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Suppose $K=\mathbb{Q}(\alpha)$ is a fixed number field with $[K:\mathbb{Q}]=d$ and fixed basis $b_1,b_2,..,b_d$. Define $$m(\alpha) = max \{ | p_i| , |q_i| : 1 \leq i \leq d \},$$ where the max is taken over all representatives of $\alpha$ of the form $$\alpha = \frac{p_1b_1+p_2b_2+\ldots+p_db_d}{q_1b_1+q_2b_2+\ldots+q_db_d}.$$

Are there any papers which describe techniques to compute this number?

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I'm going to reformat this to use proper TeX formatting, but you should really do this yourself. Just put dollar signs around the math, and if there's a problem, use back-ticks around the dollar signs.

As for your question, once you write down one representation, you can always replace $(p_1,\ldots,p_d,q_1,\ldots,q_d)$ with $(cp_1,\ldots,cp_d,cq_1,\ldots,cq_d)$ for an arbitrary integer $c$, so your quantity $m(\alpha)$ is equal to $\infty$. Maybe you want to specify that $p_1,\ldots,p_d,q_1,\ldots,q_d\in\mathbb{Z}$ and that $\gcd(p_1,\ldots,p_d,q_1,\ldots,q_d)=1$.

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