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Math & Stats Department Colloquium
Friday, October 18, 2024

Time: 2:30 pm

Place: Jeffery Hall, Room 234

Speaker: Hester Graves (Center for Computing Sciences)

Title: The Minimal Euclidean Function on the Gaussian Integers

Abstract: Motzkin proved that every Euclidean domain $R$ has a minimal Euclidean function, $\phi_R$ . He showed that when $R = \mathbb{Z}$, the minimal function is $\phi_{\mathbb{Z}} (x) =\ log 2_ |x|$. For over seventy years, $\phi_{\mathbb{Z}}$ was the only example of an explicitly-computed minimal function in a number field. We give the first explicitly-computed minimal function in a non-trivial number field, $\phi_{\mathbb{Z}[i]}$ . The proof introduces a new way to visualize quotients $\mathbb{Z}[i]$. This talk is accessible to undergraduates.