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6.2.11 Ideals
The following are commands and functions for ideals:
BB.BBasis Computes the border basis of a zero-dimensional ideal.
BB.BBasisForMP Computes the border basis of a zero-dimensional ideal generated by marked polynomials.
BB.BBasisForOI Computes the border basis of an ideal w.r.t. a given order ideal.
BB.Border Computes the border of an order ideal.
BB.Box Computes a box order ideal.
BB.HomASgens Computes the generators of the vanishing ideal of a homogeneous border basis scheme.
BB.HomNDgens Computes the generators of the vanishing ideal of a homogeneous border basis scheme.
BB.LiftAS Computes the border basis scheme ideal generators obtained from lifting of AS neighbours.
BB.LiftASViaServer Computes the border basis scheme ideal generators obtained from lifting of AS neighbours.
BB.LiftND Computes the border basis scheme ideal generators obtained from lifting of next-door neighbours.
BB.LiftNDViaServer Computes the border basis scheme ideal generators obtained from lifting of next-door neighbors.
BB.NDgens Computes the generators of the vanishing ideal of a border basis scheme.
BBF.Init Initializes a border basis computation.
CharP.GBasisF1024 Computing a Groebner basis of a given ideal in F_1024.
CharP.GBasisF128 Computing a Groebner Basis of a given ideal in F_128.
CharP.GBasisF16 Computing a Groebner Basis of a given ideal in F_16.
CharP.GBasisF2 Computing a Groebner Basis of a given ideal in F_2.
CharP.GBasisF2048 Computing a Groebner Basis of a given ideal in F_2048.
CharP.GBasisF256 Computing a Groebner Basis of a given ideal in F_256.
CharP.GBasisF32 Computing a Groebner Basis of a given ideal in F_32.
CharP.GBasisF4 Computing a Groebner Basis of a given ideal in F_4.
CharP.GBasisF512 Computing a Groebner Basis of a given ideal in F_512.
CharP.GBasisF64 Computing a Groebner Basis of a given ideal in F_64.
CharP.GBasisF8 Computing a Groebner Basis of a given ideal in F_8.
CharP.GBasisModSquares Computing a Groebner Basis of a given ideal intersected with x^2-x for all indeterminates x.
DA.DiffGB Computes a differential Groebner basis.
FGLM.FGLM Performs a FGLM Groebner Basis conversion using ApCoCoAServer.
NC.GB Enumerate a (partial) Groebner basis of a finitely generated two-sided ideal in a free monoid ring (using the Buchberger procedure).

Given an ideal I and an admissible ordering Ordering, a set of non-zero polynomials Gb is called a Groebner basis of I w.r.t. Ordering if the leading term set LT{Gb} (w.r.t. Ordering) generates the leading term ideal LT(I) (w.r.t. Ordering). Note that it may not exist finite Groebner bases of I w.r.t. Ordering.
NC.GGB Enumerate a (partial) Groebner basis of a finitely generated two-sided ideal in a free monoid ring over F2 (using Buchberger procedure).

Given an ideal I and an admissible ordering Ordering, a set of non-zero polynomials Gb is called a Groebner basis of I w.r.t. Ordering if the leading term set LT{Gb} (w.r.t. Ordering) generates the leading term ideal LT(I) (w.r.t. Ordering). Note that it may not exist finite Groebner bases of I w.r.t. Ordering.
NC.GHF Compute the values of the Hilbert function of a finitely generated F2-algebra.

For every i in N, we let F_{i} be the F2-vector subspace generated by the words of length less than or equal to i. Then {F_{i}} is a filtration of F2. Let I be an ideal of F2. The filtration {F_{i}} induces a filtration {F_{i}/(F_{i} intersect I)} of F2/I. The Hilbert function of F2-algebra F2/I is a map from N to N defined by mapping i to dim(F_{i}/(F_{i} intersect I))-dim(F_{i-1}/(F_{i-1} intersect I)).
NC.GIsGB Check whether a finite list (set) of non-zero polynomials in a free monoid ring over F2 is a Groebner basis.

Note that, given an ideal I and an admissible ordering Ordering, a set of non-zero polynomials Gb is called a Groebner basis of I w.r.t. Ordering if the leading term set LT{Gb} (w.r.t. Ordering) generates the leading term ideal LT(I) (w.r.t. Ordering). The function check whether a given finite set of non-zero polynomial G is a Groebner basis by using the Buchberger Criterion, i.e. G is a Groebner basis if all the S-polynomials of obstructions have the zero normal remainder w.r.t. G.
NC.GMB Macauley basis of an F2-algebra.
NC.GReducedGB Enumerate a reduced (partial) Groebner basis of a finitely generated two-sided ideal in a free monoid ring over F2.

Note that, given an ideal I and an admissible ordering Ordering, a set of non-zero polynomials Gb is called a Groebner basis of I w.r.t. Ordering if the leading term set LT{Gb} (w.r.t. Ordering) generates the leading term ideal LT(I) (w.r.t. Ordering). A Groebner basis w.r.t. Ordering is called a reduced Groebner basis w.r.t. Ordering if (1) it is interreduced and (2) all polynomials in it are monic. Also note that each ideal has a unique reduced Groebner basis w.r.t. Ordering. However, it is not necessarily finite.
NC.GTruncatedGB Compute a degree truncated Groebner basis of a finitely generated homogenous two-sided ideal in a free monoid ring over F2.
NC.HF Compute the values of the Hilbert function of a finitely generated K-algebra.

For every i in N, we let F_{i} be the K-vector subspace generated by the words of length less than or equal to i. Then {F_{i}} is a filtration of K. Let I be an ideal of K. The filtration {F_{i}} induces a filtration {F_{i}/(F_{i} intersect I)} of K/I. The Hilbert function of K-algebra K/I is a map from N to N defined by mapping i to dim(F_{i}/(F_{i} intersect I))-dim(F_{i-1}/(F_{i-1} intersect I)).
NC.Interreduction Interreduce a list (set) of polynomials in a free monoid ring.

Note that, given an admissible ordering Ordering, a set of non-zero polynomial G is called interreduced w.r.t. Ordering if no element of Supp(g) is contained in LT(G\{g}) for all g in G.
NC.Intersection Enumerate a (partial) Groebner basis of the intersection of two finitely generated two-sided ideals in a free monoid ring.
NC.IsFinite Check whether the K-dimension dim(K/) is finite, where is an monoid ideal generated by a finite set of monoid M.
NC.IsGB Check whether a finite list (set) of non-zero polynomials in a free monoid ring is a Groebner basis.

Note that, given an ideal I and an admissible ordering Ordering, a set of non-zero polynomials Gb is called a Groebner basis of I w.r.t. Ordering if the leading term set LT{Gb} (w.r.t. Ordering) generates the leading term ideal LT(I) (w.r.t. Ordering). The function check whether a given finite set of non-zero polynomial G is a Groebner basis by using the Buchberger Criterion, i.e. G is a Groebner basis if all the S-polynomials of obstructions have the zero normal remainder w.r.t. G.
NC.IsHomog Check whether a polynomial of a list of polynomials is homogeneous in a free monoid ring.
NC.KernelOfHomomorphism Enumerate a (partial) Groebner basis of the kernel of a K-algebra homomorphism.
NC.LTIdeal Enumerate a (partial) (two-sided) leading-term ideal of a finitely generated ideal in a free monoid ring.
NC.MB Enumerate Macaulay basis of a K-algebra.
NC.MRGB Enumerate a (partial) Groebner basis of a finitely generated two-sided ideal in a finitely presented monoid ring (by the Buchberger procedure).

Given an ideal I and an admissible ordering Ordering, a set of non-zero polynomials Gb is called a Groebner basis of I w.r.t. Ordering if the leading term set LT{Gb} (w.r.t. Ordering) generates the leading term set LT{I} (w.r.t. Ordering) as a monoid ideal. Also note that it may not exist finite Groebner bases of I w.r.t. Ordering.
NC.MRIntersection Enumerate a (partial) Groebner basis of the intersection of two finitely generated two-sided ideals in a finitely presented monoid ring.
NC.MRIsGB Check whether a finite list (set) of non-zero polynomials in a finitely presented monoid ring is a Groebner basis.

Note that, given an ideal I and an admissible ordering Ordering, a set of non-zero polynomials Gb is called a Groebner basis of I w.r.t. Ordering if the leading term set LT{Gb} (w.r.t. Ordering) generates the leading term set LT{I} (w.r.t. Ordering) as a monoid ideal. The function check whether a given finite set of non-zero polynomial G is a Groebner basis by using the Buchberger Criterion, i.e. G is a Groebner basis if all the S-polynomials of obstructions have the zero normal remainder w.r.t. G.
NC.MRIsHomog Checks whether a polynomial or a set (LIST) of polynomials is homogeneous in a finitely presented monoid ring.
NC.MRKernelOfHomomorphism Enumerate a (partial) Groebner basis of the kernel of a K-algebra homomorphism.
NC.MRLTIdeal Enumerate a (partial) (two-sided) leading-term ideal of a finitely generated ideal in a finitely presented monoid ring.
NC.MRReducedGB Enumerate a reduced (partial) Groebner basis of a finitely generated two-sided ideal in a finitely presented monoid ring.

Note that, given an ideal I and an admissible ordering Ordering, a set of non-zero polynomials Gb is called a Groebner basis of I w.r.t. Ordering if the leading term set LT{Gb} (w.r.t. Ordering) generates the leading term set LT{I} (w.r.t. Ordering) as a monoid ideal. A Groebner basis w.r.t. Ordering is called a reduced Groebner basis w.r.t. Ordering if (1) it is interreduced and (2) all polynomials in it are monic. Also note that each ideal has a unique reduced Groebner basis w.r.t. Ordering. However, it is not necessarily finite.
NC.PrefixGB Prefix Groebner basis of a finitely generated (right) ideal in a finitely presented monoid ring.
NC.PrefixReducedGB Prefix reduced Groebner basis of a finitely generated (right) ideal in a finitely presented monoid ring.
NC.PrefixSaturation Prefix saturation of a polynomial in a finitely presented monoid ring.
NC.ReducedGB Enumerate a reduced (partial) Groebner basis of a finitely generated two-sided ideal in a free monoid ring.

Note that, given an ideal I and an admissible ordering Ordering, a set of non-zero polynomials Gb is called a Groebner basis of I w.r.t. Ordering if the leading term set LT{Gb} (w.r.t. Ordering) generates the leading term ideal LT(I) (w.r.t. Ordering). A Groebner basis w.r.t. Ordering is called a reduced Groebner basis w.r.t. Ordering if (1) it is interreduced and (2) all polynomials in it are monic. Also note that each ideal has a unique reduced Groebner basis w.r.t. Ordering. However, it is not necessarily finite.
NC.TruncatedGB Compute a degree truncated Groebner basis of a finitely generated homogenous two-sided ideal in a free monoid ring.
Num.SubABM Computes a border basis of an almost vanishing sub-ideal for a set of points and an ideal using the Num.ABM algorithm.
Num.SubAVI Computes a border basis of an almost vanishing sub-ideal for a set of points and an ideal using the Num.AVI algorithm.
Num.SubBBABM Computes a border basis of an almost vanishing sub-ideal for a set of points and an ideal using the Num.BBABM algorithm.
Num.SubEXTABM Computes a border basis of an almost vanishing sub-ideal for a set of points and an ideal using the Num.EXTABM algorithm.
Weyl.AnnFs Computes annihilating ideal of a polynomial F^s in Weyl algebra A_n.
Weyl.BFs Computes B-function of a polynomial F in Weyl algebra A_n.
Weyl.CharI Computes the characteristic ideal of a D-ideal I in Weyl algebra A_n.
Weyl.InIw Computes the initial ideal of a D-ideal I in Weyl algebra A_n with respect to the weight vector W=(u_i,v_i).
Weyl.IsHolonomic Checks whether an ideal in Weyl algebra A_n is holonomic or not.
Weyl.TwoWGB Computes the reduced two-sided Groebner basis of a two-sided ideal I in the Weyl algebra A_n over the field of positive characteristic.
Weyl.WDim Computes the dimension of an ideal I in Weyl algebra A_n.
Weyl.WGB Computes the Groebner basis of an ideal I in Weyl algebra A_n.
Weyl.WLT Computes the leading term ideal of a D-ideal I in Weyl algebra A_n.




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