Lower bounds on the projective heights of algebraic points

If $α_1,\ldots,α_r$ are algebraic numbers such that $N=\sum_{i=1}^rα_i \ne \sum_{i=1}^rα_i^{-1}$ for some integer $N$, then a theorem of Beukers and Zagier gives the best possible lower bound on $\sum_{i=1}^r\log h(α_i)$ where $h$ denotes the Weil Height. We will extend this result to allow $N$ to be any totally real algebraic number. Our generalization includes a consequence of a theorem of Schinzel which bounds the height of a totally real algebraic integer.

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