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D.15.17 multigrading_lib


Multigraded Rings

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

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'.

'multiDegBasis' relies on 4ti2 for computing Hilbert Bases. All groups are finitely generated Abelian


D.15.17.1 setBaseMultigrading  attach multiweights/grading group matrices to the basering
D.15.17.2 getVariableWeights  get matrix of multidegrees of vars attached to a ring
D.15.17.3 getGradingGroup  get grading group attached to a ring
D.15.17.4 getLattice  get grading group' lattice attached to a ring (or its NF)
D.15.17.5 createGroup  create a group generated by S, with relations L
D.15.17.6 createQuotientGroup  create a group generated by the unit matrix whith relations L
D.15.17.7 createTorsionFreeGroup  create a group generated by S which is torsionfree
D.15.17.8 printGroup  print a group
D.15.17.9 isGroup  test whether G is a valid group
D.15.17.10 isGroupHomomorphism  test wheter A defines a group homomrphism from L1 to L2
D.15.17.11 isGradedRingHomomorphism  test graded ring homomorph
D.15.17.12 createGradedRingHomomorphism  create a graded ring homomorph
D.15.17.13 setModuleGrading  attach multiweights of units to a module and return it
D.15.17.14 getModuleGrading  get multiweights of module units (attached to M)
D.15.17.15 isSublattice  test whether A is a sublattice of B
D.15.17.16 imageLattice  computes an integral basis for P(L)
D.15.17.17 intRank  computes the rank of the intmat A
D.15.17.18 kernelLattice  computes an integral basis for the kernel of the linear map P.
D.15.17.19 latticeBasis  computes an integral basis of the lattice B
D.15.17.20 preimageLattice  computes an integral basis for the preimage of the lattice L under the linear map P.
D.15.17.21 projectLattice  computes a linear map of lattices having the primitive span of B as its kernel.
D.15.17.22 intersectLattices  computes an integral basis for the intersection of the lattices A and B.
D.15.17.23 isIntegralSurjective  test whether the map P of lattices is surjective.
D.15.17.24 isPrimitiveSublattice  test whether A generates a primitive sublattice.
D.15.17.25 intInverse  computes the integral inverse matrix of the intmat A
D.15.17.26 integralSection  for a given linear surjective map P of lattices this procedure returns an integral section of P.
D.15.17.27 primitiveSpan  computes a basis for the minimal primitive sublattice that contains the given vectors (by A).
D.15.17.28 factorgroup  create the group G mod H
D.15.17.29 productgroup  create the group G x H
D.15.17.30 multiDeg  compute the multidegree of A
D.15.17.31 multiDegBasis  compute all monomials of multidegree d
D.15.17.32 multiDegPartition  compute the multigraded-homogeneous components of p
D.15.17.33 isTorsionFree  test whether the current multigrading is free
D.15.17.34 isPositive  test whether the current multigrading is positive
D.15.17.35 isZeroElement  test whether p has zero multidegree
D.15.17.36 areZeroElements  test whether an integer matrix M considered as a collection of columns has zero multidegree
D.15.17.37 isHomogeneous  test whether 'a' is multigraded-homogeneous
D.15.17.38 equalMultiDeg  test whether e1==e2 in the current multigrading
D.15.17.39 multiDegGroebner  compute the multigraded GB/SB of M
D.15.17.40 multiDegSyzygy  compute the multigraded syzygies of M
D.15.17.41 multiDegModulo  compute the multigraded 'modulo' module of I and J
D.15.17.42 multiDegResolution  compute the multigraded resolution of M
D.15.17.43 multiDegTensor  compute the tensor product of multigraded modules m,n
D.15.17.44 multiDegTor  compute the Tor_i(m,n) for multigraded modules m,n
D.15.17.45 defineHomogeneous  get a grading group wrt which p becomes homogeneous
D.15.17.46 pushForward  find the finest grading on the image ring, homogenizing f
D.15.17.47 gradiator  coarsens grading of the ring until h becomes homogeneous
D.15.17.48 hermiteNormalForm  compute the Hermite Normal Form of a matrix
D.15.17.49 smithNormalForm  compute matrices D,P,Q with D=P*A*Q and D is the smith normal form of A
D.15.17.50 hilbertSeries  compute the multigraded Hilbert Series of M
D.15.17.51 lll  applies LLL(.) of lll.lib which only works for lists on a matrix A