Talk:Differential algebra

elementary Leibniz differentials

I look in vain in wikipedia for an elementary explanation of the differential, (not the derivative). The Leibniz notations, ${\frac {\mathrm {d} y}{\mathrm {d} x}}$ and $\int y\cdot dx,$ assumes dx and dy. Formal differentiation can be defined: if x is a variable then so is dx, if x is constant then dx = 0, if x is an independent variable then dx is constant and d²x = 0, the sum rule d(x+y) = dx+dy, and the product rule d(x·y) = dx·y+x·dy. This is sufficient to deduce differentiation of formal power series. Differentiation of an algebraic equation (such as x²+y² = r²) gives a differential equation (such as x·dx + y·dy = 0). Perhaps this is what this technical article on differential algebra is about ? Bo Jacoby 08:06, 7 August 2007 (UTC).

Graded derivation?

Graded derivation redirects here, but it is not mentioned in the article; on the other hand there is a section Derivation (abstract algebra)#Graded derivations! Anyone want to sort this out? PJTraill (talk) 00:00, 4 February 2014 (UTC)

I propose to remove the pseudo-differential operator section.

In view of the "This section has multiple issues" statement, I propose to remove this section and place a link to the pseudo-differential operator page. If the author of this section wants this content in wikipedia, I suggest adding the content to the pseudo-differential operator page. Another option is to verify the statements and add references. However, non-referenced statements should not be included. Thanks. TMM53 (talk) 08:45, 2 January 2023 (UTC)

This is a bad idea because it disconnects areas of mathematics that need to be connected. The pseudodifferential operators page now misleads harmonic analysts and people looking to work in the algebraic theory of differential equations. 2620:104:E001:9031:A579:E7FE:27E5:1B3D (talk) 14:48, 19 June 2024 (UTC)

Proposal of a new version

I will substitute a major revision for this wikipedia page. The reasons are most content contains no citations, the language is contradictory and unclear, fundamental information is missing, some content is incorrect and the order of presentation is wrong i.e. beginning a topic by discussing an open problem in the field. If someone feels current content should be retained, please provide an appropriate citation to a credible source with page number. Thanks TMM53 17:22, 4 January 2023 (UTC)

Thank you for your interest in improving Wikipedia. The present version requires certainly to be improved, and a new version would be welcome. However, it seems that you are a new editor and that you do not master the constraints of Wikipedia editing (a witness of this is that I had to add blank lines and a heading to your post for making it readable). So I am afraid that if, you write a new version fom scratch, it will not be accepted by other editors and reverted. So, if will much better to write your version in the draft space and to provide a link here. This would allow other editors to discuss and improve the new version before moving it in the main space.

Another solution, probably easier it to solve one after the other the issues that you have listed:

• "the order of presentation is wrong": it is easy to move a section.
• "the language is contradictory and unclear": you may fixing this, one sentence or paragraph after the other.
• "fundamental information is missing": you may add it without rewrite the remainder of the article
• "most content contains no citations": I have tagged the article with {{no footnotes}}. In general when citations are lacking but exist in the literature, a content must not be deleted, but, instead, a citation or the tag {{citation needed}} must be added.

As one may presume that some of your intended modifications will be welcome, and some other will be rejected by the other editors, proceeding step by step will certainly be a much more efficient method of editing. D.Lazard (talk) 08:45, 5 January 2023 (UTC)

Thanks for your suggestions and shared interest in this subject.

Wikipedia guidelines recommend creating a user subpage for revisions.

This is a proposal for a revision: Differential algebra revision

The problems and solutions describe the reason for this revision. I am available to receive and edit suggestions for the next 2 weeks before I replace the current article with the revision. If more than 2 weeks is needed, let me know. I appreciate your feedback and experience. The revisions comply fully with Wikipedia recommendations regarding content.

Further information: WP:REMOVAL

Problems, solutions and relevant Wikipedia guidelines

Problem: This Wikipedia article fails to cite sources for any content except for a single historical item.

Solution: The revision cites reliable sources for all content other than some examples, and these examples are simple and unlikely to be challenged. The reliable sources are textbooks and peer-reviewed publications.

Further information: WP:CS

Problem: This Wikipedia article contains minimal content, lacking the universally recognized introductory content from the book chapters of recognized experts.

Solution: The revision contains the universally recognized introductory content from book chapters of recognized experts: Kolchin, Sit and Bronstein.

Further information: WP:STUB

Problem: This Wikipedia article fails to follow Wikpedia's guideline for mathematics to begin with simple content and gradually increase to complex content. The article begins with complex content (open Problems) and fails to introduce the fundamental concepts.

Solution: The revision begins with simple content (definitions, why is this important, history) and gradually adds increasingly more complex content (complex algebraic structures, challenging problems).

Further information: MOS:MATH

Problem: This Wikipedia article contains non-functioning links e.g. the Kolchin catenary conjecture and the Ritt Problem and lacks internal links where needed for mathematical terms.

Solution: The revision has internal functioning links for all mathematical terms.

Further information: WP:LINK

Problem: This Wikipedia article does not use inline option for math content.

Solution: The revision follows Wikipedia guidelines for inline format for formulas.

Further information: WP:MATH

Problem: This Wikipedia article's equations for the pseudo-differential operator and integral do not correctly reflect the contents of a prior research preprint. This content is original research.

Solution: Wikipedia prohibits original research content. The revision uses content from textbooks and peer-reviewed journals.

Further information: WP:NOR

Problem: This wikipedia article does not use italic text for emphasis.

Solution: The revision uses italic text for emphasis.

Further information: MOS:B

Problem: This Wikipedia article has inconsistent text. For example, a derivation must always follow the Leibniz product rule by definition, and this same derivation sometimes does not follow this rule.

Solution: The revision has consistent writing.

Problem: This Wikipedia article uses unexplained redundant symbols, e.g. 3 symbols for derivation ${\textstyle \partial ,d,D}$.

Solution: The revision defines each mathematical symbol.

Problem: This Wikipedia article uses 2 formats for mathematical expressions without explanation that convey no additional value e.g. ${\textstyle \partial (r_{1}r_{2})=(\partial r_{1})r_{2}+r_{1}(\partial r_{2})}$ and ${\textstyle \partial \circ M=M\circ (\partial \times \operatorname {id} )+M\circ (\operatorname {id} \times \partial )}$.

Solution: The revision uses a single consistent format for mathematical expressions.

Problem: This Wikipedia article does not follow a consistent format for introducing equations and text items.

Solution: The revision follows a consistent format with bulleted points for text statements and indents for mathematical expressions.

Problem: This Wikipedia article introduces a differential ring by definition and by examples as a noncommutative ring with noncommutative elements. This contradicts the universal approach used by recognized experts.

Solution: The revision introduces a differential ring as a commutative ring with commutative derivations consistent with recognized experts.

Problem: This Wikipedia article does not place content into appropriate sections. Examples are 1) an undefined algebraic structure, possibly a differential form, ${\textstyle d(xy)=xdy+ydx}$ in the differential ring section rather than graded algebra section, 2) K-algebra in the differential field section rather than a K-algebra section. 3) a closed differential field is defined in an example section rather than in the field section,.

Solution: The revision places content in appropriate sections.

Problem: This Wikipedia article introduces terms and statements with incomplete definitions. An example is "For a differential rings R there is an embedding of the Weyl algebra in the ring of pseudodifferential operators ${\textstyle R((\partial ^{-1}))}$ as the finite tails of these infinite series."

Solution: The revision introduces each algebraic structure with a complete definition, describes the associated derivation, provides relevant mathematical formula, provides an internal link and cites a source.

Comment: The revision has retained as much content as possible from the Wikipedia article, but I have edited and moved some content to the appropriate section. Examples include:

"A natural example of a differential field is the field of rational functions in one variable over the complex numbers ..." ${\textstyle \to }$ Examples section.

"are unary functions that are linear" ${\textstyle \to }$ Differential ring section.

"Differential algebra was introduced by Joseph Ritt in 1950" ${\textstyle \to }$ History section.

"Note that a differential ring is a (not necessarily graded) Z-differential algebra" ${\textstyle \to }$ Examples section.

==Notes==

==References==

* Bronstein, Manuel (2005). Symbolic integration I : transcendental functions (2nd ed.). Berlin: Springer. ISBN 3-540-21493-3.

• Kolchin, Ellis (1973). Differential Algebra And Algebraic Groups. Academic Press. ISBN 978-0-08-087369-5.
• Sit, William Y. (2002). "The Ritt-Kolchin theory for differential polynomials". In Guo, Li; Cassidy, Phyllis J; Keigher, William F; Sit, William Y (eds.). Differential algebra and related topics: proceedings of the International Workshop, Newark Campus of Rutgers, the State University of New Jersey, 2-3 November 2000. River Edge, NJ: World Scientific. ISBN 981-02-4703-6.

TMM53 (talk) 07:57, 6 March 2023 (UTC)

Hello, I was able to track references from many publications and verify that Parshin's preprint was ultimately published in a Russian journal. I used that reference to correct the current error in this Wikipedia article in the pseudo-differential operator ring section. The revised proposal's link is:

Differential algebra revision 2023-03-06a

• Parshin, Aleksei Nikolaevich (1999). "On a ring of formal pseudo-differential operators" (PDF). Proc. Steklov Math. Institute. 224: 266–280.

TMM53 (talk) 23:05, 6 March 2023 (UTC)

"Differential algebraic variety" listed at Redirects for discussion

The redirect Differential algebraic variety has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at Wikipedia:Redirects for discussion/Log/2024 April 13 § Differential algebraic variety until a consensus is reached. Jay 💬 10:07, 13 April 2024 (UTC)

updating stale page

Currently, the article is very disconnected from other areas in the Algebraic Theory of Differential Equations and the classical roots of the subject. This hurts people coming to this article from different areas looking for answers to their problems.

I really think we should be reverting this page or updating this page to something like what I drafted--I myself had come to this page to look up a formula I recorded only to find it deleted!

https://en.wikipedia.org/w/index.php?title=Differential_algebra&diff=prev&oldid=1225610872 2620:104:E001:9031:A579:E7FE:27E5:1B3D (talk) 15:00, 19 June 2024 (UTC)

I suppose that you are TMM53, and the draft that you are refering to is either User:TMM53/tmp/differential algebra 2023-03-06 or User:TMM53/tmp/differential algebra 2023-03-06a. I must apologize to not having read in details your very long post in § Proposal of a new version. So, I never saw that it contains a link (and even two) to a draft.

I agree that the article has many problems, and this is for this reason that, some time ago, I rewrote the beginning up to section § Differential polynomial included. I planned to continue this rewriting, but I had to pass to other things. This may explain some incoherences in the article.

Having had a quick look on your draft, it seems to be an improvement on some aspects, such as adding many references. But it is also a disimprovement on some aspects. In particular, it removes the description of the context and most explanations. Also, the guidelines recommend to not use bulleted lists when prose does the job well. Instead of replacing the excessive bulleted lists by prose, it seems that a part of your work consisted to replace prose by lists.

Also, your style is useful for people searching references, but is not suited for non specialists who need to be introduced to the subject. In particular, it does not follow the guidelines of MOS:MATH, in particular MOS:MATH#Article introduction. D.Lazard (talk) 18:39, 19 June 2024 (UTC)

Hello, I am not user 2620:104:E001:9031:A579:E7FE:27E5:1B3D and did not generate the topic "Updating stale page."

The original draft was written a long time ago, and I now recognize the deficiencies in that draft. I thank you and

appreciate you providing this feedback. Regards. TMM53 (talk) 00:59, 20 June 2024 (UTC)