Questions tagged [differential-algebra]

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

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C-planes and associated valuations in the paper of James Ax titled "On Schanuel's conjecture"

I am currently trying to understand Proposition 2. (page 257 of the JSTOR version) in the above cited paper of Ax. It says the following: Let $F\supseteq C \supseteq \mathbb{Q}$ be a tower of fields. ...
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The field of constants of a differential ring. Derivative of real and complex numbers.

Let $D$ be a derivation operator over a ring $R$: $$D(a + b) = D(a) + D(b) \\ D(ab) = D(a)b + aD(b)$$ for all elements $a,b\in R$. If the ring is the field $\mathbb{Q}$, all derivatives should be ...
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If C is a coalgebra with counit € and D=ker(€), how define a coalgebra structure on tensor algebra T(D) so that C to T(D) is inclusion coalgebra map

Let C be a coalgebra with comultiplication N and counit € and let D = ker(€). Consider the tensor algebra on D , denoted by T(D). My question is : how one can define a coalgebra structure on T(D) so ...
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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 ...
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Accessible first book/resource on differential algebra

I am very interested in learning about the conditions under which a function can be integrated in elementary terms, a topic that I understand falls within the purview of differential algebra. Having ...
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Intuition of Liouville's Theorem (differential algebra) Proof

At the end of my abstract algebra class this spring, we were given an overview of differential algebra and some differential Galois theory. We went too fast to prove anything nontrivial, but I found ...
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Solved inverse Galois problem for $\mathbb{C}(z)$ seems to contradict the theory about Liouvillian extensions.

The theory about Liouvillian extensions tells us that a Picard-Vessiot extension $L \supset k$ is Liouvillian if and only if the identity component $G^°$ of $G = Gal(L / k)$ is solvable. I think I ...
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Need help with the proposition 4.2 in R. C. Churchill's paper on Liouville's Theorem in differential algebra

I'm currently studying Liouville's theorem in differential algebra from this paper. I'm stuck on the proof of the propositions 4.2. In the chapter 4, this "logarithmic derivative identity" is ...
Which kinds of zeroing equations of elementary functions can have solutions which are elementary numbers or explicit elementary numbers? Take the equation $$F(x)=0,$$ wherein $F$ is an elementary ...