Abstract
We count algebraic numbers of fixed degree d and fixed (absolute multiplicative Weil) height {mathcal {H}} with precisely k conjugates that lie inside the open unit disk. We also count the number of values up to {mathcal {H}} that the height assumes on algebraic numbers of degree d with precisely k conjugates that lie inside the open unit disk. For both counts, we do not obtain an asymptotic, but only a rough order of growth, which arises from an asymptotic for the logarithm of the counting function; for the first count, even this rough order of growth exists only if k in {0,d} or gcd (k,d) = 1. We therefore study the behaviour in the case where 0< k < d and gcd (k,d) > 1 in more detail. We also count integer polynomials of fixed degree and fixed Mahler measure with a fixed number of complex zeroes inside the open unit disk (counted with multiplicities) and study the dynamical behaviour of the height function.
Highlights
Of fixed degree and bounded height: Schanuel first proved, in [21], an asymptotic for the number of algebraic points of bounded height that are defined over a fixed number field
The same problem has been studied for integral points, i.e. elements of Qn whose coordinates are algebraic integers: In Theorem 5.2 in Chapter 3 of [15], Lang gives an asymptotic for the number of algebraic integers of bounded height that lie in a fixed number field
In [13], Grizzard and Gunther counted algebraic integers of fixed degree, fixed norm, and bounded height. This last result is somewhat related to our work in that the d-th power of the height of an algebraic integer of degree d with no conjugate inside the open unit disk is equal to the absolute value of its norm
Summary
In [1], Barroero extended the results of Lang and Chern and Vaaler to count algebraic integers of fixed degree over any fixed number field and bounded height. In [13], Grizzard and Gunther counted (among other things) algebraic integers of fixed degree (over Q), fixed norm, and bounded height This last result is somewhat related to our work in that the d-th power of the height of an algebraic integer of degree d (over Q) with no conjugate inside the open unit disk is equal to the absolute value of its norm. B(k, d) contains arbitrarily large elements and at least the limit superior and inferior corresponding to a(k, d) certainly exist We remark that it is not clear if the conjugates inside the open unit disk are the right thing to take into account here. |{α ∈ A(k, d, H); the Galois group of the normal closure of Q(α) acts primitively on the set of conjugates of α}| ≤ CH
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