Singular

D.15.10 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.15.10.1 setBaseMultigrading		attach multiweights/grading group matrices to the basering
D.15.10.2 getVariableWeights		get matrix of multidegrees of vars attached to a ring
D.15.10.3 getGradingGroup		get grading group attached to a ring
D.15.10.4 getLattice		get grading group' lattice attached to a ring (or its NF)
D.15.10.5 createGroup		create a group generated by S, with relations L
D.15.10.6 createQuotientGroup		create a group generated by the unit matrix with relations L
D.15.10.7 createTorsionFreeGroup		create a group generated by S which is torsionfree
D.15.10.8 printGroup		print a group
D.15.10.9 isGroup		test whether G is a valid group
D.15.10.10 isGroupHomomorphism		test whether A defines a group homomrphism from L1 to L2
D.15.10.11 isGradedRingHomomorphism		test graded ring homomorph
D.15.10.12 createGradedRingHomomorphism		create a graded ring homomorph
D.15.10.13 setModuleGrading		attach multiweights of units to a module and return it
D.15.10.14 getModuleGrading		get multiweights of module units (attached to M)
D.15.10.15 isSublattice		test whether A is a sublattice of B
D.15.10.16 imageLattice		computes an integral basis for P(L)
D.15.10.17 intRank		computes the rank of the intmat A
D.15.10.18 kernelLattice		computes an integral basis for the kernel of the linear map P.
D.15.10.19 latticeBasis		computes an integral basis of the lattice B
D.15.10.20 preimageLattice		computes an integral basis for the preimage of the lattice L under the linear map P.
D.15.10.21 projectLattice		computes a linear map of lattices having the primitive span of B as its kernel.
D.15.10.22 intersectLattices		computes an integral basis for the intersection of the lattices A and B.
D.15.10.23 isIntegralSurjective		test whether the map P of lattices is surjective.
D.15.10.24 isPrimitiveSublattice		test whether A generates a primitive sublattice.
D.15.10.25 intInverse		computes the integral inverse matrix of the intmat A
D.15.10.26 integralSection		for a given linear surjective map P of lattices this procedure returns an integral section of P.
D.15.10.27 primitiveSpan		computes a basis for the minimal primitive sublattice that contains the given vectors (by A).
D.15.10.28 factorgroup		create the group G mod H
D.15.10.29 productgroup		create the group G x H
D.15.10.30 multiDeg		compute the multidegree of A
D.15.10.31 multiDegBasis		compute all monomials of multidegree d
D.15.10.32 multiDegPartition		compute the multigraded-homogeneous components of p
D.15.10.33 isTorsionFree		test whether the current multigrading is free
D.15.10.34 isPositive		test whether the current multigrading is positive
D.15.10.35 isZeroElement		test whether p has zero multidegree
D.15.10.36 areZeroElements		test whether an integer matrix M considered as a collection of columns has zero multidegree
D.15.10.37 isHomogeneous		test whether 'a' is multigraded-homogeneous
D.15.10.38 equalMultiDeg		test whether e1==e2 in the current multigrading
D.15.10.39 multiDegGroebner		compute the multigraded GB/SB of M
D.15.10.40 multiDegSyzygy		compute the multigraded syzygies of M
D.15.10.41 multiDegModulo		compute the multigraded 'modulo' module of I and J
D.15.10.42 multiDegResolution		compute the multigraded resolution of M
D.15.10.43 multiDegTensor		compute the tensor product of multigraded modules m,n
D.15.10.44 multiDegTor		compute the Tor_i(m,n) for multigraded modules m,n
D.15.10.45 defineHomogeneous		get a grading group wrt which p becomes homogeneous
D.15.10.46 pushForward		find the finest grading on the image ring, homogenizing f
D.15.10.47 gradiator		coarsens grading of the ring until h becomes homogeneous
D.15.10.48 hermiteNormalForm		compute the Hermite Normal Form of a matrix
D.15.10.49 smithNormalForm		compute matrices D,P,Q with D=P*A*Q and D is the smith normal form of A
D.15.10.50 hilbertSeries		compute the multigraded Hilbert Series of M
D.15.10.51 lll		applies LLL(.) of lll.lib which only works for lists on a matrix A