6.2.11 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 |
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 |
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 |
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. |