Timeline for Analytic analogue of implicit functions for differential operators
Current License: CC BY-SA 4.0
8 events
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| Jun 5 at 15:38 | comment | added | HASouza | @Z.M yes, but this was just for the sake of the argument - when the coefficients and solutions are both polynomials we can just express everything in terms of the Weyl algebra. For analytic solutions one just changes all the above in accordance with the proper ring of differential operators $\mathcal{D}$ and module of coefficients. The point is that even in the polynomial case the answer is not clear to me. | |
| Jun 5 at 11:25 | comment | added | Z. M | You cannot restrict to polynomials for solution space, since after restriction, you are only looking at algebraic solutions, which rarely exist. | |
| Jun 5 at 10:12 | history | edited | HASouza |
Added new tag that became relevant in the comments
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| Jun 5 at 10:09 | comment | added | HASouza | In fact, if $e(M)$ denotes the degree of the Hilbert polynomial of the solution space and $d(M)$ the dimension of the characteristic variety, the statement I'm looking after would be something like this: $e(M) + n = d(M)$. For holonomic $M$ we have $d(M) = n$ and the solution space is finite dimensional, so $e(M) = 0$. For $M$ presented by killing only one derivative $\partial_i$, the characteristic variety is a hyperplane, so $d(M) = 2n -1$, and the solution space is a space of polynomials in $n-1$-variables, so $e(M) = n-1$. | |
| Jun 5 at 7:17 | comment | added | HASouza | I've been thinking a bit more about this, and if $R$ denotes the coordinate ring $\mathbb{R}[x,y,\partial_x,\partial_y]/J$, then the graded object associated to the degree function on the solution space embeds in the group $\operatorname{Hom}_{\operatorname{gr} \mathcal{D}}(R, \mathbb{R}[x,y])$ of graded homomorphisms. But playing around with some examples, I don't think this says much, as for $T = \partial_i$ this graded $\operatorname{Hom}$ group is just isomorphic to $\mathbb{R}[x,y]$! | |
| Jun 4 at 18:44 | comment | added | HASouza | @Z.M restricting from power series to polynomials, to say that $\mathcal{D}/\mathcal{D}T$ is regular holonomic, it means that the graded ideal $J = \operatorname{gr} \mathcal{D}T$ of the polynomial ring $\mathbb{R}[x,y,\partial_x,\partial_y]$ is radical and the associated characteristic variety is 2-dimensional in $\mathbb{R}^4$ (ignoring any issue coming from $\mathbb{R}$ not being algebraically closed). How does this relate to the solution space $\operatorname{Hom}(\mathcal{D}/\mathcal{D}T, \mathbb{R}[x,y])$? Can one read the dimension from this vector space? | |
| Jun 4 at 11:06 | comment | added | Z. M | It seems to be equivalent to ask whether the $\mathcal D$-module $\mathcal D/\mathcal DT$ is regular holonomic. I am not familiar with $\mathcal D$-modules thus I hope that an expert would explain this. | |
| Jun 3 at 17:07 | history | asked | HASouza | CC BY-SA 4.0 |