# 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, ...
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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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### 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 ...
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### 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 ...
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 $(f(x) \frac{d}{dx})^n f(x)$? ...
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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 ...
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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 ...
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 ...