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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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How can you prove that a function has no closed form integral?

In the past, I've come across statements 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 multiplication/division raising to powers and…
Simon Nickerson
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How to determine with certainty that a function has no elementary antiderivative?

Given an expression such as $f(x) = x^x$, is it possible to provide a thorough and rigorous proof that there is no function $F(x)$ (expressible in terms of known algebraic and transcendental functions) such that $ \frac{d}{dx}F(x) = f(x)$? In other…
hesson
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Why isn't differential Galois theory widely used?

Ellis Kolchin developed differential Galois theory in the 1950s. It seems to be a powerful tool that can decide the solvability and the form of the solutions to a given differential equation. Why isn't differential Galois theory widely used in…
Henry.L
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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, I have never studied Lie groups. Is it at all…
Islands
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Is there a solvable differential equation with a nonsolvable lie group of symmetries?

For a polynomial equation in one variable over $\mathbb{Q}$, it is well known that the equation is solvable by radicals if and only if the equation's Galois group (which is a finite group) is solvable. The 'only if' part is important - we need it to…
roymend
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How do you prove that something can't be expressed in terms of elementary functions?

It happens quite often that something like an integral or the solution to a certain equation cannot be expressed in terms of elementary functions. For example, the solution to the equation $\cos x=x$ or $xe^x=\alpha$, or the integral of $x^x$ with…
Elvis
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Theorem: Anti-differentiation is harder than differentiation

The question of why anti-differentiation is "harder" than differentiation was the topic of an earlier question, and some of the answers are interesting, but I'm not sure they fully answer it, and this question will not be exactly the same. Someone…
Michael Hardy
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Can anyone show an example of going through Liouville's differential algebra theorem?

WARNING: This is long and layman-like, you may have a difficult time withstanding reading this if you consider yourself a seasoned mathematician. At one point I came across Liouville's theorem of differential algebra, but I don't understand the…
StackQuest
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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 Liouville's theorem on elementary antiderivatives…
JFox
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Inconsistencies in the definition of derivative of a polynomial over a field

A problem I came across defines a particular differentiation operator $D$ over the set of polynomials $\{P\}$ over a field $F$ with "the normal formula; that is $D(\sum_{i=0}^n a_nx^i) = \sum_{i=1}^n na_nx^{i-1}$." However, there seem to be some…
CJ Dowd
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Differential algebra and differential-algebraic equations

Could you give me some information about differential algebra? What is it about? Differential-algebraic equations (DAEs) are polynomials with complex coefficients and the unknown variables are $z, x, x'$. Is this correct? What is the difference…
Mary Star
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Proof that the solutions are algebraic functions

I am looking at the following: $$$$ $$$$ I haven't really understood the proof... Why do we consider the differential equation $y'=P(x)y$ ? Why does the sentence: "If $(3)_{\mathfrak{p}}$ has a solution in…
Mary Star
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derivations of the ring of germs of $C^{\infty}$ functions

Let $\mathcal{O}_{\mathbb{R},0}$ be the ring of germs of $C^{\infty}$ funcitons on the real line. A derivation of $\mathcal{O}_{\mathbb{R},0}$ is a $\mathbb{R}$-linear map $\partial:\mathcal{O}_{\mathbb{R},0}\to\mathcal{O}_{\mathbb{R},0}$ that…
DobryZiom
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Can we find a simple basis for the cokernel of this derivation?

Let $K$ be a field of characteristic zero. Let $R$ be the $K$-algebra $K[x_0,x_1,\ldots]$ of polynomials in countably infinitely many variables. Consider the $K$-linear derivation $\delta:R\to R$ following the Leibniz rule $\delta(ab)=\delta(a)\cdot…
Jyrki Lahtonen
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Soft Question - Generalizations of the Derivative

This is a soft question. I'm asking for any interesting and rather unknown generlizations of the derivative. I know it is generalized through derivations which are functions $\delta$ satisfying $$\delta(uv) = v\delta(u) + u\delta(v)$$ Some of them…
Mako
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