers of degree 6

A Salem number

of degree 6 has a minimal polynomial of the following form:

where a,b and c are integers. The trace of

, where the sum is over the six conjugates of

, i.e., the six roots of (2.1). Two of these conjugates are

and

and the remaining conjugates satisfy

, so a bound on |a| implies a bound on

and hence on |b| and |c|. So, for fixed a, there is a finite set of

with

. Observe that

for all Salem numbers of degree 6. In fact, there are no such numbers with trace -1, 4 with trace 0, 15 with trace 1, and 39 with trace 2.

It is not difficult to recognize Salem numbers of degree 6. Since is reciprocal, we can write

where

The zeros of U are the numbers

and hence U must have two roots in the open interval -2 < x < 2 and one root with x > 2. This is equivalent to the following requirements: (i)

, (ii)

has real roots, the smaller of which,

satisfies

, and

. We used this criterion in compiling our list of Salem numbers.

We also require U to be irreducible. Since U is cubic, it suffices that for any integer n. Only a finite set of n need be checked since, by a well-known estimate of Cauchy, for . It is easy to see that P can only factor into factors of even degree since the roots and must belong to the same factor, or else there would be a factor the product of whose roots is in absolute value less than 1, which is clearly impossible. Thus, P is irreducible if and only if U is irreducible. The following useful result is an elementary deduction from the above discussion.

Lemma 2.1

Let P be as in (2.1) and U as in (2.3). A necessary condition for P to be the minimal polynomial of a Salem number is that

for . A sufficient condition for P to be the minimal polynomial of a Salem number is, in addition to (2.4), any one of , or .

In general, if P is the minimal polynomial of a Salem number of degree d = 2s, then there is a monic polynomial U with integer coefficients for which . The numbers are the zeros of U repeated twice each. The following computation gives a useful expression for :

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