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Questions tagged [differential-algebra]

Differential algebra is the study of differential rings and fields and related structures.

In mathematics, differential rings, differential fields, and differential algebras are rings, fields, and algebras equipped with finitely many derivations, which are unary functions that are linear and satisfy the Leibniz product rule. Differential algebra refers then to the area of mathematics consisting in the study of these algebraic objects and their use for an algebraic study of the differential equations.

One of the main objects of differential algebra is the algebra of differential polynomials $\mathscr{F}(Y_1,\ldots, Y_n)$, which is the analogue of the ring of polynomials in commutative algebra.

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What are differential algebras?

On the bottom of page 10 of this paper (/!\ It's in french !) they talk about the 'differential $\mathbb{C}$-algebra of convergent series $\{C(x);x^2d/dx\}$' and about the 'differential $\mathbb{C}$-algebra of the germ of functions holomorphic at…
Hippalectryon
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What is "Cohen’s book of the twenties" about differential algebra Gian-Carlo Rota mentions in his TEN LESSONS?

I am reading "Ten lessons I wish I had learned before I started teaching differential equations" by Gian-Carlo Rota. On item 4, he says: Should we, then, let the students remain blissfully unaware of the existence of linear differential equations…
Red Banana
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Differential algebra in Wikipedia and nlab

The difference between the two definitions is clear in Wikipedia and nlab regarding the definition of a graduated algebra. How to explain this nuance? From Wikipedia: A differential ring is a ring R equipped with one or more derivations, that…
Zbigniew
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Showing that a differential Ideal is prime

Consider $\mathbb{R}$ with trivial derivation, $\mathbb{R}\{x\}$ the ring of differential polynomials in $x,$ and let $J$ be the differential ideal generated by $x''+4x.$ In the quotient $\mathbb{R}\{x\}/J$ we take the ideal generated by…
user558035
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Problem with a proof where algebraic extensions are assumed to be finite extensions

I'm reading the article "Integration in Finite Terms" by Maxwell Rosenlicht and I have a problem with one step in a proof. Rosenlicht wants to prove the following: If $F$ is a differential field of characteristic zero and $K$ an algebraic extension…
Frunobulax
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Algebras vs. rings in algebraic differential calculus

Vector fields (and differential operators of higher order) on a real manifold are often defined in terms of $\mathbb{R}$-algebra $C^\infty(M)$. However, it is not clear to me why is the $\mathbb{R}$-algebra structure so important. For example, the…
akater
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Can a differential algebraist corroborate what this article means?

I'm reading through this article on Liouville's differential algebra theorem which is very tedious and advanced with respect to my background. I'm looking at Propositions 1.13 and 4.2 trying to break down what they mean to make sure I understand.…
StackQuest
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Is the differential operator the only operator that satisfies the product rule?

Sorry, I'm not an advanced abstract algebra person, so I haven't seen this situation addressed before. Suppose you abstract the differential operator to just any ol' linear operator $L$ over a field which has the conditions $L[c_1 u + c_2 v] = c_1…
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Example of how to classify a function's field in Liouville's theorem?

In building off of this question Can anyone show an example of going through Liouville's differential algebra theorem? I'm slowly starting to understand more components of Liouville's theorem after organizing what I do and don't know, writing notes…
StackQuest
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Joint Continuity implies being a Hausdorff space(in the given case)?

If we consider G to be the group of non-singular n by n matrices carrying Zariski topology. Then the joint continuity of the multiplication map from G x G to G would make G Hausdorff.(We consider the topology in G x G to be the Cartesian product…
Jagdeep Singh
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Do these differential operators commute?

Let $D$ denote the differentiation operator on the real functions in $C^1$. For $\alpha\in \Bbb{R}$ and $f\in C^1$ define ${1\over D-\alpha}$ as $${1\over D-\alpha} \,f(t)=e^{\alpha t}\Bigl( \int_0^tf(u)\, e^{-\alpha u} \, du + C\Bigr)$$ where $C\in…
Atom
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Use Liouville-Ostrowsi theorem

I don't know how to answer this questions : 1) $e^x/x, exp(exp(x))$ as no elementary primitive. 2) $1/(1+x^2)$ as no elementary primitive in $R(x)$ but have one in $C(x)$. If someone could help me i will be happy :D. Thanks !
user383659