Package borderbasis
From ApCoCoAWiki
| This article is about a function from ApCoCoA-2. If you are looking for the ApCoCoA-1 version of it, see Category:ApCoCoA-1:Package borderbasis. |
This page describes the borderbasis package. The package contains various functions for computing with border bases of order ideals in a polynomial ring P=K[x_1,...,x_n] over a field K. We refer the book [M. Kreuzer and L. Robbiano, Computational Commutative Algebra 2, Springer-Verlag, Berlin, 2005] for more details about border bases. For a complete list of functions, see also Category:Package borderbasis.
The Global Alias of the package is BB.
List of the main functions
IsOrderIdeal(OO): checks whether OO is an order ideal.
input: OO a non-empty set of terms in K[x[1..N]]
output: boolean value for checking OO being an order ideal
IndexO(P,T,OO): returns index of a term in K[x[1..N]]w.r.t an order ideal OO.
input: T a term in P= K[x[1..N]], OO an oder ideal in P
output: the index of T w.r.t. OO
Border(OO): computes the border of an order ideal.
input: list of terms
output: list of terms in ascending order
Box(P,D): computes the 'box' order ideal of type D=[D1,..,DN].
input: list of integers D of length NumIndets(P), P=K[x[1..N]]
output: list of terms (sorted w.r.t. current TO)
BBasisForOI(F,OO): computes the border basis of the ideal I=<F> with respect to the order ideal OO,
gives an error messages if no border basis exists, uses the O_sigma(I) border basis and the BB transformation.
input: list of poly F, list of terms OO
output: list of poly
BorderDivAlg(P,F,OO,Prebasis): applies the Border Division Algorithm w.r.t. the order ideal OO and the border prebasis
Prebasis to the polynomial F and returns a record with fields
Quotients and Remainder where Remainder is the normal OO-remainder of F.
input: poly F, list of terms OO, list of poly Prebasis
output: record with two fields Quotients and Remainder
BorderDivAlgForCoeffs(P,F,OO,Prebasis): applies BorderDivAlg to form
a list La in P such that F has a presentation of form
F=La[1]*OO[1]+...+La[Mu]*OO[Mu]+H, H in <Prebasis>.
input: poly F, list of terms OO, list of poly Prebasis
output: list of polys of length Mu=len(OO)
BBRing(OO): creates the (standard) bb poly ring of OO.
input: list of terms OO in K[x[1..N]]
output: the ring K[c_ij] of OO
GenMultMat(BBS,OO): computes the generic multiplication matrices with respect to the order ideal OO.
input: BBS the bb poly ring of OO, OO order ideal in K[x[1..N]]
output: matrices of size Mu x Mu over the ring BBS=K[c_{ij}]
IthGenMultMat(BBS,OO,I): computes the generic multiplication matrix
for x[I] with respect to the order ideal OO.
input: I pos integer, OO order ideal in K[x[1..N]], BBS the bb poly ring of OO
output: matrix of size Mu x Mu over the ring BBS=K[c_{ij}]
GenHomMultMat(BBS,OO): computes the generic homog. mult. matrices with respect to the order ideal OO.
input: BBS the bb poly ring of OO, OO order ideal in K[x[1..N]]
output: matrices of size Mu x Mu over the ring BBS=K[c_{ij}]
IthGenHomMultMat(BBS,OO,I): computes the generic homog. mult. matrix
for x[I] with respect to the order ideal OO.
input: I pos integer, OO order ideal in K[x[1..N]], BBS the bb poly ring of OO
output: matrix of size Mu x Mu over the ring BBS=K[c_{ij}]
GenDfMultMat(BBS,OO): computes the generic deg-filt mult. matrices with respect to the order ideal OO.
input: BBS the bb poly ring of OO, OO order ideal in K[x[1..N]]
output: matrices of size Mu x Mu over the ring BBS=K[c_{ij}]
IthGenDfMultMat(BBS,OO,I): computes the generic deg-filt mult. matrix
for x[I] with respect to the order ideal OO.
input: I pos integer, OO order ideal in K[x[1..N]], BBS the bb poly ring of OO
output: matrix of size Mu x Mu over the ring BBS=K[c_{ij}]
BBscheme(BBS,OO): computes the defining equations of the border basis scheme
using the commutators of the multiplication matrices.
input: OO is an order ideal, BBS is the bb poly ring of OO
output: an ideal in the ring BBS = K[c_{ij}]
IdealOfBBScheme(BBS,OO): the same as BBscheme(BBS,OO).
DfBBscheme(BBS,OO): computes the defining equations of the deg-filt BB scheme
using the commutators of the multiplication matrices.
input: OO is an order ideal, BBS is the bb poly ring of OO
output: an ideal in the ring BBS = K[c_{ij}]
IdealOfDfBBscheme(BBS,OO): the same as DfBBscheme(BBS,OO).
HomBBscheme(BBS,OO): compute the defining equations of the homog. BB scheme
using the commutators of the generic homog mult matrices.
input: OO order ideal, BBS is the bb poly ring of OO
output: an ideal in the ring BBS = K[c_{ij}]
IdealOfHomBBscheme(BBS,OO): the same as HomBBscheme(BBS,OO).
RingOfFamily(OO): forms the ring of universal bb family.
input: OO is an order ideal in K[x[1..N]]
output: the ring of univ bb family K[c_ij,x[1..N]]
GenericBB(UF,OO): computes the 'generic' border prebasis w.r.t. OO
i.e. the polys g_j = b_j - sum_i c_{ij} t_i.
input: OO is the order ideal in K[x[1..N]]";
UF=K[c_ij,x[1..N]] is the ring of universal bb family of OO
output: list of Poly in UF
GenericHomBB(UF,OO): computes the 'generic' homog. border basis w.r.t. OO
input: OO is the order ideal in K[x[1..N]]";
UF=K[c_ij,x[1..N]] is the ring of universal bb family of OO
output: list of Poly in UF
MultMat(I,OO,BB): returns the multiplication matrix associated to the
border basis BB with respect to the I-th indet of the poly ring.
input: integer index I, list of terms OO, list of poly BB
output: matrix
CoeffOfBB(BB,OO): returns the coefficient matrix of the border basis BB.
input: list of poly BB, list of terms OO
output: matrix
NDneighbors(BBS,OO): computes the list of next-door neighbors w.r.t. OO.
input: OO is an order ideal, BBS is the bb poly ring of OO
output: list of triples [i,j,k] s.t. b_i = x_k * b_j
ARneighbors(BBS,OO): computes the list of across-the-rim neighbors w.r.t. OO.
input: OO is an order ideal, BBS is the bb poly ring of OO
output: List of [i,j,k,l,m] s.t. x_k*b_i = x_l*b_j, b_i = x_l*t_m
and b_j = x_k*t_m for some t_m in OO (l>k)
ASneighbors(BBS,OO): computes the list of across-the-street neighbors w.r.t. OO.
input: OO is an order ideal, BBS is the bb poly ring of OO
output: List of quadruples [i,j,k,l] s.t. x_k*b_i = x_l*b_j
LiftND(BBS,OO): computes the equations defining the border basis scheme
and coming from the lifting of next-door neighbors.
input: OO is an order ideal, BBS is the bb poly ring of OO
output: list of poly in the ring BBS=K[c_{ij}]
LiftAR(BBS,OO): computes the equations defining the border basis scheme
and coming from the lifting of across-the-rim neighbors.
input: OO is an order ideal, BBS is the bb poly ring of OO
output: list of poly in the ring BBS=K[c_{ij}]
LiftAS(BBS,OO): computes the equations defining the border basis scheme
and coming from the lifting of across-the-street neighbors.
input: OO is an order ideal, BBS is the bb poly ring of OO
output: list of poly in the ring BBS=K[c_{ij}]
LiftHomND(BBS,OO): computes the equations defining the homog. BB scheme
and coming from the lifting of next-door neighbors.
input: OO is an order ideal, BBS is the bb poly ring of OO
output: list of poly in the ring BBS=K[c_{ij}]
LiftHomAS(BBS,OO): computes the equations defining the homog. BB scheme
and coming from the lifting of across-the-street neighbors.
input: OO is an order ideal, BBS is the bb poly ring of OO
output: list of poly in the ring BBS=K[c_{ij}]
NDgens(BBS,K,OO): computes the generators of the defining ideal of the border basis scheme
corresponding to the lifting of the K-th element of NDneighbors(BBS,OO).
input: K=index of a NDneighbor, OO order ideal, BBS bb poly ring
output: list of polynomials in BBS=K[c_{ij}]
ARgens(BBS,K,OO): computes the generators of the defining ideal of the border basis scheme
corresponding to the lifting of the K-th element of ARneighbors(BBS,OO).
input: K=index of an ARneighbor, OO order ideal, BBS bb poly ring
output: list of polynomials in BBS=K[c_{ij}]
ASgens(BBS,K,OO): computes the generators of the defining ideal of the border basis scheme
corresponding to the lifting of the K-th element of ASneighbors(BBS,OO).
input: K=index of a ASneighbor, OO order ideal, BBS bb poly ring
output: list of polynomials in BBS=K[c_{ij}]
NatIdealOfBBS(BBS,OO): computes the defining ideal of border basis scheme of OO with natural generators.
input: OO is an order ideal, BBS is the bb poly ring of OO
output: A set of natural generators of I_BO
HomNDgens(BBS,K,OO): computes the generators of the vanishing ideal of the homogeneous border basis scheme
corresp. to the lifting of the K-th element of NDneighbors(BBS,OO).
input: K=index of a NDneighbor, OO order ideal, BBS bb poly ring
output: list of polynomials in BBS=K[c_{ij}]
HomASgens(BBS,K,OO): computes the generators of the vanishing ideal of the
homogeneous border basis scheme corresp. to the lifting of
the K-th element of ASneighbors(BBS,OO).
input: K=index of a ASneighbor, OO order ideal, BBS bb poly ring
output: list of polynomials in BBS=K[c_{ij}]
LiftHomND(BBS,OO): computes the equations defining the homog BBsch
and coming from the lifting of ND-neighbors (using Spoly).
input: OO order ideal, BBS bb poly ring
output: list of generators of I_BO^hom lifting of NDs
LiftHomAS(BBS,OO): computes the equations defining the homog BBsch
and coming from the lifting of AS-neighbors (using Spoly).
input: OO order ideal, BBS bb poly ring
output: list of generators of I_BO^hom lifting of ASs
List of support functions
LinPart(P,F): computes the homogeneous part of degree 1.
input: P = Poly ring, F = Poly or list of Poly
output: Poly or list of Poly
RLF(P,F): RLF of a polynomial returns its linear form which vanishes at the origin, independently of the grading.
RLF of a list of poly or an ideal I returns the reduced GB of the ideal generatd by the RLF of the Gens of I.
input: P = Poly ring, F = Poly or list of Poly or ideal
output: Poly or list of Poly
CoeffPoly(P,T,F,X): find the 'multivariate' coefficient of a term in a poly.
input: P = Poly ring, T term, F poly, X set of indets
output: polynomial coefficent of T in F such that no coefficient is in <X>
DF(P,F): degree form of a polynomial F.
input: P = Poly ring, F poly
output: Poly
Ccolumn(BBS, J): contructs the column (C[1,J],...,C[Mu,J])^{tr}.
input: BBS is the bb poly ring, J In 1..Nu
output: a (Mu x 1)-matrix of indets
HomCcolumn(BBS,J,OO): contructs the 'homogeneous' column (D[1,J],...,D[Mu,J])^{tr}
where D[I,J]=C[I,J] if Deg(t_i)=Deg(b_j) and D[I,J]=0 otherwise.
input: BBS is the bb poly ring, J In 1..Nu, OO order ideal
output: a (Mu x 1)-matrix of indets
IsListOfTerms(L): checks if a list is a list of terms.
input: non-empty LIST of POLY
output: TRUE if L is a list of terms, FALSE otherwise
ArrDeg(BBS, OO, opt L): computes the triple [indet, arrow-degree, arrow] of the indeterminates in L.
input: BBS is the bb poly ring, L list of indets of BBS, OO order ideal
output: [indet, arrow-degree, arrow]
TotArrDeg(BBS, OO, opt L): computes the triple [indet, Total arrow-degree, arrow] of the indeterminates in L.
input: BBS is the bb poly ring, L list of indets of BBS, OO order ideal
output: [indet, Total arrow-degree, arrow]
NonNegTotArrDeg(BBS, OO, opt L): computes the indets with non-negative total-arrow-degree.
input: BBS is the bb poly ring, L list of indets of BBS, OO order ideal
output: list of indets";
PositiveArrow(BBS, OO, opt L): computes the indets with positive total-arrow-degree.
input: BBS is the bb poly ring, L list of indets of BBS, OO order ideal
output: list of indets
ZeroTotArrDeg(BBS, OO, opt L): computes the indets with zero total-arrow-degree.
input: BBS is the bb poly ring, L list of indets of BBS, OO order ideal
output: list of indets
InteriorCij(BBS,OO): computes the indeterminates in BBS associated to the interior terms in OO
input: OO order ideal, BBS is the bb poly ring
output: list of interior indets
Example for computations
- See also: BB.Border
