Polynomials which take Gaussian integer values at Gaussian integers |
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Authors: | Douglas Hensley |
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Institution: | Institute for Advanced Study, Princeton, New Jersey 08540 USA |
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Abstract: | A factorial set for the Gaussian integers is a set G = {g1, g2 … gn} of Gaussian integers such that takes Gaussian integer values at Gaussian integers. We characterize factorial sets and give a lower bound for . It is conjectured that there are infinitely many factorial sets. A Gaussian integer valued polynomial (GIP) is a polynomial with the title property. A bound similar to the above is given for max∥z∥2=n ∥ G(z)∥ if G(z) is a GIP. There is a relation between factorial sets and testing for GIP's. We discuss this and close with some examples of factorial sets, and speculate on how to find more. |
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