### D.15.14 multigrading_lib

Todos/Issues:

- See http://code.google.com/p/convex-singular/wiki/Multigrading

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.15.14.1 setBaseMultigrading |  | attach multiweights/grading group matrices to the basering
> D.15.14.2 getVariableWeights |  | get matrix of multidegrees of vars attached to a ring
> D.15.14.3 getGradingGroup |  | get grading group attached to a ring
> D.15.14.4 getLattice |  | get grading group' lattice attached to a ring (or its NF)
> D.15.14.5 createGroup |  | create a group generated by S, with relations L
> D.15.14.6 createQuotientGroup |  | create a group generated by the unit matrix with relations L
> D.15.14.7 createTorsionFreeGroup |  | create a group generated by S which is torsionfree
> D.15.14.8 printGroup |  | print a group
> D.15.14.9 isGroup |  | test whether G is a valid group
> D.15.14.10 isGroupHomomorphism |  | test whether A defines a group homomrphism from L1 to L2
> D.15.14.11 isGradedRingHomomorphism |  | test graded ring homomorph
> D.15.14.12 createGradedRingHomomorphism |  | create a graded ring homomorph
> D.15.14.13 setModuleGrading |  | attach multiweights of units to a module and return it
> D.15.14.14 getModuleGrading |  | get multiweights of module units (attached to M)
> D.15.14.15 isSublattice |  | test whether A is a sublattice of B
> D.15.14.16 imageLattice |  | computes an integral basis for P(L)
> D.15.14.17 intRank |  | computes the rank of the intmat A
> D.15.14.18 kernelLattice |  | computes an integral basis for the kernel of the linear map P.
> D.15.14.19 latticeBasis |  | computes an integral basis of the lattice B
> D.15.14.20 preimageLattice |  | computes an integral basis for the preimage of the lattice L under the linear map P.
> D.15.14.21 projectLattice |  | computes a linear map of lattices having the primitive span of B as its kernel.
> D.15.14.22 intersectLattices |  | computes an integral basis for the intersection of the lattices A and B.
> D.15.14.23 isIntegralSurjective |  | test whether the map P of lattices is surjective.
> D.15.14.24 isPrimitiveSublattice |  | test whether A generates a primitive sublattice.
> D.15.14.25 intInverse |  | computes the integral inverse matrix of the intmat A
> D.15.14.26 integralSection |  | for a given linear surjective map P of lattices this procedure returns an integral section of P.
> D.15.14.27 primitiveSpan |  | computes a basis for the minimal primitive sublattice that contains the given vectors (by A).
> D.15.14.28 factorgroup |  | create the group G mod H
> D.15.14.29 productgroup |  | create the group G x H
> D.15.14.30 multiDeg |  | compute the multidegree of A
> D.15.14.31 multiDegBasis |  | compute all monomials of multidegree d
> D.15.14.32 multiDegPartition |  | compute the multigraded-homogeneous components of p
> D.15.14.33 isTorsionFree |  | test whether the current multigrading is free
> D.15.14.34 isPositive |  | test whether the current multigrading is positive
> D.15.14.35 isZeroElement |  | test whether p has zero multidegree
> D.15.14.36 areZeroElements |  | test whether an integer matrix M considered as a collection of columns has zero multidegree
> D.15.14.37 isHomogeneous |  | test whether 'a' is multigraded-homogeneous
> D.15.14.38 equalMultiDeg |  | test whether e1==e2 in the current multigrading
> D.15.14.39 multiDegGroebner |  | compute the multigraded GB/SB of M
> D.15.14.40 multiDegSyzygy |  | compute the multigraded syzygies of M
> D.15.14.41 multiDegModulo |  | compute the multigraded 'modulo' module of I and J
> D.15.14.42 multiDegResolution |  | compute the multigraded resolution of M
> D.15.14.43 multiDegTensor |  | compute the tensor product of multigraded modules m,n
> D.15.14.44 multiDegTor |  | compute the Tor_i(m,n) for multigraded modules m,n
> D.15.14.45 defineHomogeneous |  | get a grading group wrt which p becomes homogeneous
> D.15.14.46 pushForward |  | find the finest grading on the image ring, homogenizing f
> D.15.14.47 gradiator |  | coarsens grading of the ring until h becomes homogeneous
> D.15.14.48 hermiteNormalForm |  | compute the Hermite Normal Form of a matrix
> D.15.14.49 smithNormalForm |  | compute matrices D,P,Q with D=P*A*Q and D is the smith normal form of A
> D.15.14.50 hilbertSeries |  | compute the multigraded Hilbert Series of M
> D.15.14.51 lll |  | applies LLL(.) of lll.lib which only works for lists on a matrix A

User manual for Singular version 4.4.1, 2025, generated by texi2html.