Questions tagged [differential-algebra]
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40 questions
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When could a diligent calculus student compute all Picard iterates algebraically?
As is well known, in the typical proof of the Picard–Lindelöf theorem, one shows the existence of a solution of the initial value problem $y'(t) = f(t,y(t))$, $y(t_0) = y_0$ by considering the Picard ...
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Can the differential field of d.c.e. reals be nicely construed as a field of functions?
This question is basically a special case of this older question of mine, which is still unanswered.
Let $\mathcal{D}$ be the field of d.c.e. reals; these turn out to be exactly the reals $\alpha$ for ...
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5
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0
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Differential equations analogue of fundamental theorem of symmetric functions
In Gian-Carlo Rota's article "Ten Lessons I Wish I Had Learned Before I Started Teaching Differential Equations", at the end of the third lesson he states a theorem:
"Every differential ...
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7
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1
answer
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Which CAS can do basic non-commutative differential algebra?
This is a repost of my question at MSE from 7 months ago, to which I haven't been able to find an answer yet.
I am looking for a CAS (possibly incl. additional packages/libraries) that can compute ...
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2
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0
answers
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Open problems in differential algebra and affine algebraic geometry
I am currently doing a PhD in differential algebra and affine algebraic geometry at the University of Buenos Aires. I've been struggling to find a list of interesting and big open problems in these ...
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2
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Does there exist a complete implementation of the Risch algorithm?
Is there a generally available (commercial or not) complete implementation of the Risch algorithm for determining whether an elementary function has an elementary antiderivative?
The Wikipedia article ...
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16
votes
6
answers
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A Leibniz-like formula for $(f(x) \frac{d}{dx})^n f(x)$?
Let $f(x)$ be sufficiently regular (e.g. a smooth function or a formal power series in characteristic 0 etc.). In my research the following recursion made a surprising entrance
$$
f_1(x) = f(x),\ f_{n+...
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11
votes
0
answers
300
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Can every "not-too-big" differential field be thought of as actually consisting of functions?
Previously asked and bountied at MSE without success:
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)$ ...
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85
votes
2
answers
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Why is differential Galois theory 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.(e.g. ...
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4
votes
0
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Reference request, or maybe not really a reference request, on differential algebra
Of differential algebra, Gian-Carlo Rota wrote:
No elementary presentation of this beautiful subject has ever been attempted, to the best of my knowledge; Cohen’s book of the twenties is the closest, ...
2
votes
1
answer
141
views
Is there a bound on the number of connected components of a zero set of an integrable function?
If $f$ is a real-analytic function on $[0,1]^n$, and $f$ has finite differential transcendence degree, is there some way to bound the number of connected components of its zero set or the set where it ...
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25
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2
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When can an invertible function be inverted in closed form?
The Risch algorithm answers the question:
"When can a function be integrated in closed form?", see:
https://en.wikipedia.org/wiki/Symbolic_integration
Is anyone aware of any work that answers the ...
- 351
7
votes
0
answers
191
views
How can Gröbner bases be generalized to differential algebra?
I'm aware of the Rosenfeld-Gröbner algorithm "for computing a regular decomposition of a radical differential ideal generated by a set of polynomial differential equations, ordinary or with ...
- 415
34
votes
2
answers
2k
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Is it always possible to calculate the limit of an elementary function?
I already asked this question on https://math.stackexchange.com/questions/2691331/is-it-always-possible-to-calculate-the-limit-of-an-elementary-function but as I received no answer; maybe it is not as ...
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2
votes
1
answer
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The combinatorics of $(f \partial)^n$ in the noncommutative setting?
This is a noncommutative version of these three previous questions:
differential operator power coefficients
Сlosed formula for $(g\partial)^n$
A Leibniz-like formula for $(f(x) \frac{d}{dx})^n f(x)$?
...
- 7,127
6
votes
1
answer
245
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Reference request for results that involve the transcendence degree
Currently I'm reading "On the Decidability of the Real Exponential Field" by Macintyre and Wilkie and the Proof of Theorem 1.1 (page 462-464) uses two algebraic results that involve the ...
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4
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0
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Jet Nestruev's proof that the exterior derivative $d$ on a real line is not a Kähler differential $d_{C^\infty(\mathbb{R})/\mathbb{R}}$
The relation between the exterior derivative $d:C^\infty(\mathbb{R})\to\Omega^1\mathbb(\mathbb{R})$ and the Kähler differential $d_{C^\infty(\mathbb{R})/\mathbb{R}}:C^\infty(\mathbb{R})\to\Omega_{C^\...
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1
vote
1
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References on function fields over imperfect fields in positive characteristic
There are many references (good books, papers, ...) available that treat global function fields (over finite fields) of one independent variable. To name a few, Stichtenoth's book "Algebraic ...
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7
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2
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385
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Books/Lecture notes which contrast Risch algorithm with basic standard procedure of finding an antiderivative
I vaguely remember a book/some lecture notes which introduce integration algorithms such as Risch algorithm by first giving a list of quasi-algorithmic way of evaluating symbolic integrals. (For ...
- 1,755
6
votes
0
answers
867
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How to extend Ritt's theorem on elementary invertible bijective elementary functions?
The elementary functions according to Liouville and Ritt are the functions of a complex variable built up by applying exponentiation, logarithms and/or algebraic operations finitely often. That means, ...
- 1,053
8
votes
1
answer
771
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Non-normal numbers definable without parameters in the langauge of differential rings with composition
Background: It is currently unknown whether $e$ is normal. A natural way to approach this question is to find a class to which $e$ belongs, and prove all members of that class are normal. For example, ...
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17
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5
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Differential Algebra Book
I'm looking for a couple good textbooks covering differential algebra. I'm a prospective Ph.D. student, and this is potentially applicable to my specialization. As such, I'm not afraid of depth; I've ...
8
votes
1
answer
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Semantics of derivations as derivatives
My understanding of how derivations on commutative rings are like derivatives is that a derivation on $R$ is differentiation with respect to a vector field on $\text{Spec}(R)$. But derivations are ...
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23
votes
1
answer
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Does a theory of stochastic differential algebras exist?
My question is motivated primarily by finance, where a non-technical student will learn how to approach SDEs using the symbolic manipulation of Itô calculus and the few basic rules of Brownian motion, ...
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7
votes
2
answers
815
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Differential ideal membership problem
We know that in the ordinary algebra ideal case, the ideal membership problem can be solved by the Grobner Base theory, then, is there a counterpart theory in the differential ideal case?
To be ...
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2
votes
1
answer
2k
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What is the index of a given DAE system of equations?
I have very simple multi-body dynamic system from which I have to solve following DAE:
$ \textbf{q}(t) - 3 \times1 \text{ vector of known state variables} $
$ \phi(\textbf{q}(t))=0 - 2 \times 1 - \...
- 131
1
vote
0
answers
1k
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Are all solutions to an ordinary differential equation continuous solutions to the associated implied differential equation and vice versa?
Now I have to heavily emphasize the fact that I have never studied differential algebra or the concept of other types of differentiation (which is what I believe is the concept behind a differential ...
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3
votes
1
answer
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"Canonical" form for gauge equivalence classes of matrices in $\mathfrak{gl}_n(x)$
Let $\mathfrak{gl}_n(x)= \mathfrak{gl}_n \otimes_\mathbb{C}\mathbb{C}(x)$ be the algebra of matrices taking values in rational functions.
Definition: Two matrices $A, B \in \mathfrak{gl}_n(x)$ are ...
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6
votes
1
answer
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Levelt-Turrittin Theorem over p-adics (or the monodromy theorem)
Let $V$ be a finite dimensional vector space over $\mathbb{C}((t))$. Let $D:V\rightarrow V$ be a differential operator; i.e., an additive $\mathbb{C}$-linear map satisfying
$$
D(a.v)=(t\frac{d}{dt}a)....
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15
votes
2
answers
2k
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Solvability in differential Galois theory
It is well known that the function $f(x) = e^{-x^2}$ has no elementary anti-derivative.
The proof I know goes as follows:
Let $F = \mathbb{C}(X)$. Let $F \subseteq E$ be the Picard-Vessiot extension ...
- 1,214
2
votes
0
answers
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views
Is there an extension of the Kovacic algorithm to handle algebraic coefficients?
Kovacic's algorithm solves second-order linear homogeneous differential equations with rational function coefficients.
I'm wondering if anybody has extended this algorithm to handle algebraic ...
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7
votes
2
answers
471
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Decidability of differential equations
Is there anything well-known about the algorithmic decidability of the satisfiability of an ODE $\dot{x}=f(x)$, $x: [0,1]\to R^n$ with an initial condition $x(0)=x_0$, given that $f(x)$ belongs to ...
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3
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1
answer
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Complications barring differential rings with an infinite number of derivations
In all the texts I have seen on differential algebra, differential rings/fields/algebras/etc. are always specified as having a finite number of derivations (some books e.g. those specifically on ...
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2
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1k
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Why do we need admissible isomorphisms for differential Galois theory?
Background: In Kaplansky's Introduction to Differential Algebra, an isomorphism between differential fields $K, L$ is defined to be admissible if $K,L$ are contained in a larger differential field $M$....
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7
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Differentially closed fields
Let F be a field. Recall that an additive map $d: F\rightarrow F$ is said to be a derivation if $d(ab)=ad(b)+d(a)b$.
Now let $F$ be a ring and let $d$ be a derivation of $F$. Examples I have in mind ...
- 2,751
3
votes
0
answers
204
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Exponential analogue of formal connections
Let $F=\mathbb{C}((t))$. Let $G=GL_n$. Then $G(F)$ acts on $\mathfrak{g}(F)$ by gauge transformation:
$$
g.x:=gxg^{-1} + \dot{g}g^{-1},\quad \quad g\in G(F), \quad x\in \mathfrak{g}(F).
$$
Here, $\...
- 2,751
2
votes
0
answers
193
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Factorisation of twisted polynomials
Let $K=\mathbb{C}((t))$ and let $K_m=\mathbb{C}((t^{1/m}))$. let $K\{x\}$ denote the ring of twisted polynomials. The addition in this ring is defined as usual, but the multiplication is adjusted by ...
- 2,751
0
votes
1
answer
484
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algebraic extensions of (differential) function fields
Let $K$ be a differential field with algebraically closed constant field $C$ (Think $K=\mathbb{C}(x)$ here). I am looking for an example of a simple algebraic extension $L = K[t]$, such, that $t' \...
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Can it be decided whether $\int\root 3 \of{\cos^2(t)}\,dt$ is expressible by elementary functions? [closed]
I would like to decide by methods of Differential Algebra whether the integral $\int\root 3 \of{\cos^2(t)}\,dt$ contrary to the output of CAS Mathematica Online Integrator might
be expressible by ...
1
vote
0
answers
142
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Logarithms in an algebraic differential field
I have this problem: let $Y_1,\dots,Y_n$ be real analytic functions $\mathbb{R}^+\to\mathbb{R}^+$ such that all the $Y_1,\dots,Y_n$ and all their derivatives are algebraically independent over $\...
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