Oaku, Toshinori

Kyoto University Research Information Repository

TitleA localization algorithm for $D$-modules and its application(Microlocal Analysis and PDE in the Complex Domain)Author(s)Oaku, ToshinoriCitation数理解析研究所講究録 (2000), 1159: 74-80Issue Date2000-06URL http://hdl.handle.net/2433/64211RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityA localization algorithm for $D$-modules and itsapplicationToshinori OakuTokyo Woman’s Christian University( )First, we review the localization algorithm given in a joint paper with N. Takayama(Kobe) and U. Walther $(\mathrm{M}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{o}\mathrm{t}\mathrm{a}/\mathrm{M}\mathrm{S}\mathrm{R}\mathrm{I})$ [6] with slightly different reasoning of thecorrectness. The latter part applies this algorithm to the problem of finding the annihilatorideal of some elementary functions.1 A localization algorithmWe work entirely in the algebraic category. Put $X=\mathbb{C}^{n}$ and $Y=\{x=(x_{1}, \ldots, x_{n})\in$$X|f(x)=0\}$ with a nonzero polynomial $f\in \mathbb{C}[x]$ . We denote by $D_{X}$ the sheafof algebraic differential operators on $X$ . Let $\mathcal{M}$ be a coherent left $D_{X}$-module on $X$such that $\mathcal{M}$ is holonomic on $X\backslash Y$ . Then Kashiwara ([2]) proved that the localization$\mathcal{M}[1/f]:=\mathcal{M}\otimes_{\mathcal{O}_{X}}O_{X}[1/f]$ of $\mathcal{M}$ by $f$ is a holonomic $D_{X}$-module on $X$ , where $\mathcal{O}_{X}$ is thesheaf of regular functions on X. (In fact he proved this fact in the analytic category, whichis a stronger statement.) Here we remark that starting from an algebraic (i.e., a $D_{X}-$ )module, the localization is the same both in the algebraic and in the analytic category.More precisely, if we denote by $O_{X}^{\mathrm{a}\mathrm{n}}$ and $D_{X}^{\mathrm{a}\mathrm{n}}$ the sheaves of holomorphic functions and ofholomorphic differential operators on $X$ respectively, then we have an isomorphism$\mathcal{O}_{X}^{\mathrm{a}\mathrm{n}}[1/f]\otimes o_{X}^{\mathrm{a}}\mathrm{n}(D^{\mathrm{a}\mathrm{n}}\otimes D_{X}\mathcal{M})xD\mathcal{M}\simeq D_{Xx}^{\mathrm{a}}\mathrm{n}\otimes[1/f]$.Our claim is that $\mathcal{M}[1/f]$ is computable if the input, i.e. both $\mathcal{M}$ and $f$ are defined overa computable subfield of $\mathbb{C}$ (e.g. over $\mathbb{Q}$).Now let us explain our algorithm. Introcducing an auxiliary variable $t$ , put$W:=\{(t, x)\in \mathbb{C}\cross X|tf(x)=1\}$and let $\iota$ : $Warrow \mathbb{C}^{n+1}$ be the natural embedding. Let $p:Warrow X$ be the projection$p(t, x)=x$ and $\varphi$ : $X\backslash Yarrow W$ be the isomorphism defined by $\varphi(x)=(1/f(x), x)$ . Let1159 2000 74-80 74$j$ : $X\backslash Yarrow X$ be the natural embedding. Thus we have a commutative diagram$W$ $arrow\iota \mathbb{C}\cross X$$\varphi\uparrow$ $p\downarrow$$X\backslash Yrightarrow j$ $X$Then by using the integration functor (in the algebraic category) we get$\mathcal{M}[1/f]=\int_{j}j^{-1}\mathcal{M}=\int_{p}\int_{\iota}\int_{\varphi}j-1\mathcal{M}$ .(See e.g. [7].) Our algorithm simply performs the rightmost successive integration stepby step. (For the sake of simplicity we describe the algorithm in the case where $\mathcal{M}$ isgenerated by one element.) We denote by $D_{n}=D_{n}(\mathbb{C})$ the Weyl algebra on $n$ variables$x$ with coefficients in $\mathbb{C}$ . Then we can regard $D_{n}$ as the set of global sections $\Gamma(X, D_{X})$of $D_{X}$ . In general, for a left coherent $D_{X}$ -module $\mathcal{M}$ , its global sections $M:=\mathrm{r}(x, \mathcal{M})$is a finitely generated left $D_{n}$-module and its correspondence yields an equivalence ofthe category of left coherent $D_{X}$ -modules on $X$ and that of finitely generated left $D_{n^{-}}$modules since $X$ is affine. The converse correspondence is given by the sheafification (orthe ‘localization’) $\mathcal{M}=D_{X}\otimes_{D_{n}}M$.Algorithm-Theorem 1Input: A polynomial $f\in \mathbb{Q}[x]$ and a finite subset $\{P_{1}, \ldots, P_{r}\}$ of $D_{n}(\mathbb{Q})$ which generatesa left ideal $I$ of $D_{n}$ such that the sheafification $\mathcal{M}$ of $M:=D_{n}/I$ is holonomic on $X\backslash Y$ .(1)(a) Put $\theta_{i}:=\partial_{i^{-}}t^{2}fi\partial_{t}$ with $f_{i}:=\partial f/\partial x_{i},$ $\partial i:=\frac{\partial}{\partial x_{i}}$ , and $\partial_{t}:=\frac{\partial}{\partial t}$ .(b) Compute $\overline{P}_{i}:=P_{i}(x, \theta_{1}, \ldots, \theta n)$ . More precisely, Writing $P_{i}$ in the normal form(i.e. make the derivations first and then multiply by polynomials) and substitute$\theta_{i}$ for $\partial_{i}$ in $P_{i}$ . (Note that $\theta_{1},$ $\ldots,$ $\theta_{n}$ commute with one another.)(c) Let $J$ be the left ideal of $D_{n+1}$ generated $\mathrm{b}\mathrm{y}.\overline{P}_{1},$ $\ldots,\overline{P}_{r}$ and 1 $-tf(x)$ and put$N:=D_{n+1}/J$(2) Compute $N/\partial_{t}N$ as left $D_{n}$-module as follows:(a) Let $G$ be an involutive basis of $J$ with respect to the weight vector$w=(1,0, \ldots, 0;-1,0, \ldots , 0)$for $(t, x_{1}, \ldots , x_{n};\partial_{t,1}\partial, \ldots, \partial_{n})$ .(b) Compute a generator $b(s)$ of the ideal{ $b(s)\in \mathbb{C}[s]|b(t\partial_{t})+Q\in J$ with some $Q\in D_{n+1}$ such that $\mathrm{o}\mathrm{r}\mathrm{d}_{w}(Q)\leq-1$ }75of $\mathbb{C}[s]$ , where $\mathrm{o}\mathrm{r}\mathrm{d}_{w}(Q)$ denotes the maximum weight of the terms of $Q$ with respectto the weight $w$ . $b(s)$ can be computed by eliminating $x$ and $\partial_{1},$ $\ldots,$ $\partial_{n}$ from thehighest weight parts (w.r.t. $w$ ) of elements of $G$ . Find the largest nonnegativeinteger root $k_{1}$ of $b(s)=0$ . If there is no nonnegative integer root, then we have$M[1/f]=0$.(c) In general, for $P\in D_{n+1}$ , there exist unique $Q\in D_{n+1}$ and $R\in D_{n}[t]$ such that$P=\partial_{t}Q+R$ .Let us denote this $R$ by $R=\rho(P)$ . Then $R$ can be regarded as a relation amongthe residue classes $\overline{1},$ $\overline{t},$ $\overline{t^{2}},$ $\ldots$ in $N/\partial_{t}N$ . Let $L$ be the left $D_{n}$-submodule of $D_{n}+$$tD_{n}+\cdots+t^{k_{1}}D_{n}\simeq D_{n^{1}}^{k+1}$ generated by$\{\rho(t^{j}P)|P\in G, \mathrm{o}\mathrm{r}\mathrm{d}_{w}(P)+j\leq k_{1}\}$ .Then $L$ defines a system of linear differentieal equations for $\overline{1},$ $\overline{t},$ $\cdots\overline{tk_{1}}$ in $N/\partial_{t}N$ .(d) Eliminate $\overline{1},\overline{t},$ $\cdot$ , . $\overline{t^{k_{1}-1}}$ from $L$ and obtain an ideal $L_{0}$ of $D_{n}$ which annihilates $\overline{tk_{1}}$.Output: $\mathcal{M}[1/f]$ is isomorphic to the sheafification of $D_{n}/L_{0}$ . More precisely the ideal$L_{0}$ is the annihilator ideal of $f^{-k_{1}-2}u$ , which generates $M[1/f]$ (here $u$ is the residue classof 1 in $M$).Proof: First we have$j^{-1}\mathcal{M}=D_{X}[1/f]/(D_{X}[1/f]P_{1}+\cdots+D_{X}[1/f]P_{r})$ ,which is a holonomic $D_{X}[1/f]$ -module. Let $A_{W}$ be the subring of $D_{n+1}$ generated by$\mathbb{C}[t, x]$ and $\theta_{1},$ $\ldots,$ $\theta_{n}$ . Then $A_{W}(tf(x)-1)$ is a two-sided ideal of $A_{W}$ and $D_{W}$ $:=$$A_{W}/A_{W}(1-tf(X))$ is the set of global sections of the sheaf $D_{X}$ of algebraic differentialoperators on $W$ (note that $W$ is affine). Then we have an isomorphism (see [4])$\int_{\varphi}j^{-1}\mathcal{M}\simeq D_{W}/(D_{W}\overline{P}1+\cdots+DW\overline{P}_{r})$.Next, the integration along $\iota$ is nothing but the so-called Kashiwara equivalence and inview of Proposition A.l of [4] we have$\int_{\iota}\int_{\varphi}j^{-}1\mathcal{M}=D\mathbb{C}\mathrm{x}x\otimes_{D}n+1N$,which is a holonomic $D_{\mathbb{C}\cross X}$ -module. Next by the definition of the integration we have$\int_{p}N=N/\partial_{t}N$ .76Let $\mathcal{F}(J)$ be the partial Fourier transform of $J$ with respect to $t$ , which is the ringisomorphism of $D_{n+1}$ that sends $t$ to $-\partial_{t},$ $\partial_{t}$ to $t$ , and leaves $x_{i},$ $\partial_{i}$ unchanged. Put$\mathcal{F}(N):=D_{n+1}/\mathcal{F}(J)$ . Then we have$N/\partial_{t}N\simeq \mathcal{F}(N)/t\mathcal{F}(N)$and the step (2) is nothing but (the Fourier transform of) the restriction algorithm (The-orem 5.7) of [5]. Note that $N/\partial_{t}N$ is a holonomic $D_{n}$-module since $N$ is holonomic on$\mathbb{C}\cross X$ . Thus we have proved that$M[1/f]\simeq N/\partial_{t}N\simeq(D_{n})^{k_{1}}+1/L$ . (1)Let us describe the first isomorphism of (1) more explicitly. First note the isomorphism$N/\partial_{t}N\simeq D_{n+1}/(J+\partial_{t}D_{n+1})$ .For an arbitrary $P\in D_{n+1}$ , there exist unique $R_{0},$ $R_{1},$ $\ldots\in D_{n}$ and $S\in D_{n+1}$ such that$P= \sum_{j\geq 0}tjRj(X, \theta_{1}, \ldots, \theta_{n})+\partial_{t}s$. (2)Then we define$\psi(P)$ $:=$$\sum_{j\geq 0}f^{-j}-2Rj(X, \partial_{1}, \ldots, \partial_{n})u$$=$ $\sum_{j\geq 0}R_{j}(x,$ $\partial_{1}+(j+2)\frac{f_{1}}{f},$ $\ldots,$$\partial n+(j+2)\frac{f_{n}}{f})f^{-j}-2u$ $\in M[1/f]$ .Note that since the commutation relation of $x_{i}$ and $\theta_{i}$ is the same as that of $x_{i}$ and $\partial_{i}$ , theabove (non-commutative) substitution makes sense irrespective of the actual expressionof $R_{j}$ . This defines a left $D_{n}$-homomorphism $\psi$ : $D_{n+1}arrow M[1/f]$ . In fact, we have$\psi(\partial_{i}P)=\partial_{i}\psi(P)$ since$\psi(\partial_{i}t^{j}R_{j}(x, \theta_{1}, \ldots, \theta_{n}))$ $=$ $.\psi(t^{j}(\theta_{i}+t^{2}f_{i}\partial t)R_{j}(x, \theta 1, \ldots, \theta_{n}))$$=$ $\psi(t^{j}(\theta_{i}-(j+2)tf_{i})+\partial_{t}t^{j+2}fi)R_{j(\theta_{n}))}x,$$\theta 1,$ $\ldots$ ,$=$ $\psi(t^{j}(\theta_{i}-(j+2)tfi)R_{j(x,\theta\theta_{n}))}1,$$\ldots$ ,$=$ $f^{-j-2}\partial_{i}Rj(x, \partial 1, \ldots, \partial_{n})u-(j+2)f^{-}j-\mathrm{s}f_{i}R_{j}(x, \partial 1, \ldots, \partial n)$$=$ $\partial_{i}f^{-j-2}Rj(X, \partial 1, \ldots, \partial_{n})u$ .Since $P_{1},$ $\ldots$ , $P_{r}$ annihilate $u$ , we get$\psi(t^{j}\overline{P}_{i})=f^{-j-2}P_{i}(X, \partial 1, \ldots, \partial n)u=0$ .77It is easy to see that $\psi(t^{j}(1-tf(X))=0$ . Hence $J+\partial_{t}D_{n+1}$ is contained in the kernel of$\psi$ .Conversely, suppose that $P$ of the form (2) is contained in the kernel of $\psi$ . Then thereexist $Q_{1}(t, x, \partial),$ $\ldots,$ $Qr(t,$ $x,$ $\partial \mathrm{I}\in D_{n}[t]$ such that$\sum_{j\geq 0}f^{-j}Rj(x, \partial_{1}, \ldots, \partial n)=\sum_{i=1}^{r}f^{2}Qi(1/f, X, \partial_{1,\ldots,n}\partial)Pi(x, \partial_{1}, \ldots, \partial n)$holds in $D_{n}[1/f]$ . Then the Hilbert Nullstellensatz assures that$\sum_{j\geq 0}t^{j}Rj(x, \theta 1, \ldots, \theta n)-\sum^{r}f^{2}Q_{i}(t, X, \theta 1, \ldots, \theta i=1n)Pi(x, \theta_{1}, \ldots, \theta n)$ $\in(1-tf(X))Dn+1$since $1-tf(x)$ is irreducible. Noting $(1-tf(x))\theta_{i}\in D_{n+1}(1-tf(X))$ , we conclude that$P\in J+\partial_{t}D_{n+1}$ . Thus $\psi$ gives the first isomorphism of (1). This implies that $M[1/f]$ is$\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{d}\sim$ by $f^{-2}u,$ $\ldots,$ $f-k_{1}-2u$ , and hence only by $f^{-k_{1}-2}u$ . This completes the proof.2 An application to holonomic functionsLet $u$ be a (possibly multivalued) analytic function defined on $\mathbb{C}^{n}$ minus an algebraicset. Supppose that $u$ is hyperexponential ([1]); i.e., $g_{i}:=\partial_{i}u/u$ is a rational function forany $i=1,$ $\ldots,$ $n$ . For example, if $f_{1},$ $\ldots,$ $f_{m},$ $g$ are rational functions and $\alpha_{1},$ $\ldots,$ $\alpha_{m}$ arecomplex numbers, then$u=f_{1}^{\alpha_{1}}\cdots f_{m}^{\alpha_{m}}\exp(g(_{X))}$is a hyperexponential function. Then we can find the annihilator ideal$\mathrm{A}\mathrm{n}\mathrm{n}(u):=\{P\in D_{n}| Pu=0\}$of $u$ exactly by applying the localization algorithm.Algorithm-Theorem 2Input: Let $u$ be a (possibly) multi-valued analytic function such that $g_{i}:=\partial_{i}u/u\in \mathbb{Q}(x)$for any $i=1,$ $\ldots,$ $n$ .(1) Let $g\in \mathbb{Q}[x]$ be the least common multiple of the denominators of $g_{1},$ $\ldots,$ $g_{n}$ . Let$f(x)$ be the square-free part of $g$ .(2) Put $I:=D_{n}(g\partial_{1^{-g}}g1)+\cdots+D_{n}(g\partial_{n}-gg_{n})$ .(3) Apply Algorithm-Theorem 1 with input $D_{n}/I$ and $f$ , and let $L_{0}$ be the output idealwith the integer $k_{1}$ .78(4) Compute the ideal quotient$L_{1}:=L_{0}$ : $(f^{k_{1}+2})=\{P\in D_{n}|Pf^{k_{1}+2}\in L_{0}\}$by syzygy computation through Gr\"obner basis.Output: $L_{1}=\mathrm{A}\mathrm{n}\mathrm{n}(u)$ . In particular, $u$ is a holonomic function, i.e., $D_{n}/\mathrm{A}\mathrm{n}\mathrm{n}(u)$ is aholonomic system.Proof: Put $\mathcal{L}:=D_{X}u$ , which is a sheaf of multivalued analytic functions, and definethe sheaf$Ann(u):=\{P\in D_{X}| Pu=0\}$ ,which is the sheafification of $\mathrm{A}\mathrm{n}\mathrm{n}(u)$ . Then we have $\mathcal{L}\simeq D_{X}/Ann(u)$ . Let $\mathcal{M}$ be thesheafification of $M=D_{n}/I$ . It is easy to see that $\mathcal{M}$ is a holonomic system of rank oneoutside of $Y:=\{x\in X=\mathbb{C}^{n}|f(x)=0\}$ . This implies that the two sheaves $\mathcal{M}$ and $\mathcal{L}$coincide on $X\backslash Y$ . Hence in view of the Hilbert Nullstellensatz, we have$\mathcal{M}[1/f]=\mathcal{L}[1/f]$ . (3)By Algorithm-Theorem 1, $M[1/f]$ is generated by $f^{-k_{1}-2}\overline{1}$ whose annihilator ideal is $L_{0}$ .Hence $L_{1}$ is the annihilator ideal of $\overline{1}$ in $M[1/f]$ .On the other hand, since $\mathcal{L}$ is a set of analytic functions, the natural homomorphism$\mathcal{L}arrow \mathcal{L}[1/f]=O_{X}[1/f]\otimes \mathit{0}_{X}\mathcal{L}$induced by the embbeding of $\mathcal{O}_{X}$ to $\mathcal{O}_{X}[1/f]$ is injective. In fact, this follows from thefact that $f$ . : $\mathcal{L}arrow \mathcal{L}$ is injective. By the isomorphism (3), $\overline{1}\in M[1/f]$ correspondsto $u\in \mathcal{L}$ , and its annihilator ideal in $\mathcal{L}[1/f]$ is given by $L_{1}$ . Since $\mathcal{L}$ is a submodule of$\mathcal{L}[1/f]$ , the annihilator ideal of $u$ in $\mathcal{L}[1/f]$ coincides with that in $\mathcal{L}$ . This implies that$L_{1}=\mathrm{A}\mathrm{n}\mathrm{n}(u)$ . This completes the proof.Example 1 Put $X:=\{(x, y, z)\in \mathbb{C}^{3}\}$ and $f(x, y, z):=x^{3}-y^{2}z^{2}$ . Let us find theannihilator ideal of the function $u:=\exp(1/f(x))$ . The following computations wereperformed by computer algebra systems $\mathrm{k}\mathrm{a}\mathrm{n}/\mathrm{s}\mathrm{m}\mathrm{l}[9]$ and $\mathrm{R}\mathrm{i}\mathrm{s}\mathrm{a}/\mathrm{A}\mathrm{s}\mathrm{i}\mathrm{r}[8]$ which are connectedvia open xxx protocol [10]. First let $I$ be the left ideal of $D_{3}$ generated by$f^{2}\partial_{x}-f_{x}$ , $f^{2}\partial_{y}-f_{y}$ , $f^{2}\partial_{z}-f_{z}$with $\partial_{x}=\partial/\partial x,$ $f_{x}=\partial f/\partial x$ , and so on. By computing the characteristic variety Char $(M)$of $M:=D_{3}/I$ (see [3] for an algorithm) and by decomposing it to prime (or primary)factors, we know thatChar $(M)\supset\{(x, y, z;\xi, \eta, \zeta)\in T^{*}\mathbb{C}^{3}|x=y=0\}\cup\{x=z=0\}$ .79In particular, $M$ is not holonomic on $\mathbb{C}^{3}$ . Next by using Algorithm-Theorem 1 with $I$ and$f$ as input, we know that $\mathrm{A}\mathrm{n}\mathrm{n}(u)$ is generated by the following eight operators:$36y\partial_{y}-36_{Z}\partial_{z}$ ,$-24y_{Z^{2}\partial_{x}36}-x^{2}\partial_{y}$ ,$-24y^{2_{Z}}\partial_{x}-36x^{2}\partial_{z}$ ,$-24z^{3}\partial x\partial \mathcal{Z}-36x\partial^{2}2-y24z^{2}\partial_{x}$ ,$24x^{4}\partial^{2}x+72x^{3}Z\partial_{x}\partial z+54xz22\partial_{z}2+96x^{3}\partial_{x}+162x^{2_{Z}}\partial_{z}+72\partial_{x}$ ,$36y^{2_{Z}3}\partial_{z}-24X^{4}\partial_{x}-72x^{3_{Z\partial_{z}}}-72$,$-36y_{Z^{4}}\partial_{z}^{2}+24x^{4}\partial_{xy}\partial+72_{X^{3}Z}\partial_{yz}\partial-108yZ^{3}\partial_{z}+72\partial_{y}$ ,$36Z^{5}\partial 3-zy24X^{4}\partial_{x}\partial 2-72x^{3}Z\partial_{y}^{2}\partial z+216_{Z^{4}}\partial_{z}2+216Z^{3}\partial_{z}-72\partial_{y}^{2}$.We can verify that $D_{3}/\mathrm{A}\mathrm{n}\mathrm{n}(u)$ is in fact holonomic and that $I$ is contained in $\mathrm{A}\mathrm{n}\mathrm{n}(u)$ .[1] Almkvist, G., Zeilberger, D., The method of differentiating under the integral sign.J. Symbolic Computation 10 (1990), 571-591.[2] Kashiwara, M., On the holonomic systems of linear differential equations, II. Invent.Math. 49 (1978), 121-135.[3] Oaku, T., Computation of the characteristic variety and the singular locus of a systemof differential equations with polynomial coefficients. Japan J. Indust. Appl. Math.11 (1994), 485-497.[4] Oaku, T., Gr\"obner bases for $D$-modules on a non-singular affine algebraic variety.T\^ohoku Math. J. 48 (1996), 575-600.[5] Oaku, T., Algorithms for $b$-functions, restrictions, and algebraic local cohomologygroups of $D$-modules. Advances in Appl. Math. 19 (1997), 61-105.[6] Oaku, T., Takayama, N., Walther, U.: A localization algorithm for $D$-modules. toappear in J. Symbolic Computation.[7] , $D$ , 1995.[8] $\mathrm{f}\mathrm{t}\mathrm{p}://\mathrm{e}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{a}\mathrm{V}\mathrm{o}\mathrm{r}$ . fuj itsu. $\mathrm{c}\mathrm{o}.\mathrm{j}\mathrm{p}/\mathrm{p}\mathrm{u}\mathrm{b}/\mathrm{i}\mathrm{S}\mathrm{i}\mathrm{S}/\mathrm{a}\mathrm{S}\mathrm{i}\mathrm{r}$[9] $\mathrm{W}\mathrm{W}\mathrm{W}.\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{h}.\mathrm{k}\mathrm{o}\mathrm{b}\mathrm{e}-_{\mathrm{u}}.\mathrm{a}\mathrm{C}.\mathrm{j}\mathrm{p}/\mathrm{K}\mathrm{A}\mathrm{N}$[10] $\mathrm{w}\mathrm{w}\mathrm{w}.\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{h}.\mathrm{k}\mathrm{o}\mathrm{b}\mathrm{e}^{-}\mathrm{u}.\mathrm{a}\mathrm{C}.\mathrm{i}\mathrm{P}/\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{n}\mathrm{x}\mathrm{X}\mathrm{X}$80

A localization algorithm for $D$-modules and its application (Microlocal Analysis and PDE in the Complex Domain)

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A localization algorithm for $D$ -modules and its application (Microlocal Analysis and PDE in the Complex Domain)

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A localization algorithm for DDD-modules and its application (Microlocal Analysis and PDE in the Complex Domain)

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A localization algorithm for $D$ -modules and its application (Microlocal Analysis and PDE in the Complex Domain)