Changeset 972fb1 in git for Singular/LIB/rinvar.lib
 Timestamp:
 Feb 19, 2002, 1:30:16 PM (21 years ago)
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 (u'spielwiese', '91fdef05f09f54b8d58d92a472e9c4a43aa4656f')
 Children:
 900cea1b1461e5dcef8e13aba10034edc5914c65
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 ebcda0c47fce53217b409edb6303bb31e69e3ef4
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Singular/LIB/rinvar.lib
rebcda0 r972fb1 1 1 // Last change 10.12.2000 (TB) 2 2 /////////////////////////////////////////////////////////////////////////////// 3 version="$Id: rinvar.lib,v 1. 7 20010202 16:34:03 mschulzeExp $";3 version="$Id: rinvar.lib,v 1.8 20020219 12:30:16 Singular Exp $"; 4 4 category="Invariant theory"; 5 5 info=" … … 41 41 proc EquationsOfEmbedding(ideal embedding, int nrs) 42 42 "USAGE: EquationsOfEmbedding(embedding, s); ideal embedding; int s; 43 PUR OPSE: compute the ideal of the variety parameterized by 'embedding' by43 PURPOSE: compute the ideal of the variety parameterized by 'embedding' by 44 44 implicitation and change the variables to the old ones. 45 45 RETURN: ideal … … 71 71 proc ImageGroup(ideal Grp, ideal Gaction) 72 72 "USAGE: ImageGroup(G, action); ideal G, action; 73 PUR OPSE: compute the ideal of the image of G in GL(m,K) induced by the linear73 PURPOSE: compute the ideal of the image of G in GL(m,K) induced by the linear 74 74 action 'action', where G is an algebraic group and 'action' defines 75 75 an action of G on K^m (size(action) = m). … … 199 199 proc HilbertWeights(ideal I, wt) 200 200 "USAGE: HilbertWeights(I, w); ideal I, intvec wt 201 PUR OPSE: compute the weights of the "slack" variables needed for the201 PURPOSE: compute the weights of the "slack" variables needed for the 202 202 computation of the algebraic relations of the generators of 'I' s.t. 203 203 the Hilbert driven 'std' can be used. … … 218 218 proc HilbertSeries(ideal I, wt) 219 219 "USAGE: HilbertSeries(I, w); ideal I, intvec wt 220 PUR OPSE: compute the polynomial p of the Hilbert Series,represented by p/q, of220 PURPOSE: compute the polynomial p of the Hilbert Series,represented by p/q, of 221 221 the ring K[t_1,...,t_m,y_1,...,y_r]/I1 where 'w' are the weights of 222 222 the variables, computed, e.g., by 'HilbertWeights', 'I1' is of the … … 244 244 proc HilbertSeries1(wt) 245 245 "USAGE: HilbertSeries1(wt); ideal I, intvec wt 246 PUR OPSE: compute the polynomial p of the Hilbert Series represented by p/q of246 PURPOSE: compute the polynomial p of the Hilbert Series represented by p/q of 247 247 the ring K[t_1,...,t_m,y_1,...,y_r]/I where I is a complete inter 248 248 section and the generator I[i] has degree wt[i] … … 273 273 proc ImageVariety(ideal I, F, list #) 274 274 "USAGE: ImageVariety(ideal I, F [, w]);ideal I; F is a list/ideal, intvec w. 275 PUR OPSE: compute the Zariski closure of the image of the variety of I under275 PURPOSE: compute the Zariski closure of the image of the variety of I under 276 276 the morphism F. 277 277 NOTE: if 'I' and 'F' are quasihomogenous w.r.t. 'w' then the Hilbertdriven … … 359 359 proc LinearizeAction(ideal Grp, Gaction, int nrs) 360 360 "USAGE: LinearizeAction(G,action,r); ideal G, action; int r 361 PUR OPSE: linearize the group action 'action' and find an equivariant embedding361 PURPOSE: linearize the group action 'action' and find an equivariant embedding 362 362 of K^m where m = size(action). 363 363 ASSUME: G contains only variables var(1..r) (r = nrs) … … 504 504 proc LinearActionQ(Gaction, int nrs) 505 505 "USAGE: LinearActionQ(action,nrs,nrt); ideal action, int nrs 506 PUR OPSE: check if the action defined by 'action' is linear w.r.t. the variables506 PURPOSE: check if the action defined by 'action' is linear w.r.t. the variables 507 507 var(nrs + 1...nvars(basering)). 508 508 RETURN: 0 action not linear … … 542 542 proc LinearCombinationQ(ideal I, poly f) 543 543 "USAGE: LinearCombination(I, f); ideal I, poly f 544 PUR OPSE: test if f can be written as a linear combination of the generators of I.544 PURPOSE: test if f can be written as a linear combination of the generators of I. 545 545 RETURN: 0 f is not a linear combination 546 546 1 f is a linear combination … … 584 584 proc InvariantRing(ideal G, ideal action, list #) 585 585 "USAGE: InvariantRing(G, Gact [, opt]); ideal G, Gact; int opt 586 PUR OPSE: compute generators of the invariant ring of G w.r.t. the action 'Gact'586 PURPOSE: compute generators of the invariant ring of G w.r.t. the action 'Gact' 587 587 ASSUME: G is a finite group and 'Gact' is a linear action. 588 588 RETURN: polynomial ring over a simple extension of the groundfield of the … … 706 706 proc InvariantQ(poly f, ideal G, action) 707 707 "USAGE: InvariantQ(f, G, action); poly f; ideal G, action 708 PUR OPSE: check if the polynomial f is invariant w.r.t. G where G acts via708 PURPOSE: check if the polynomial f is invariant w.r.t. G where G acts via 709 709 'action' on K^m. 710 710 ASSUME: basering = K[s_1,...,s_m,t_1,...,t_m] where K = Q of K = Q(a) and … … 731 731 proc MinimalDecomposition(poly f, int nrs, int nrt) 732 732 "USAGE: MinimalDecomposition(f,a,b); poly f; int a, b. 733 PUR OPSE: decompose f as a sum M[1,1]*M[2,1] + ... + M[1,r]*M[2,r] where M[1,i]733 PURPOSE: decompose f as a sum M[1,1]*M[2,1] + ... + M[1,r]*M[2,r] where M[1,i] 734 734 contains only s(1..a), M[2,i] contains only t(1...b) s.t. r is minimal 735 735 ASSUME: f polynomial in K[s(1..a),t(1..b)], K = Q or K = Q(a) and minpoly != 0 … … 827 827 proc NullCone(ideal G, action) 828 828 "USAGE: NullCone(G, action); ideal G, action 829 PUR OPSE: compute the ideal of the nullcone of the linear action of G on K^n,829 PURPOSE: compute the ideal of the nullcone of the linear action of G on K^n, 830 830 given by 'action', by means of Deksen's algorithm 831 831 ASSUME: basering = K[s(1..r),t(1..n)], K = Q or K = Q(a) and minpoly != 0, … … 902 902 proc ReynoldsOperator(ideal Grp, ideal Gaction, list #) 903 903 "USAGE: ReynoldsOperator(G, action [, opt); ideal G, action; int opt 904 PUR OPSE: compute the Reynolds operator of the group G which act via 'action'904 PURPOSE: compute the Reynolds operator of the group G which act via 'action' 905 905 RETURN: polynomial ring R over a simple extension of the groundfield of the 906 906 basering (the extension might be trivial), containing a list … … 961 961 proc ReynoldsImage(list reynoldsOp, poly f) 962 962 "USAGE: ReynoldsImage(RO, f); list RO, poly f 963 PUR OPSE: compute the Reynolds image of the polynomial f where RO represents963 PURPOSE: compute the Reynolds image of the polynomial f where RO represents 964 964 the Reynolds operator 965 965 RETURN: poly … … 980 980 static proc SimplifyCoefficientMatrix(matrix coefMatrix) 981 981 "USAGE: SimplifyCoefficientMatrix(M); M matrix coming from coef(...) 982 PUR OPSE: simplify the matrix, i.e. find linear dependencies among the columns982 PURPOSE: simplify the matrix, i.e. find linear dependencies among the columns 983 983 RETURN: matrix M, f = M[1,1]*M[2,1] + ... + M[1,n]*M[2,n] 984 984 " … … 1065 1065 static proc TransferIdeal(R, string name, poly newA) 1066 1066 " USAGE: TransferIdeal(R, name, newA); ring R, string name, poly newA 1067 PUR OPSE: Maps an ideal with name 'name' in R to the basering, s.t. all1067 PURPOSE: Maps an ideal with name 'name' in R to the basering, s.t. all 1068 1068 variables are fixed but par(1) is replaced by 'newA'. 1069 1069 RETURN: ideal
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