Speaker: Travor Liu, May Jiang, Qing Su, Hantang Guo
Location: Gordon Street (25) Maths 500
Abstract:
p-adic numbers arise in the study of polynomial congruence $f(x)\equiv 0\pmod{p^k}$. By Hensel's lemma, an integer sequence $\langle x_n\rangle$ can be constructed such that $x_n$ solves the congruence at $k=n$ and $x_n\equiv x_{n-1} \pmod{p^{n-1}}$. This raises the question of interpreting the meaning of $x_n$ as $n\to\infty$.
To characterize p-adic convergence, a new absolute value is defined and completes $\mathbb Q$ to the p-adic number field $\mathbb Q_p$. In this talk, we investigate the analytic properties of sequences and functions in $\mathbb Q_p$ and demonstrate how analysis and number theory, two apparently unrelated areas of mathematics, are linked together. Finally, we present numerical algorithms to visualize Hensel's lemma.
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Slides: https://ucl-ug-col.github.io/past/slides/28feb24.pdf
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Website: https://ucl-ug-col.github.io/
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