# Tagged Questions

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### Does the “Leibniz multicategory over $R$” have an accepted name?

Let $R$ denote a commutative ring. Definition. The "Leibniz multicategory" over $R$ is given as follows: Objects. $R[D]$-modules (where $D$ is a formal symbol; an 'indeterminate'). ...
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### What is a constant field?

I am looking at the following: Could you explain to me what a constant field is?  P.S. I found this in the paper of T. Honda, "Algebraic differential equation" (pages 170-176).
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### A tricky Differential Equation

How do you solve $$\frac{dy}{dx} = \frac{y^3}{e^{2x} + y^2}$$ I just need a hint. Its not an exact differential nor a linear D. E which I can solve...
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### The solutions are linearly independent and algebraic

The Grothendieck problem for differential equations (Grothendieck-Katz conjecture) is the following: $$\alpha_n(x)y^{(n)}(x)+\dots +a_1 (x)y'(x)+a_0(x)y(x)=0, a_i \in \mathbb{Z}[x]\ \ \ \ (*)$$ We ...
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### Finite extension of a closed differential field is closed.

Let G be the differential Galois group of a differential field extension M of K. Then any finite dimensional extension of a closed intermediate differential field is closed. I can not realise this ...
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### What are all types of elementary second order ordinary differential equation that can not be expressed in closed form?

Can we define all types of elementary second order ordinary differential equation that can not be expressed in closed form as opposed to the one that we can solve? In differential algebra, Picardâ€“...
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### Why differential Galois theory is not widely used?

E.R.Kolchin has developed the differential Galois theory in 1950s. And it seems powerful a tool which can decide the solvability and the form of solutions to a given differential equation. My ...
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### Prerequisites for Differential Galois theory

I would like to know the prerequisites for Differential Galois theory. I have taken Rings, Fields, Groups, Galois theory, and Algebraic Geometry + Commutative Algebra. Looking at the wikipedia page, ...
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### How can you prove that a function has no closed form integral?

I've come across statements in the past along the lines of "function $f(x)$ has no closed form integral", which I assume means that there is no combination of the operations: addition/subtraction ...
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### What are elementary field extensions?

While reading about symbolic integration I encountered some concepts of Differential Algebra. I do not know much of D.A and Fields in general also I have encountered as an extension of Rings. I haven'...
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### Is tetration a transcendental function?

Is tetration a transcendental function? If so are there any papers with a proof? I suspect that it is because I have not seen any algebraic situations where tetration is the answer and the fact that ...
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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 $\mathbb{R}$-...
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### What is the connection between Grothendieck's Differential Operators and Hochschild Cohomology

For a given commutative algebra $A$ over a field $\mathbb{K}$(with char=0) the algebra of differential operators on $A$ is the set of endomorphism $D$ of $A$ such for some $n$ we have that for any ...
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### Determination of inverse laplace transform using primitive functions

In How can you prove that a function has no closed form integral?, the accepted answer points to http://www.sci.ccny.cuny.edu/~ksda/PostedPapers/liouv06.pdf where one can find a corollary by Liouville ...