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The precise question is the following:

Question: Can one reasonably bound the number of algebraic integers $\alpha$ of degree at most $k$ - that means there exists a monic integer polynomial $p$ with $\deg(p) \leq k$ and $p(\alpha)=0$ - and such that $p(\beta)=0$ implies $|\beta| \leq n$, i.e. all Galois conjugates of $\alpha$ have a modulus bounded by $n$.

Obviously, the number of relevant polynomials $p(t) = \sum_{i = 0}^k a_{k-i} t^i$ is bounded since $$|a_i| \leq {{k}\choose {i}} \cdot n^i$$ In particular, the number of such $\alpha$ is finite and one obtains a crude upper bound. One can also make a packing argument by observing that the distance between any two such algebraic integers cannot be too small. I am basically asking whether there are better bounds.

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1 Answer

Schanuel, S. On heights in number fields. Bull. Amer. Math. Soc. 70 1964 262–263.

Masser, David; Vaaler, Jeffrey D., Counting algebraic numbers with large height. II. Trans. Amer. Math. Soc. 359 (2007), no. 1, 427–445

Edit: As per Kevin's request, more context. The first reference is the basic result in the field and the second is the state of the art. The bound can be improved as expected. The second reference also has lots of additional references. Link to abstract and references (and also paper if your institution subscribes):

http://www.ams.org/journals/tran/2007-359-01/S0002-9947-06-04115-8/home.html

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Are these related? Do they answer the question affirmatively, or suggest that nothing better is known? A little indication of what's in the articles would make the answer much more useful to us lurkers. –  Kevin O'Bryant Nov 26 '10 at 15:28
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Both papers count algebraic numbers of bounded height which is slightly different from what you want. (The height of $\alpha$, with conjugates $\alpha_j$ ($1\leq j\leq d$) is the logarithm of the Mahler measure of its minimal polynomial; for an algebraic integer, $h(\alpha)= d^ {-1} \sum \log\max(1,|\alpha_j|)$. However, while Schanuel counts elements of a given number field, Masser and Vaaler count elements of degree $k$ over a given number field. For small $k$, or over $\mathbf Q$, one even has asymptotic expansions but basically nothing more precise seems to be known. –  ACL Nov 26 '10 at 19:02
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