New algorithms to compute the characteristic polynomial of the Frobenius endomorphism of a Drinfeld module #38174
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Documentation preview for this PR (built with commit 3dd3c04; changes) is ready! 🎉 |
| Check that ``var`` inputs are taken into account for cached | ||
| characteristic polynomials:: | ||
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| sage: Fq = GF(2) | ||
| sage: A.<T> = Fq[] | ||
| sage: K.<z> = Fq.extension(2) | ||
| sage: phi = DrinfeldModule(A, [z, 0, 1]) | ||
| sage: phi.frobenius_charpoly() | ||
| X^2 + X + T^2 + T + 1 | ||
| sage: phi.frobenius_charpoly(var='Foo') | ||
| Foo^2 + Foo + T^2 + T + 1 | ||
| sage: phi.frobenius_charpoly(var='Bar') | ||
| Bar^2 + Bar + T^2 + T + 1 |
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I feel like there a little bit too much emphasis on this optional flag and the documentation of this method is already quite lenghty. I think this could either: be rephrased into a more "software reference manual documentation" way OR put into a TEST block in order to hide it from the manual completely.
What I mean by the first suggestion is: rephrase the sentence like "Use the optional var parameter in order to change the variable name of the resulting univariate polynomial" or somthing like this. Also, I would reuse the last phi to make it a one-liner doctest.
Anyway, this is only my preference as someone who don't really like to read documentation and prefer to have the most straightforward info.
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But this block (lines 408 to 420) are already in a TESTS:: block, no? See line 359.
src/sage/rings/function_field/drinfeld_modules/finite_drinfeld_module.py
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…_module.py Co-authored-by: David Ayotte <34245930+DavidAyotte@users.noreply.github.com>
| if self._frobenius_charpoly is not None: | ||
| return self._frobenius_charpoly | ||
| return self._frobenius_charpoly.change_variable_name(var) |
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With these lines, you use the cached value even if the user explicitly asks for another algorithm. I think that it is not what we want.
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Agreed. Will also take care of it.
(Wasn't it already the case tho? I remember having to use some tricks in the past to not use the cached data, but maybe I was mistaken.)
| @cached_method | ||
| def frobenius_trace(self): |
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In this method, you use the crystalline matrix in all cases. I think it is okay but aren't there any case where the motive matrix is faster to compute?
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Yup. Will take care of that.
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Any objection to making DrinfeldModuleMorphism._motive_matrix public? It's interesting in its own right and would be useful in our case.
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What do you mean by public? Without the initial underscore?
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Well, eventually, I plan to have a method motive to create the motive of a Drinfeld module or a morphism between those, and a method matrix to get the matrix of a morphism of Anderson motives.
So, if you make the method _motive_matrix public now, it means that it will need to be deprecated afterwards. Therefore, I prefer keeping it as it is.
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Is it better now @vbraun? To be fair, as pointed out by Xavier, many fails seem unrelated to this PR. |
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Strong +1. |
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This time, for instance, I got the error for the check I assume this is not due to this PR. With Should I care about this? All In any case, it seems that the issue with the documentation is now fixed. So I give again a positive review. I'm sorry if I'm wrong but I'm completely lost. |
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Why is the git commit history so ugly? It's quite hard to follow what has been done in this PR. |
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I think it's my fault but I don't know exactly what happened: at some point, I merged with the latest version of |
…mial of the Frobenius endomorphism of a Drinfeld module
<!-- ^ Please provide a concise and informative title. -->
<!-- ^ Don't put issue numbers in the title, do this in the PR
description below. -->
<!-- ^ For example, instead of "Fixes sagemath#12345" use "Introduce new method
to calculate 1 + 2". -->
<!-- v Describe your changes below in detail. -->
<!-- v Why is this change required? What problem does it solve? -->
<!-- v If this PR resolves an open issue, please link to it here. For
example, "Fixes sagemath#12345". -->
This pull request implements two new algorithms to compute the
characteristic polynomial of the Frobenius endomorphism of a Drinfeld
$\mathbb F_q[T]$-module over a finite field $K$. Previously, only the
algorithms based on crystalline cohomology (see [Musleh-Schost
2023](https://dl.acm.org/doi/10.1145/3597066.3597080)) or on Anderson
motives (see [Caruso-Leudière 2023](https://arxiv.org/abs/2307.02879))
were implemented. We propose two new algorithms:
- The algorithm based on central simple algebras described in Chapter 4
of [Caruso-Leudière 2023](https://arxiv.org/abs/2307.02879).
- The algorithm described by Gekeler in [Gekeler 1991](https://www.scien
cedirect.com/science/article/pii/002186939190211P).
**Acknowledgement.** This implementation was originally due to @xcaruso
(see [here](https://github.com/xcaruso/sage/blob/d2e36bd18b51c93806b7a3b
5c8261da7dc98c494/src/sage/rings/function_field/drinfeld_modules/finite_
drinfeld_module.py)), and after a private discussion, I took the liberty
of creating this PR.
I also propose to change the formula computed by `frobenius_norm`.
Before, it computed
$$(-1)^n \mathrm{N}_{K/\mathbb F_q}(\Delta) p^{n / \mathrm{deg}(p)},$$
where $K$ is the ground field, $n$ is the degree of $K$ over $\mathbb
F_q$, and $p$ is the monic generator of the $\mathbb
F_q[T]$-characteristic of $K$. The docstring claimed this was $(-1)^r$
times the constant coefficient of the characteristic polynomial of the
Frobenius endomorphism, $r$ being the rank of the Drinfeld module. I
believe this was a mistake, and instead changed the formula, following
Theorem 4.2.7 of [Papikian's
book](https://link.springer.com/book/10.1007/978-3-031-19707-9), to
$$(-1)^{nr - n - r} \mathrm{N}_{K/\mathbb F_q}(\Delta) p^{n /
\mathrm{deg}(p)}.$$
P.S. @DavidAyotte, given that you now work in the industry, please tell
us if you still want to be tagged on these Drinfeld module stuff. Same
question for you @ymusleh given that you defended your thesis
(congratulations!).
### 📝 Checklist
<!-- Put an `x` in all the boxes that apply. -->
- [X] The title is concise and informative.
- [X] The description explains in detail what this PR is about.
- [ ] I have linked a relevant issue or discussion.
- [X] I have created tests covering the changes.
- [X] I have updated the documentation and checked the documentation
preview.
<!-- List all open PRs that this PR logically depends on. For example,
-->
<!-- - sagemath#12345: short description why this is a dependency -->
<!-- - sagemath#34567: ... -->
(Dédicace à ABLCCN)
URL: sagemath#38174
Reported by: Antoine Leudière
Reviewer(s): Antoine Leudière, David Ayotte, Xavier Caruso
|
The error in the conda workflow at least seems to be introduced by this PR or not? Could this be fixed please? Thanks! (Maybe as a follow-up since its already in volkers release branch) |
…mial of the Frobenius endomorphism of a Drinfeld module
<!-- ^ Please provide a concise and informative title. -->
<!-- ^ Don't put issue numbers in the title, do this in the PR
description below. -->
<!-- ^ For example, instead of "Fixes sagemath#12345" use "Introduce new method
to calculate 1 + 2". -->
<!-- v Describe your changes below in detail. -->
<!-- v Why is this change required? What problem does it solve? -->
<!-- v If this PR resolves an open issue, please link to it here. For
example, "Fixes sagemath#12345". -->
This pull request implements two new algorithms to compute the
characteristic polynomial of the Frobenius endomorphism of a Drinfeld
$\mathbb F_q[T]$-module over a finite field $K$. Previously, only the
algorithms based on crystalline cohomology (see [Musleh-Schost
2023](https://dl.acm.org/doi/10.1145/3597066.3597080)) or on Anderson
motives (see [Caruso-Leudière 2023](https://arxiv.org/abs/2307.02879))
were implemented. We propose two new algorithms:
- The algorithm based on central simple algebras described in Chapter 4
of [Caruso-Leudière 2023](https://arxiv.org/abs/2307.02879).
- The algorithm described by Gekeler in [Gekeler 1991](https://www.scien
cedirect.com/science/article/pii/002186939190211P).
**Acknowledgement.** This implementation was originally due to @xcaruso
(see [here](https://github.com/xcaruso/sage/blob/d2e36bd18b51c93806b7a3b
5c8261da7dc98c494/src/sage/rings/function_field/drinfeld_modules/finite_
drinfeld_module.py)), and after a private discussion, I took the liberty
of creating this PR.
I also propose to change the formula computed by `frobenius_norm`.
Before, it computed
$$(-1)^n \mathrm{N}_{K/\mathbb F_q}(\Delta) p^{n / \mathrm{deg}(p)},$$
where $K$ is the ground field, $n$ is the degree of $K$ over $\mathbb
F_q$, and $p$ is the monic generator of the $\mathbb
F_q[T]$-characteristic of $K$. The docstring claimed this was $(-1)^r$
times the constant coefficient of the characteristic polynomial of the
Frobenius endomorphism, $r$ being the rank of the Drinfeld module. I
believe this was a mistake, and instead changed the formula, following
Theorem 4.2.7 of [Papikian's
book](https://link.springer.com/book/10.1007/978-3-031-19707-9), to
$$(-1)^{nr - n - r} \mathrm{N}_{K/\mathbb F_q}(\Delta) p^{n /
\mathrm{deg}(p)}.$$
P.S. @DavidAyotte, given that you now work in the industry, please tell
us if you still want to be tagged on these Drinfeld module stuff. Same
question for you @ymusleh given that you defended your thesis
(congratulations!).
### 📝 Checklist
<!-- Put an `x` in all the boxes that apply. -->
- [X] The title is concise and informative.
- [X] The description explains in detail what this PR is about.
- [ ] I have linked a relevant issue or discussion.
- [X] I have created tests covering the changes.
- [X] I have updated the documentation and checked the documentation
preview.
<!-- List all open PRs that this PR logically depends on. For example,
-->
<!-- - sagemath#12345: short description why this is a dependency -->
<!-- - sagemath#34567: ... -->
(Dédicace à ABLCCN)
URL: sagemath#38174
Reported by: Antoine Leudière
Reviewer(s): Antoine Leudière, David Ayotte, Xavier Caruso
This bug is not even in the current version; I merged So no, I don't think so, and I can hardly find it realistic that Xavier or me would have touched this file. |
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Yo! All required tests pass! 🎉 |
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Ah bah non attends. (Meaning: oh but wait but no.) Some stuff related to coverage is failing. Oh well. |
…mial of the Frobenius endomorphism of a Drinfeld module
<!-- ^ Please provide a concise and informative title. -->
<!-- ^ Don't put issue numbers in the title, do this in the PR
description below. -->
<!-- ^ For example, instead of "Fixes sagemath#12345" use "Introduce new method
to calculate 1 + 2". -->
<!-- v Describe your changes below in detail. -->
<!-- v Why is this change required? What problem does it solve? -->
<!-- v If this PR resolves an open issue, please link to it here. For
example, "Fixes sagemath#12345". -->
This pull request implements two new algorithms to compute the
characteristic polynomial of the Frobenius endomorphism of a Drinfeld
$\mathbb F_q[T]$-module over a finite field $K$. Previously, only the
algorithms based on crystalline cohomology (see [Musleh-Schost
2023](https://dl.acm.org/doi/10.1145/3597066.3597080)) or on Anderson
motives (see [Caruso-Leudière 2023](https://arxiv.org/abs/2307.02879))
were implemented. We propose two new algorithms:
- The algorithm based on central simple algebras described in Chapter 4
of [Caruso-Leudière 2023](https://arxiv.org/abs/2307.02879).
- The algorithm described by Gekeler in [Gekeler 1991](https://www.scien
cedirect.com/science/article/pii/002186939190211P).
**Acknowledgement.** This implementation was originally due to @xcaruso
(see [here](https://github.com/xcaruso/sage/blob/d2e36bd18b51c93806b7a3b
5c8261da7dc98c494/src/sage/rings/function_field/drinfeld_modules/finite_
drinfeld_module.py)), and after a private discussion, I took the liberty
of creating this PR.
I also propose to change the formula computed by `frobenius_norm`.
Before, it computed
$$(-1)^n \mathrm{N}_{K/\mathbb F_q}(\Delta) p^{n / \mathrm{deg}(p)},$$
where $K$ is the ground field, $n$ is the degree of $K$ over $\mathbb
F_q$, and $p$ is the monic generator of the $\mathbb
F_q[T]$-characteristic of $K$. The docstring claimed this was $(-1)^r$
times the constant coefficient of the characteristic polynomial of the
Frobenius endomorphism, $r$ being the rank of the Drinfeld module. I
believe this was a mistake, and instead changed the formula, following
Theorem 4.2.7 of [Papikian's
book](https://link.springer.com/book/10.1007/978-3-031-19707-9), to
$$(-1)^{nr - n - r} \mathrm{N}_{K/\mathbb F_q}(\Delta) p^{n /
\mathrm{deg}(p)}.$$
P.S. @DavidAyotte, given that you now work in the industry, please tell
us if you still want to be tagged on these Drinfeld module stuff. Same
question for you @ymusleh given that you defended your thesis
(congratulations!).
### 📝 Checklist
<!-- Put an `x` in all the boxes that apply. -->
- [X] The title is concise and informative.
- [X] The description explains in detail what this PR is about.
- [ ] I have linked a relevant issue or discussion.
- [X] I have created tests covering the changes.
- [X] I have updated the documentation and checked the documentation
preview.
<!-- List all open PRs that this PR logically depends on. For example,
-->
<!-- - sagemath#12345: short description why this is a dependency -->
<!-- - sagemath#34567: ... -->
(Dédicace à ABLCCN)
URL: sagemath#38174
Reported by: Antoine Leudière
Reviewer(s): Antoine Leudière, David Ayotte, Xavier Caruso
|
The test is not required :-). So the most likely is that there is a problem somewhere in the configuration of the container, but not in this PR. |
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Not all tests are required?! |
|
I'm not sure. But some of them are marked |
|
coverage ci will be fixed by #38687. You may ignore the ci failure here. |
Many thanks for the heads-up! |
This pull request implements two new algorithms to compute the characteristic polynomial of the Frobenius endomorphism of a Drinfeld$\mathbb F_q[T]$ -module over a finite field $K$ . Previously, only the algorithms based on crystalline cohomology (see Musleh-Schost 2023) or on Anderson motives (see Caruso-Leudière 2023) were implemented. We propose two new algorithms:
The algorithm based on central simple algebras described in Chapter 4 of Caruso-Leudière 2023.
The algorithm described by Gekeler in Gekeler 1991.
Acknowledgement. This implementation was originally due to @xcaruso (see here), and after a private discussion, I took the liberty of creating this PR.
I also propose to change the formula computed by
frobenius_norm. Before, it computedwhere$K$ is the ground field, $n$ is the degree of $K$ over $\mathbb F_q$ , and $p$ is the monic generator of the $\mathbb F_q[T]$ -characteristic of $K$ . The docstring claimed this was $(-1)^r$ times the constant coefficient of the characteristic polynomial of the Frobenius endomorphism, $r$ being the rank of the Drinfeld module. I believe this was a mistake, and instead changed the formula, following Theorem 4.2.7 of Papikian's book, to
$$(-1)^{nr - n - r} \mathrm{N}_{K/\mathbb F_q}(\Delta) p^{n / \mathrm{deg}(p)}.$$
P.S. @DavidAyotte, given that you now work in the industry, please tell us if you still want to be tagged on these Drinfeld module stuff. Same question for you @ymusleh given that you defended your thesis (congratulations!).
📝 Checklist
(Dédicace à ABLCCN)