Arithmetical Applications of an Identity for the Vandermonde Determinant

Abstract

When $\{{\alpha }_i{\}}_{1\le i\le m}$ is a sequence of distinct non-zero elements of an integral domain $A$ and $\gamma$ is a common multiple of the ${\alpha }_i$ in $A$ we obtain, by means of a simple identity for the Vandermonde determinant, a lower bound for ${\sup }_{1\le i<j\le m}\varphi ({\alpha }_i-{\alpha }_j)$ in terms of $\varphi (\gamma )$, where $\varphi$ is a function from the nonzero elements of $A$ to ${\mathbf{R}}_{+}$ satisfying certain natural conditions. We describe several applications of this bound.

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Cited by in corpus (1)

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