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Simultaneous approximations to p-adic numbers and algebraic dependence via multidimensional continued fractions.
| Content Provider | Semantic Scholar |
|---|---|
| Author | Murru, Nadir Terracini, Lea |
| Copyright Year | 2019 |
| Abstract | Unlike the real case, there are not many studies and general techniques for providing simultaneous approximations in the field of $p$--adic numbers $\mathbb Q_p$. Here, we study the use of multidimensional continued fractions (MCFs) in this context. MCFs were introduced in $\mathbb R$ by Jacobi and Perron as a generalization of continued fractions and they have been recently defined also in $\mathbb Q_p$. We focus on the dimension two and study the quality of the simultaneous approximation to two $p$-adic numbers provided by $p$-adic MCFs, where $p$ is an odd prime. Moreover, given algebraically dependent $p$--adic numbers, we see when infinitely many simultaneous approximations satisfy the same algebraic relation. This also allows to give a condition that ensures the finiteness of the $p$--adic Jacobi--Perron algorithm when it processes some kinds of $\mathbb Q$--linearly dependent inputs. |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | https://arxiv.org/pdf/1906.09570v1.pdf |
| Alternate Webpage(s) | http://export.arxiv.org/pdf/1906.09570 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
