Questions tagged [differential-algebra]

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

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The antiderivative of $e^{x^2}$

I am asked to show that show that the antiderivative of $e^{x^2}$ cannot be expressed as $R(e^x)$, where $R$ is a rational function over the reals. Can this be shown without using transcendental ...
FireWolf15G8's user avatar
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What non-algebraic methods do we have to solve this equation?

I'm recently studying homogeneous linear second order ODEs, and got into the integrating factors technique. For every equation, if the integrating factor is known, we can reduce its order provided ...
Simón Flavio Ibañez's user avatar
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Magma - Differential field extension over a differential field

I want to construct the differential field $\mathbb{Q}(x,\,\log x,\,\log(\log x))$ in Magma. I have tried the following: ...
Mitchell Holt's user avatar
3 votes
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Can the formation of Kähler differentials be seen as a functor into the category of modules (without specifying the ring of scalars)?

In an Algebraic Geometry course, I've seen the definition of the $A$-module of Kähler differentials $\Omega_A^1$ given a $k$-algebra $A$ ($k$ is a field). Then if $X$ is a variety, we defined the ...
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q-Leibniz Rule?

The famous Leibniz rule: $(uv)' = u'v+uv'$ can be generalized for any order derivative using the following formula $$(uv)^{(n)} = \sum_{k=0}^n{n\choose k}u^{(n-k)}v^{(k)}$$ This is a known formula ...
Mako's user avatar
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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 $$\...
Mako's user avatar
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Which CAS can do non-commutative differential algebra?

I am looking for a CAS (possibly incl. additional packages/libraries) that can compute generic non-commutative differential expressions. Let me illustrate what I mean by two examples. Let $(R,\partial)...
M.G.'s user avatar
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Let $D:\mathbb{R}(x)\to\mathbb{R}(x)$ be the algebraic(!) differential operator such that $D(x)=1$. Is it true that $D(c)=0$ for all $c\in\mathbb{R}$?

Define the field of rational functions $\mathbb{R}(x)$ with algebraic differential operator $D:\mathbb{R}(x)\to\mathbb{R}(x)$ such that $D(x)=1$. By algebraic differential operator, we mean a mapping ...
Math Enjoyer's user avatar
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Is there really a need to prove both cases for logarithmic and exponential extensions in the theory of Symbolic Integration?

I've been researching Symbolic Integration in Finite Terms using a Differential Algebra and Computer Algebra perspective, reading papers by Risch, Rothstein, Trager, Ritt, Bronstein, etc., as well as ...
Math Enjoyer's user avatar
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how do you know when there is no antiderivative written in elementary functions? [duplicate]

I know that there are integrals that cannot be written in this way such as $e^{-x^2}$ but there is a general rule that can be used to recognize these types of integrals?
marcoturbo's user avatar
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Decomposition of the differential graded algebra

Let $(V,d)$ be the finite dimensional differential graded algebra. Let $V=V_{1}\oplus V_{2},$ $d(V_{1})\subseteq V_{1}.$ If $d(v)=x+y,$ where $v$ belongs to bases of $V_{2}$ and $x\in V_{1},$ $0\neq ...
King Khan's user avatar
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Can every "not-too-big" differential field be thought of as actually consisting of functions?

Now also asked at MO: Let $\sim$ denote the "no-disagreement" relation between partial functions: $f\sim g$ iff there is no $x$ such that $f(x)$ and $g(x)$ are each defined and are distinct. ...
Noah Schweber's user avatar
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Solving Riccati differential equation in general case

What I am trying to do is to solve Riccati equation: $$y'=a(x)y^2+b(x)y+c(x)$$ I introduced new function and called it $Ri(x)$, which satisfy the following ODE: $$y'=y^2+x$$ Is it possible that using ...
Darek's user avatar
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Is the derivative of an algebraic function algebraic? (Solved)

I'm studying Liouville's theory of elementary integration and I came across an unproven fact that the derivative of an algebraic function is algebraic. I can see this intuitively, but I can't ...
Allan Kenedy's user avatar
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If $\delta \colon R[x_1, \dotsc, x_r] \to M$ is an $R$-derivation, then $\delta P = \sum_{i = 1}^r \frac{\partial P}{\partial x_i} dx_i$

Let $\phi \colon R \rightarrow S$ be a ring homomorphism and $M$ an $S$-module. An $R$-derivation $\delta \colon S \rightarrow M$ is a $R$-module homomorphism such that: $\phi(R) \subset \ker(\delta)$...
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Is the integral of a non elementary integral also non elementary? [closed]

Consider a non elementary integral with an integrand like $\frac{\exp(x)}{x}$ or $\exp(\exp(x))$. Is the indefinite integral of such functions necessarily non elementary ?
userrandrand's user avatar
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Is the differential equation of the pendulum unsolvable?

The differential equation for the pendulum is $$\ddot{\theta}=-\frac{g}{L}\sin\theta$$ But physics professors (on youtube at least) turn this equation into $\ddot{\theta}=-\frac{g}{L}\theta$ and $\...
Kamal Saleh's user avatar
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Free sources to learn Field theory.

I want to learn all the theories that are required to learn Differential Algebra. One of them is Field theory. I wonder if there are any free credible sources on the internet about this field where I ...
Kamal Saleh's user avatar
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For a geom. integral curve $X/k$ of char $p$, do the constants $\mathrm{ker}\left[K(X) \to \Omega^1_{X/k}\otimes K(X)\right]$ equal $K(X)^{(p)}$?

The motivations for this question are related to Constants of universal derivation on an $R$-algebra $A$ under localization ($d: S^{-1}A \to S^{-1}\Omega_{A/R}$) Let $X$ be a geometrically integral ...
Somatic Custard's user avatar
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Derivations of algebraic independence

I was wondering if this is this is true or not: If $D$ is a derivation and $x_1,x_2,...,x_n$ are algebraically independent, and $p(x_1,...,x_n)$ is a homogeneous polynomial with all of its monomials ...
F.H.A's user avatar
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Does an elementary antiderivative of $e^{\sin x} \sin x$ exist?

I wonder if an elementary antiderivative of the function $e^{\sin x} \sin x$ exist? If so, could anyone help me to derive this certain antiderivative step by step? If not, is a strict proof of the ...
zyy's user avatar
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Constants of universal derivation on an $R$-algebra $A$ under localization ($d: S^{-1}A \to S^{-1}\Omega_{A/R}$)

Suppose that $A$ is an $R$-algebra, with both $R, A$ integral domains. Let $B = S^{-1}A$ be a localization of $A$. Let $$\partial_A: A \to M$$ $$\partial_B: B \to S^{-1}M$$ be an $R$-derivation on $A$ ...
Somatic Custard's user avatar
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How can one formalize elementary-ness?

I'm trying to decipher what general principals generalize all elementary functions. Does the following hold?: All elementary functions $y = f(x)$ are a solution an equation of the form $P(x,e^x,\ln(x),...
StackQuest's user avatar
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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 ...
StackQuest's user avatar
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174 views

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 ...
StackQuest's user avatar
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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 ...
StackQuest's user avatar
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Is the abstraction of continuity in a general differential field implied?

I'll preface this by saying I don't have an advanced background in abstract algebra, so I'm sorry if these concepts already exist but I just have never heard of them. So, in my study of real analysis, ...
StackQuest's user avatar
7 votes
1 answer
553 views

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 ...
StackQuest's user avatar
4 votes
1 answer
212 views

Showing that exponential of a derivation is automorphism on formal power series ring

Let $R$ be a $\mathbb{Q}-$algebra and let $D_0: R \to R$ be some non-trivial derivation on $R$. Moreover let $R[[T]]$ be formal power series ring. We define $D$ - extension of $D_0$ on $R[[T]]$ as $D\...
MI00's user avatar
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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. ...
Andry's user avatar
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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 ...
Kubrick's user avatar
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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 ...
Red Banana's user avatar
7 votes
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238 views

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}_{\...
DobryZiom's user avatar
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When does $\sqrt{f(x)}\exp{g(x)}$ have an elementary antiderivative?

Liouville's original criterion for elementary anti-derivatives states: If $f,g$ are rational, nonconstant functions, then the antiderivative of $f(x)\exp{g(x)}$ can be expressed in terms of ...
Semiclassical's user avatar
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1 answer
198 views

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....
roymend's user avatar
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3 votes
1 answer
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Loewy decomposition of differential operators

The paper by Fritz Schwarz, "Loewy decomposition of linear differential equations", contains the following lemma, which I try to prove in order to understand the algorithm which Schwarz ...
Roland Salz's user avatar
1 vote
1 answer
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What are the factors of the exterior product of this exterior differential algebra?

I have questions about [Risch 1979] (see the citation and reference below). 1.) What are the factors of the exterior product of the exterior differential algebra in the citation below? I already know ...
IV_'s user avatar
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How to apply Risch's algorithm to $\int\frac{x}{\sqrt{x^4-2x^3+3x^2+4x+1}}\,\mathrm{d}x$?

It is known that $$\int\frac{x}{\sqrt{x^4-2x^3+3x^2+4x+1}}\,\mathrm{d}x=$$ $$-\frac{1}{6}\log\Big((2x^4-10x^3+24x^2-28x+14)\sqrt{x^4-2x^3+3x^2+4x+1}-2x^6+12x^5-36x^4+56x^3-42x^2+13\Big)+C.$$ To get ...
Yizhen Chen's user avatar
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449 views

Prove that $\int \sin(x^2)dx$ is not elementary

See edit It is known that the anti derivative of $\sin(x^2)$ is not an elementary function, and one can represent it using a power series by term-by-term integration of its Taylor series. However, is ...
user12986714's user avatar
7 votes
1 answer
189 views

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 $\delta:R\to R$ ...
Jyrki Lahtonen's user avatar
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1 answer
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Bounded function and Derivatives [closed]

Does eventually all the successive derivatives of a bounded functions become bounded if one of them becomes bounded?(for entire number line case)
Anshul Agrawal's user avatar
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A priori structural identifiability analysis of a system of ordinary differential equation using differential algebra

Consider an ordinary differential equation model of a dynamic system: $\dot{x} = f(x,u,p)$ $y = g(x,p)$ where $x$ is the n-dimensional state vector, $u$ is the r-dimensional input vector, $p$ is ...
Kaushal Kamal Jain's user avatar
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What the orbits of the following DS look like?

Let us consider the following DS: $$x'(t) = y(t)$$ $$y'(t) = −ax(t) + bx^2(t)$$ I have to describe the orbits of the system when $a>0$ and $b=0$. The eigenvalues of the matrix associated to the ...
Alchemy's user avatar
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Find the equation of the orbit of following ODE

We call orbit associated to the initial datum $(t_0,x_0)$, the set of points: $C=\{$x$(t;t_0,x_0), t \in T\}$. Given the Matrix A: \begin{pmatrix}  0 & 1 \\ -1 & 0 \end{pmatrix} Write the ...
Alchemy's user avatar
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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 ...
Zbigniew's user avatar
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1 answer
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Definition of $\mathfrak{g}$-differential graded algebra

I am reading Group actions on manifolds by Eckhard Meinrenken (Lecture Notes, University of Toronto, Spring 2003). In page $45$, definition $5.2$, author introduce the notion of $\mathfrak{g}$-...
Praphulla Koushik's user avatar
4 votes
1 answer
119 views

Are all unital endormorphisms of a Weyl algebra automorphisms?

Given $k$ a field of characteristic $0$, let $A_n(k)$ be the $n$-th Weyl algebra over $k$ – i.e. the unital algebra generated by elements $p_1, ..., p_n$ and $q_1, ..., q_n$ modulo the relations $[p_i,...
Atticus Stonestrom's user avatar
1 vote
1 answer
119 views

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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8 votes
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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 ...
JFox's user avatar
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1 answer
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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 ...
red_trumpet's user avatar
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