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D.15.12 multigrading_lib
Todos/Issues:
- Library:
- multigrading.lib
- Purpose:
- Multigraded Rings
- Authors:
- Benjamin Bechtold, benjamin.bechtold@googlemail.com
Rene Birkner, rbirkner@math.fu-berlin.de
Lars Kastner, lkastner@math.fu-berlin.de
Simon Keicher, keicher@mail.mathematik.uni-tuebingen.de
Oleksandr Motsak, U@D, where U={motsak}, D={mathematik.uni-kl.de}
Anna-Lena Winz, anna-lena.winz@math.fu-berlin.de - Overview:
- This library allows one to virtually add multigradings to Singular:
grade multivariate polynomial rings with arbitrary (fin. gen. Abelian) groups.
For more see http://code.google.com/p/convex-singular/wiki/Multigrading
For theoretical references see:
E. Miller, B. Sturmfels: 'Combinatorial Commutative Algebra' and
M. Kreuzer, L. Robbiano: 'Computational Commutative Algebra'. - Note:
- 'multiDegBasis' relies on 4ti2 for computing Hilbert Bases.
All groups are finitely generated Abelian
Procedures:
D.15.12.1 setBaseMultigrading attach multiweights/grading group matrices to the basering D.15.12.2 getVariableWeights get matrix of multidegrees of vars attached to a ring D.15.12.3 getGradingGroup get grading group attached to a ring D.15.12.4 getLattice get grading group' lattice attached to a ring (or its NF) D.15.12.5 createGroup create a group generated by S, with relations L D.15.12.6 createQuotientGroup create a group generated by the unit matrix with relations L D.15.12.7 createTorsionFreeGroup create a group generated by S which is torsionfree D.15.12.8 printGroup print a group D.15.12.9 isGroup test whether G is a valid group D.15.12.10 isGroupHomomorphism test whether A defines a group homomrphism from L1 to L2 D.15.12.11 isGradedRingHomomorphism test graded ring homomorph D.15.12.12 createGradedRingHomomorphism create a graded ring homomorph D.15.12.13 setModuleGrading attach multiweights of units to a module and return it D.15.12.14 getModuleGrading get multiweights of module units (attached to M) D.15.12.15 isSublattice test whether A is a sublattice of B D.15.12.16 imageLattice computes an integral basis for P(L) D.15.12.17 intRank computes the rank of the intmat A D.15.12.18 kernelLattice computes an integral basis for the kernel of the linear map P. D.15.12.19 latticeBasis computes an integral basis of the lattice B D.15.12.20 preimageLattice computes an integral basis for the preimage of the lattice L under the linear map P. D.15.12.21 projectLattice computes a linear map of lattices having the primitive span of B as its kernel. D.15.12.22 intersectLattices computes an integral basis for the intersection of the lattices A and B. D.15.12.23 isIntegralSurjective test whether the map P of lattices is surjective. D.15.12.24 isPrimitiveSublattice test whether A generates a primitive sublattice. D.15.12.25 intInverse computes the integral inverse matrix of the intmat A D.15.12.26 integralSection for a given linear surjective map P of lattices this procedure returns an integral section of P. D.15.12.27 primitiveSpan computes a basis for the minimal primitive sublattice that contains the given vectors (by A). D.15.12.28 factorgroup create the group G mod H D.15.12.29 productgroup create the group G x H D.15.12.30 multiDeg compute the multidegree of A D.15.12.31 multiDegBasis compute all monomials of multidegree d D.15.12.32 multiDegPartition compute the multigraded-homogeneous components of p D.15.12.33 isTorsionFree test whether the current multigrading is free D.15.12.34 isPositive test whether the current multigrading is positive D.15.12.35 isZeroElement test whether p has zero multidegree D.15.12.36 areZeroElements test whether an integer matrix M considered as a collection of columns has zero multidegree D.15.12.37 isHomogeneous test whether 'a' is multigraded-homogeneous D.15.12.38 equalMultiDeg test whether e1==e2 in the current multigrading D.15.12.39 multiDegGroebner compute the multigraded GB/SB of M D.15.12.40 multiDegSyzygy compute the multigraded syzygies of M D.15.12.41 multiDegModulo compute the multigraded 'modulo' module of I and J D.15.12.42 multiDegResolution compute the multigraded resolution of M D.15.12.43 multiDegTensor compute the tensor product of multigraded modules m,n D.15.12.44 multiDegTor compute the Tor_i(m,n) for multigraded modules m,n D.15.12.45 defineHomogeneous get a grading group wrt which p becomes homogeneous D.15.12.46 pushForward find the finest grading on the image ring, homogenizing f D.15.12.47 gradiator coarsens grading of the ring until h becomes homogeneous D.15.12.48 hermiteNormalForm compute the Hermite Normal Form of a matrix D.15.12.49 smithNormalForm compute matrices D,P,Q with D=P*A*Q and D is the smith normal form of A D.15.12.50 hilbertSeries compute the multigraded Hilbert Series of M D.15.12.51 lll applies LLL(.) of lll.lib which only works for lists on a matrix A