Questions tagged [differential-algebra]
The differential-algebra tag has no usage guidance.
24
questions
2
votes
0answers
14 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 ...
8
votes
1answer
230 views
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, ...
3
votes
0answers
180 views
How to extend Ritt's theorem on elementary invertible bijective elementary functions
The elementary functions of a complex variable $z$ according to Liouville and Ritt are those functions built up from the rational functions of $z$ by exponentiation, taking logarithms, and algebraic ...
28
votes
2answers
2k views
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 ...
6
votes
1answer
526 views
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 ...
18
votes
2answers
960 views
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 ...
3
votes
1answer
166 views
“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 ...
1
vote
0answers
401 views
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 ...
2
votes
1answer
802 views
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 - \...
3
votes
1answer
81 views
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 ...
6
votes
1answer
257 views
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
$$
D(a.v)=(t\frac{d}{dt}a)....
2
votes
0answers
186 views
Factorisation of twisted polynomials
Let $K=\mathbb{C}((t))$ and let $K_m=\mathbb{C}((t^{1/m}))$. let $K\{x\}$ denote the ring of twisted polynomials. The addition in this ring is defined as usual, but the multiplication is adjusted by ...
3
votes
0answers
187 views
Exponential analogue of formal connections
Let $F=\mathbb{C}((t))$. Let $G=GL_n$. Then $G(F)$ acts on $\mathfrak{g}(F)$ by gauge transformation:
$$
g.x:=gxg^{-1} + \dot{g}g^{-1},\quad \quad g\in G(F), \quad x\in \mathfrak{g}(F).
$$
Here, $\...
6
votes
0answers
152 views
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 ...
20
votes
1answer
499 views
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, ...
55
votes
2answers
11k views
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. ...
14
votes
5answers
2k views
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 ...
2
votes
0answers
133 views
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 $\...
2
votes
0answers
296 views
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 ...
5
votes
2answers
242 views
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 ...
0
votes
1answer
337 views
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' \...
13
votes
2answers
2k views
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 ...
6
votes
2answers
714 views
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 ...
15
votes
2answers
1k views
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 $M$....
