Using geometric continued fractions for universal quadratic forms

Abstract: Already in the 18th century, Lagrange proved that every positive integer can be expressed as the sum of four squares of integers. Nowadays we say that the sum of four squares is a universal quadratic form. In the friendly and accessible talk, I’ll discuss some results on universal quadratic forms over Z and over totally real number fields. In particular, I'll focus on the role of continued fractions in many of the advances in the area over the last 10 years. First, I'll talk about the connection between classical continued fractions and quadratic forms over real quadratic fields. Then I'll turn to the more complicated situation of higher degree fields where we use geometric continued fractions as developed by Klein, Arnold, Karpenkov and many others. The talk is based on recent joint works with Valentin Blomer, Siu Hang Man, Magdalena Tinkova, Robin Visser, and Pavlo Yatsyna.

dynamical systemsnumber theory

Audience: researchers in the topic

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One World Numeration seminar

Series comments: Description: Online seminar on numeration systems and related topics

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