Timeline for Polynomials leaving invariant the Gaussian integers
Current License: CC BY-SA 3.0
9 events
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| Jul 24, 2017 at 21:48 | comment | added | Salvo Tringali | @MarcoSan If your question means, "Do you know of an algorithm to compute the coefficients $d_n$ in Remark II.1.5(ii)?", the answer is, "No, not off the top of my head." But if you insist to have a "more explicit description" of a basis, then take a look at Proposition II.3.14 and Exercise 24 on p. 48 of the book: This is perhaps closer to the spirit of what you're asking for, and may convince you that the kind of "nice closed form for a basis" you're implicitly hoping for is not so likely. | |
| Jul 24, 2017 at 20:10 | vote | accept | MarcoSan | ||
| Jul 24, 2017 at 20:09 | comment | added | MarcoSan | Thank you @SalvoTringali ! do you have any idea on how to describe the generators of the ideals $\mathfrak{J}_n$ (using the notations of the book you quoted)? (say induction relation, or growth of the modules, as FedorPetrov pointed out) I have shown that for instance $ \mathfrak{J}_2 = \frac {1+i}{2}\mathbb{Z}[i] $ | |
| Jul 24, 2017 at 6:21 | history | edited | Salvo Tringali | CC BY-SA 3.0 |
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| Jul 24, 2017 at 6:13 | comment | added | Salvo Tringali | @FedorPetrov. I wouldn't either, that's why I wrapped the term explicit basis in quotation marks. However, I'm not sure that what the OP is asking for is possible if taken literally (whatever it means to have a closed form for a basis), and on the other hand, the construction in Remark II.1.5(ii) of Cahen & Chabert's book qualifies, for me, as "a precise description of such polynomials" (I'm quoting the OP). | |
| Jul 24, 2017 at 6:07 | comment | added | Fedor Petrov | I would not call this basis explicit. It is constructed inductively, and studying concrete properties of this basis (say, the growth of coefficients) is non-trivial. | |
| Jul 24, 2017 at 5:29 | history | edited | Salvo Tringali | CC BY-SA 3.0 |
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| Jul 24, 2017 at 5:14 | history | edited | Salvo Tringali | CC BY-SA 3.0 |
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| Jul 24, 2017 at 5:08 | history | answered | Salvo Tringali | CC BY-SA 3.0 |