Changeset 4f461c in git
 Timestamp:
 Mar 5, 2010, 2:28:28 PM (13 years ago)
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 (u'spielwiese', '0d6b7fcd9813a1ca1ed4220cfa2b104b97a0a003')
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 24c3680c7cb6e7c29e4766bc33d10684a0c2e1f2
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 8a7e767173900998af700bab01a16d332d191b3d
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Singular/LIB/dmodvar.lib
r8a7e76 r4f461c 4 4 info=" 5 5 LIBRARY: dmodvar.lib Algebraic Dmodules for varieties 6 6 7 AUTHORS: Daniel Andres, daniel.andres@math.rwthaachen.de 7 @*Viktor Levandovskyy, levandov@math.rwthaachen.de8 @*Jorge MartinMorales, jorge@unizar.es8 Viktor Levandovskyy, levandov@math.rwthaachen.de 9 Jorge MartinMorales, jorge@unizar.es 9 10 10 11 THEORY: Let K be a field of characteristic 0. Given a polynomial ring 11 @*R = K[x_1,...,x_n] and a set of polynomial f_1,..., f_r in R, define12 @*F = f_1 * ... * f_r and F^s:=f_1^s_1*...*f_r^s_r for symbolic discrete13 @*(that is shiftable) variables s_1,..., s_r.14 @* The module R[1/F]*F^s has a structure of a D<S>module, where 15 @*D<S> := D(R) tensored with S over K, where16 @* D(R) is an nth Weyl algebra K<x_1,...,x_n,d_1,...,d_n  d_j x_j = x_j d_j +1>17 @* S is the universal enveloping algebra of gl_r, generated by s_{ij}, where s_{ii}=s_i.18 @*One is interested in the following data:19 @* the left ideal Ann F^s in D<S>, usually denoted by LD in the output20 @* global Bernstein polynomial in one variable s = s_1 + ...+ s_r, denoted by bs,21 @* its minimal integer root s0, the list of all roots of bs, which are known22 @*to be rational, with their multiplicities, which is denoted by BS23 @* an rtuple of operators in D<S>, denoted by PS, such that the functional equality24 @*sum(k=1 to k=r) P_k*f_k*F^s = bs*F^s holds in R[1/F]*F^s.12 R = K[x_1,...,x_n] and a set of polynomial f_1,..., f_r in R, define 13 F = f_1 * ... * f_r and F^s:=f_1^s_1*...*f_r^s_r for symbolic discrete 14 (that is shiftable) variables s_1,..., s_r. 15 The module R[1/F]*F^s has a structure of a D<S>module, where 16 D<S> := D(R) tensored with S over K, where 17  D(R) is an nth Weyl algebra K<x_1,...,x_n,d_1,...,d_n  d_j x_j = x_j d_j +1> 18  S is the universal enveloping algebra of gl_r, generated by s_{ij}, where s_{ii}=s_i. 19 One is interested in the following data: 20  the left ideal Ann F^s in D<S>, usually denoted by LD in the output 21  global Bernstein polynomial in one variable s = s_1 + ...+ s_r, denoted by bs, 22  its minimal integer root s0, the list of all roots of bs, which are known 23 to be rational, with their multiplicities, which is denoted by BS 24  an rtuple of operators in D<S>, denoted by PS, such that the functional equality 25 sum(k=1 to k=r) P_k*f_k*F^s = bs*F^s holds in R[1/F]*F^s. 25 26 26 27 REFERENCES: 27 @*(BMS06) Budur, Mustata, Saito: BernsteinSato polynomials of arbitrary varieties (2006).28 @*(ALM09) Andres, Levandovskyy, MartinMorales : Principal Intersection and BernsteinSato Polynomial of an Affine Variety (2009).28 (BMS06) Budur, Mustata, Saito: BernsteinSato polynomials of arbitrary varieties (2006). 29 (ALM09) Andres, Levandovskyy, MartinMorales : Principal Intersection and BernsteinSato Polynomial of an Affine Variety (2009). 29 30 30 31 MAIN PROCEDURES: … … 166 167 // the case: given ORD, given engine 167 168 eng = int(#[2]); 168 } 169 } 169 170 else 170 171 { … … 197 198 int ppl = printlevelvoice+2; 198 199 // returns a list with a ring and an ideal LD in it 199 // save, N, P and the indices are already defined 200 // save, N, P and the indices are already defined 200 201 int Nnew = 2*N+P+P^2; 201 202 list RL = ringlist(basering); … … 328 329 intvec iv = P+1..Nnew; 329 330 tmpM = imap(@R@,@D); 330 kill @R@; 331 kill @R@; 331 332 LordM = submat(tmpM,iv,iv); 332 333 matrix @D2 = LordM; … … 548 549 // Name, Dname will be used further 549 550 kill NName, TName, Name, DTName, DName; 550 // ORD already set, default ord dp; 551 // ORD already set, default ord dp; 551 552 L[3] = ORDstr2list(ORD,Nnew); 552 553 // we are done with the list … … 605 606 @* Varnames of the basering do not include t(1),...,t(r) and 606 607 @* Dt(1),...,Dt(r), where r is the number of entries of the input ideal. 607 BACKGROUND: In this proc, the initial ideal of the multivariate Malgrange ideal 608 BACKGROUND: In this proc, the initial ideal of the multivariate Malgrange ideal 608 609 @* defined by I is computed and then a system of linear equations is solved 609 610 @* by linear reductions following the ideas by Noro. … … 619 620 @* time. 620 621 @* If b<>0, @code{std} is used for GB computations in characteristic 0, 621 @* otherwise, and by default, @code{slimgb} is used. 622 @* otherwise, and by default, @code{slimgb} is used. 622 623 @* If c<>0, a matrix ordering is used for GB computations, otherwise, 623 624 @* and by default, a block ordering is used. … … 838 839 I = std(I); 839 840 //ideal I = z(6)^2z(3)*z(7), z(5)*z(6)z(2)*z(7), z(5)^2z(1)*z(7), 840 // z(4)*z(5)z(3)*z(6), z(3)*z(5)z(2)*z(6), z(2)*z(5)z(1)*z(6), 841 // z(4)*z(5)z(3)*z(6), z(3)*z(5)z(2)*z(6), z(2)*z(5)z(1)*z(6), 841 842 // z(3)^2z(2)*z(4), z(2)*z(3)z(1)*z(4), z(2)^2z(1)*z(3); 842 843 bfctVarIn(I,1); // no result yet
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