6.1.9 Package gbmr |
NC.Add | Addition of two polynomials in a free monoid ring. |
NC.Deg | (Standard) degree of a polynomial in a free monoid ring. |
NC.FindPolynomials | Find polynomials with specified alphabet (set of indeterminates) from a list of polynomials in monoid rings. |
NC.GAdd | Addition of two polynomials in a free group ring over F2. |
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.GDeg | (Standard) degree of a polynomial in a free monoid ring over F2. |
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.GInterreduction |
Interreduce a list (set) of polynomials in a free monoid ring over F2.
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.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.GLC | Leading coefficient of a polynomial in a free monoid ring over F2. |
NC.GLT | Leading term of a polynomial in a free monoid ring over F2. |
NC.GMB | Macauley basis of an F2-algebra. |
NC.GMultiply | Multiplication of two polynomials in a free monoid ring over F2. |
NC.GNR | Normal remainder of polynomial with respect to a list of polynomials in a free monoid ring over F2. |
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.GSubtract | Subtraction of two polynomials in a free monoid ring over F2. |
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.LC | Leading coefficient of a polynomial in a free monoid ring. |
NC.LT | Leading term of a polynomial in a free monoid ring. |
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.MinimalPolynomial | Compute a minimal polynomial of an element over the quotient of a free monoid ring if it exists. |
NC.MRAdd | Addition of two polynomials in a finitely presented monoid ring. |
NC.MRDeg | (Standard) degree of a polynomial in a finitely presented monoid ring. |
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.MRHF |
Hilbert function of a 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.MRInterreduction |
Interreduce a list (set) of polynomials in a finitely presented monoid ring.
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.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.MRLC | Leading coefficient of a polynomial in a finitely presented monoid ring. |
NC.MRLT | Leading term of a polynomial in a finitely presented monoid ring. |
NC.MRLTIdeal | Enumerate a (partial) (two-sided) leading-term ideal of a finitely generated ideal in a finitely presented monoid ring. |
NC.MRMB | Enumerate Macaulay basis of a K-algebra. |
NC.MRMinimalPolynomial | Compute a minimal polynomial of an element over the quotient of a finitely presented monoid ring if it exists. |
NC.MRMultiply | Multiplication of two polynomials in a finitely presented monoid ring. |
NC.MRNR | Normal remainder of a polynomial with respect to a list of polynomials 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.MRSubtract | Subtraction of two polynomials in a finitely presented monoid ring. |
NC.Multiply | Multiplication of two polynomials in a free monoid ring. |
NC.NR | Normal remainder of polynomial with respect to a list of polynomials in a free monoid ring. |
NC.PrefixGB | Prefix Groebner basis of a finitely generated (right) ideal in a finitely presented monoid ring. |
NC.PrefixInterreduction | Prefix interreduction of a list of polynomials in a finitely presented monoid ring. |
NC.PrefixNR | Prefix normal remainder of a polynomial with respect to a list of polynomials 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.SetFp | Set coefficient field to a finite field. |
NC.SetOrdering |
Set an admissible ordering on |
NC.SetRelations | Set relations for a finitely presented monoid ring. |
NC.SetRules | Set the (rewriting) rules over monoid rings. |
NC.SetX | Set alphabet (set of indeterminates). |
NC.Subtract | Subtraction of two polynomials in a free monoid ring. |
NC.TruncatedGB | Compute a degree truncated Groebner basis of a finitely generated homogenous two-sided ideal in a free monoid ring. |
NC.UnsetFp | Set coefficient field to the default coefficient field Q, i.e. RAT in CoCoAL. |
NC.UnsetOrdering | Set the current admissible ordering to the default ordering LLEX (length-lexicographic ordering). |
NC.UnsetRelations | Set relations to the empty set. |
NC.UnsetRules | Set rewriting rules to the empty set. |
NC.UnsetX | Set alpbabet (set of inderminates) to the empty string. (It might be a useless function.) |