Questions tagged [differential-algebra]

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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, ...
Michael Hardy's user avatar
10 votes
0 answers
224 views

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 $g(x)$ ...
Noah Schweber's user avatar
7 votes
0 answers
141 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 ...
Brent Baccala's user avatar
2 votes
1 answer
119 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 ...
L.C. Brown's user avatar
2 votes
1 answer
219 views

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 $(f(x) \frac{d}{dx})^n f(x)$? ...
M.G.'s user avatar
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16 votes
6 answers
1k views

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+...
M.G.'s user avatar
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1 vote
1 answer
119 views

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 ...
Andry's user avatar
  • 93
6 votes
1 answer
229 views

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 ...
Bytegear's user avatar
  • 123
4 votes
0 answers
284 views

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^\...
Fallen Apart's user avatar
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63 votes
2 answers
22k views

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 ...
Timothy Chow's user avatar
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7 votes
2 answers
312 views

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 ...
Ma Joad's user avatar
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2 votes
0 answers
35 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 ...
Brent Baccala's user avatar
8 votes
1 answer
723 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, ...
James's user avatar
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5 votes
0 answers
818 views

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, ...
IV_'s user avatar
  • 1,063
34 votes
2 answers
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 ...
Olivier Esser's user avatar
8 votes
1 answer
736 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 ...
Alex Mennen's user avatar
  • 2,080
25 votes
2 answers
2k 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 ...
P. Carr's user avatar
  • 351
3 votes
1 answer
216 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 ...
Saal Hardali's user avatar
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1 vote
0 answers
928 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 ...
user64742's user avatar
  • 111
2 votes
1 answer
2k 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 - \...
Maverick's user avatar
  • 131
3 votes
1 answer
100 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 ...
BHT's user avatar
  • 191
6 votes
1 answer
376 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)....
Dr. Evil's user avatar
  • 2,519
2 votes
0 answers
191 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 ...
Dr. Evil's user avatar
  • 2,519
3 votes
0 answers
200 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, $\...
Dr. Evil's user avatar
  • 2,519
7 votes
0 answers
276 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 ...
Dr. Evil's user avatar
  • 2,519
23 votes
1 answer
923 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, ...
user85875's user avatar
  • 231
82 votes
2 answers
18k 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. ...
Henry.L's user avatar
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16 votes
5 answers
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 ...
Benjamin Roycraft's user avatar
1 vote
0 answers
142 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 $\...
Andre's user avatar
  • 11
2 votes
0 answers
311 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 ...
Wolfgang Tintemann's user avatar
7 votes
2 answers
429 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 ...
Peter Franek's user avatar
0 votes
1 answer
456 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' \...
goens's user avatar
  • 103
15 votes
2 answers
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 ...
the L's user avatar
  • 1,194
7 votes
2 answers
795 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 ...
Jiang's user avatar
  • 1,508
15 votes
2 answers
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 $M$....
Akhil Mathew's user avatar
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