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We offer a combinatorial method of evaluating Vandermonde's determinant,

We prove that the two smallest values of h(α)+h(1 1−α)+h(1− 1 α) are 0 and 0.4218. .. , for α any algebraic integer.

The cyclotomic polynomials n for n = 1, 2, 3,. .. (familiar to every student of algebra) are the minimal polynomials for the primitive nth roots of unity: n (x) = (k,n)=1 x − e 2πik/n. Clearly n has degree φ(n), where φ signifies Euler's totient function. These monic polynomials can be defined recursively as 1 (x) = x − 1 and i|n i (x) = x n − 1 for n > 1.… (More)

For At(x) = f (x) − t g(x), we consider the set { At(α)=0 h(α) : t ∈ Q}. The polynomials f (x), g(x) are in Z[x], with only mild restrictions, and h(α) is the Weil height of α. We show that this set is dense in [d, ∞) for some effectively computable limit point d.

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