# Questions tagged [d-modules]

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38 questions
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### Bernstein polynomial of $x_1^2+\cdots+x_n^2$

According to S. C. Coutinho’s A Primer of Algebraic $D$-modules (see p. 95), the Bernstein polynomial of $p=x_1^2+x_2^2+\cdots+x_n^2\in K[x_1,\dots,x_n]$ (with $K$ an arbitrary field of characteristic ...
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### Characteristic Variety of the Principal Symbol solves PDE system?

In the study of partial differential equations, it is often considered enough to analyze the principal symbols and their characteristic variety (see, https://en.wikipedia.org/wiki/...
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### When is a D-module holonomic?

Suppose that I have a small family of partial differential equations such that $y(x)$ must be a solution to $$P_i(x,D) y(x)=0,\ \ \ \ \ \ i=1,...,m$$ for all $i=1,...,m$, where I am using multi-...
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### Defining a sheaf of differential operators

I have two questions related to the coordinate-free definition of $\mathcal{D}$-modules provided in Ginzburg's notes (see pages 24-25): Exercise 2.1.13 says that Diff($M,M$) is almost-commutative, i....
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### A question on Weyl algebra $A_1$

This question is from A Primer of Algebraic D-Modules by S.C. Coutinho, Ex 2.4.10. In the following $A_1$ means the Weyl algebra generated by $x$ and $d$. Let $f: A_1^2\to A_1$ be the map defined ...
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### Tensor product of $A_n$ modules/ localisation at ring of differentials

I'm working through Coutinho's "A Primer of Algebraic D-Modules" and I've gotten stuck on the following question: Let $p \in K[x_1, \ldots ,x_n]$ be non-zero, and let $A_n$ be the Weyl Algebra. Show ...
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### A problem in elementary calculus

Let $P(x), Q(x)$ be two polynomials with real coefficients and set $F(x) = \frac{P(x)}{Q(x)}$. Consider a table which has the function $e^{\int_0^x F \, dx}$ on it. The table has the set of rules that ...
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### Is it really unknown that every endomorphism of the Weyl algebra $A_1$ is an isomorphism?

Here $A_1 := K\{x\cdot-, \frac{d}{dx}\} \subset \operatorname{End}_K(K[x])$ for some characteristic-zero field $K$. I found this claim in Coutinho's "A Primer of Algebraic D-Modules." If this is ...
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### Perverse sheaves (or D-modules) on vector spaces, constructible with respect to a hyperplane arrangement

Let $V$ be a finite dimensional complex vector space and let $\mathcal{A}$ be a finite collection of hyperplanes in $V$. Stratify $V$ by the intersections of elements of $\mathcal{A}$, and consider ...
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### Recommendation textbooks on D-module

I am going to take part in a seminar on D-module and applications, the textbooks that will be used are : D-modules, Perverse Sheaves, and Representation Theory, and A Primer of Algebraic D-Modules ...
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### Holonomic ideals and D-finite power series

I would like to understand the connection between the term D-finite power series (in n variables) and the term of a holonomic module over the Weyl algebra A_n. A power series $f \in K[[x_1,...,x_n]]$ ...
I am reading the lecture notes of geometric representation theory: http://www.math.harvard.edu/~gaitsgde/267y/catO.pdf. I have a question on coroot. In general, if we have a root $\alpha$, then the ...