Gregory P. Dresden

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that is as easy as playing cards. Let Vn denote the Vandermonde matrix with (i, j)th entry vi j = x j i (0 ≤ i, j ≤ n). Since the determinant of Vn is a polynomial in x0, x1, . . . , xn , it suffices to prove the identity for positive integers x0, x1, . . . , xn with x0 ≤ x1 ≤ · · · ≤ xn . We define a Vandermonde card to possess a suit and a value, where a(More)
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) = xn − 1 for n > 1. The first few are easily calculated to be x − 1, x + 1, x2 + x + 1, x2 + 1, . . . . For these and other basic facts, see an algebra text such as [5]. While it might appear that the(More)
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. Introduction For K an algebraic number field, let Kv be the completion of K at the place v and let | |v be the absolute value associated with this completion Kv (more precise definitions are given below). For α ∈ K, we define the(More)
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