document { Key => [DHom,Strategy], "Option is passed to Dresolution. See ", TO [Dresolution,Strategy] } document { Key => {DHom, (DHom,Module,Module), (DHom,Module,Module,List), (DHom,Ideal,Ideal)}, Headline=>"D-homomorphisms between holonomic D-modules", Usage => "DHom(M,N), DHom(M,N,w), DHom(I,J)", Inputs => { "M" => Module => {"over the Weyl algebra ", EM "D"}, "N" => Module => {"over the Weyl algebra ", EM "D"}, "I" => Ideal => {"which represents the module ", EM "M = D/I"}, "J" => Ideal => {"which represents the module ", EM "N = D/J"}, "w" => List => "a positive weight vector" }, Outputs => { HashTable => {" a basis of D-homomorphisms between holonomic D-modules ", EM "M", " and ", EM "N" } }, "The set of D-homomorphisms between two holonomic modules ", EM "M", " and ", EM "N", " is a finite-dimensional vector space over the ground field. Since a homomorphism is defined by where it sends a set of generators, the output of this command is a list of matrices whose columns correspond to the images of the generators of ", EM "M", ". Here the generators of ", EM "M", " are determined from its presentation by generators and relations.", PARA { "The procedure calls ", TO "Drestriction", ", which uses ", EM "w", " if specified." }, PARA { "The algorithm used appears in the paper 'Computing homomorphisms between holonomic D-modules' by Tsai-Walther(2000). The method is to combine isomorphisms of Bjork and Kashiwara with the restriction algorithm."}, EXAMPLE lines /// W = QQ[x, D, WeylAlgebra=>{x=>D}] M = W^1/ideal(D-1) N = W^1/ideal((D-1)^2) DHom(M,N) ///, Caveat => {"Input modules ", EM "M", ", ", EM "N", ", ", EM "D/I", " and ", EM "D/J", " should be holonomic."}, SeeAlso => {"DExt", "Drestriction"} } document { Key => [DExt,Strategy], "Option is passed to Dresolution. See ", TO [Dresolution,Strategy] } document { Key => [DExt,Special] } document { Key => Special, SeeAlso => "DExt" } document { Key => None, Headline => "an option for DExt=>Special", SeeAlso => "DExt" } document { Key => [DExt,Output]} document { Key => Output } document { Key => [DExt,Info] } document { Key => Info } document { Key => {DExt, (DExt, Module, Module), (DExt, Module, Module, List)}, Headline => "Ext groups between holonomic modules", Usage => "DExt(M,N), DExt(M,N,w)", Inputs => { "M" => Module => {"over the Weyl algebra ", EM "D"}, "N" => Module => {"over the Weyl algebra ", EM "D"}, "w" => List => "a positive weight vector" }, Outputs => { HashTable => {" the ", TT "Ext"," groups between holonomic D-modules", EM "M", " and ", EM "N" } }, "The ", TEX "Ext", " groups between D-modules ", EM "M"," and ", EM "N", " are the derived functors of ", TEX "Hom", ", and are finite-dimensional vector spaces over the ground field when ", EM "M", " and ", EM "N", " are holonomic.", PARA { "The procedure calls ", TO "Drestriction", ", which uses ", EM "w", " if specified." }, PARA { "The algorithm used appears in the paper 'Polynomial and rational solutions of holonomic systems' by Oaku-Takayama-Tsai (2000). The method is to combine isomorphisms of Bjork and Kashiwara with the restriction algorithm."}, EXAMPLE lines /// W = QQ[x, D, WeylAlgebra=>{x=>D}] M = W^1/ideal(x*(D-1)) N = W^1/ideal((D-1)^2) DExt(M,N) ///, Caveat =>{ "Input modules M, N should be holonomic.", "Does not yet compute explicit representations of Ext groups such as Yoneda representation." }, SeeAlso => {"DHom", "Drestriction"} } document { Key => {Ddual, (Ddual,Module), (Ddual,Ideal)}, Headline => "holonomic dual of a D-module", Usage => "Ddual M, Ddual I", Inputs => { "M" => Module => {"over the Weyl algebra ", EM "D"}, "I" => Ideal => {"which represents the module ", EM "M = D/I"} }, Outputs => { Module => {"the holonomic dual of ", EM "M"} }, "If M is a holonomic left D-module, then ", BOLD "Ext", SUP "n", SUB "D", "(", EM "M,D", ")", " is a holonomic right D-module. The holonomic dual is defined to be the left module associated to ", BOLD "Ext", SUP "n", SUB "D", "(", EM "M,D", ")", ". The dual is obtained by computing a free resolution of ", EM "M", ", dualizing, and applying the standard transposition to the ", EM "n", "-th homology.", EXAMPLE lines /// I = AppellF1({1,0,-3,2}) Ddual I ///, Caveat =>{"The input module ", EM "M", " should be holonomic. The user should check this manually with the script ", TT "Ddim", "."}, SeeAlso => {"Ddim", "Dtransposition"} } document { Key => [polynomialExt,Strategy], "Option is passed to Dresolution. See ", TO [Dresolution,Strategy] } document { Key => {polynomialExt, (polynomialExt,Module), (polynomialExt,ZZ,Ideal), (polynomialExt,ZZ,Module), (polynomialExt,Ideal)}, Headline => "Ext groups between a holonomic module and a polynomial ring", Usage => "polynomialExt M, polynomialExt I; rationalFunctionExt(i,M), rationalFunctionExt(i,I)", Inputs => { "M" => Module => {"over the Weyl algebra ", EM "D"}, "I" => Ideal => {"which represents the module ", EM "M = D/I"}, "i" => ZZ => "nonnegative" }, Outputs => { { ofClass HashTable, " or ", ofClass Module, ", the ", TEX "Ext^i"," group(s) between holonomic ", EM "M", " and the polynomial ring" } }, "The ", TT "Ext", " groups between a D-module ", EM "M", " and the polynomial ring are the derived functors of ", TT "Hom", ", and are finite-dimensional vector spaces over the ground field when ", EM "M", " is holonomic.", PARA { "The algorithm used appears in the paper 'Polynomial and rational solutions of holonomic systems' by Oaku-Takayama-Tsai (2000). The method is to combine isomorphisms of Bjork and Kashiwara with the restriction algorithm."}, EXAMPLE lines /// W = QQ[x, D, WeylAlgebra=>{x=>D}] M = W^1/ideal(x^2*D^2) polynomialExt(M) ///, Caveat =>{"Does not yet compute explicit representations of Ext groups such as Yoneda representation."}, SeeAlso => {"polynomialSolutions", "rationalFunctionExt", "DExt", "Dintegration"} } document { Key => [rationalFunctionExt,Strategy], "Option is passed to Dresolution. See ", TO [Dresolution,Strategy] } document { Key => {rationalFunctionExt, (rationalFunctionExt,Module), (rationalFunctionExt,ZZ,Ideal,RingElement), (rationalFunctionExt,ZZ,Ideal), (rationalFunctionExt,Ideal,RingElement), (rationalFunctionExt,Ideal),(rationalFunctionExt,ZZ,Module,RingElement), (rationalFunctionExt,ZZ,Module), (rationalFunctionExt,Module,RingElement)}, Headline => "Ext(holonomic D-module, polynomial ring localized at the singular locus)", Usage => "rationalFunctionExt M, rationalFunctionExt I; rationalFunctionExt(M,f), rationalFunctionExt(I,f); rationalFunctionExt(i,M), rationalFunctionExt(i,I); rationalFunctionExt(i,M,f), rationalFunctionExt(i,I,f)", Inputs => { "M" => Module => {"over the Weyl algebra ", EM "D"}, "I" => Ideal => {"which represents the module ", EM "M = D/I"}, "f" => RingElement => "a polynomial", "i" => ZZ => "nonnegative" }, Outputs => { { ofClass HashTable, " or ", ofClass Module, ", the ", TEX "Ext^i"," group(s) between holonomic ", EM "M", " and the polynomial ring localized at the singular locus of ", EM "M", " (or at ", EM "f", " if specified)" } }, "The Ext groups between M and N are the derived functors of Hom, and are finite-dimensional vector spaces over the ground field when M and N are holonomic.", PARA { "The algorithm used appears in the paper 'Polynomial and rational solutions of holonomic systems' by Oaku-Takayama-Tsai (2000). The method is to combine isomorphisms of Bjork and Kashiwara with the restriction algorithm."}, EXAMPLE lines /// W = QQ[x, D, WeylAlgebra=>{x=>D}] M = W^1/ideal(x*D+5) rationalFunctionExt M ///, Caveat =>{"Input modules M or D/I should be holonomic."}, SeeAlso => {"Dresolution", "Dintegration"} } doc /// Key polynomialSolutions (polynomialSolutions,Module) (polynomialSolutions,Ideal,List) (polynomialSolutions,Module,List) (polynomialSolutions,Ideal) Headline polynomial solutions of a holonomic system Usage polynomialSolutions I polynomialSolutions M polynomialSolutions(I,w) polynomialSolutions(M,w) Inputs M:Module over the Weyl algebra $D$ I:Ideal holonomic ideal in the Weyl algebra $D$ w:List a weight vector Outputs :List a basis of the polynomial solutions of $I$ (or of $D$-homomorphisms between $M$ and the polynomial ring) using $w$ for Groebner deformations. If no $w$ is given, then it is taken to be the all ones vector. Description Text The polynomial solutions of a holonomic system form a finite-dimensional vector space. There are two algorithms implemented to get these solutions. The first algorithm is based on Gröbner deformations and works for ideals $I$ of PDE's - see the paper {\em Polynomial and rational solutions of a holonomic system} by Oaku, Takayama and Tsai (2000). The second algorithm is based on homological algebra - see the paper {\em Computing homomorphisms between holonomic D-modules} by Tsai and Walther (2000). Example makeWA(QQ[x]) I = ideal(dx^2, (x-1)*dx-1) polynomialSolutions I SeeAlso rationalFunctionSolutions Dintegration /// document { Key => Alg } document { Key => [polynomialSolutions, Alg], Headline => "algorithm for finding polynomial solutions", UL { {BOLD "GD", " -- uses Groebner deformations"}, {BOLD "Duality", " -- uses homological duality"} } } document { Key => GD, Headline => "an option for polynomialSolutions=>Alg", SeeAlso => "polynomialSolutions" } document { Key => Duality, Headline => "an option for polynomialSolutions=>Alg", SeeAlso => "polynomialSolutions" } doc /// Key rationalFunctionSolutions (rationalFunctionSolutions,Ideal,List,List) (rationalFunctionSolutions,Ideal,RingElement,List) (rationalFunctionSolutions,Ideal,List) (rationalFunctionSolutions,Ideal,RingElement) (rationalFunctionSolutions,Ideal) Headline rational solutions of a holonomic system Usage rationalFunctionSolutions I rationalFunctionSolutions(I,f) rationalFunctionSolutions(I,f,w) rationalFunctionSolutions(I,ff) rationalFunctionSolutions(I,ff,w) Inputs I:Ideal holonomic ideal in the Weyl algebra @EM "D"@ f:RingElement a polynomial ff:List a list of polynomials w:List a weight vector Outputs :List a basis of the rational solutions of @EM "I"@ with poles along @EM "f"@ or along the polynomials in @TT "ff"@ using @EM "w"@ for Groebner deformations Description Text The rational solutions of a holonomic system form a finite-dimensional vector space. The only possibilities for the poles of a rational solution are the codimension one components of the singular locus. An algorithm to compute rational solutions is based on Gröbner deformations and works for ideals $I$ of PDE's - see the paper {\em Polynomial and rational solutions of a holonomic system} by Oaku, Takayama and Tsai (2000). Example makeWA(QQ[x]) I = ideal((x+1)*dx+5) rationalFunctionSolutions I Caveat The most efficient method to find rational solutions of a system of differential equations is to find the singular locus, then try to find its irreducible factors. With these, call rationalFunctionSolutions(I, ff, w), where w should be generic enough so that the polynomialSolutions routine will not complain of a non-generic weight vector. SeeAlso polynomialSolutions rationalFunctionExt DHom ///

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