6.2.14 ApCoCoAServer |
BB.BBasis | Computes the border basis of a zero-dimensional ideal. |
BB.BorderDivAlg | Applies the border division algorithm. |
BB.LiftASViaServer | Computes the border basis scheme ideal generators obtained from lifting of AS neighbours. |
BB.LiftNDViaServer | Computes the border basis scheme ideal generators obtained from lifting of next-door neighbors. |
BB.TransformBBIntoGB | Transforms a border basis into a Groebner basis. |
BB.TransformGBIntoBB | Transforms a Groebner basis into a border basis. |
BBSGen.BBFinder | Let LF be a list of indeterminates from the ring K[t[k,l,i,j] that is the subset of the ring XX:=K[c[1..Mu,1..Nu],t[1..N,1..N,1..Mu,1..Mu]] . This function finds the defining equations of border basis scheme and their degrees that correspond to the elements of the list LF. |
BBSGen.JacobiFull | Let R:=K[x_1,...,x_N]. This function computes the entries of the Jacobi identity matrix J^klm [ A_m[A_k,A_l]]+[ A_k[ A_l,A_m]] +[ A_l[A_m,A_k ] ], where m,k,l is from {1...N}. |
BBSGen.JacobiLin | This function computes the K[c]-linear polynomial entries of the Jacobi identity matrix [ A_m[A_k,A_l ] ]+[ A_k[ A_l,A_m]] +[ A_l[A_m,A_k ] ] where m,k,l is from {1,...,N}. |
BBSGen.JacobiStep | Let R:=K[x_1,...,x_N] and let OO be an order ideal. This function computes the entry in the position (I,J) of the Jacobi identity matrix J^klm [ A_m[A_k,A_l ] ]+[ A_k[ A_l,A_m]] +[ A_l[A_m,A_k ] ] where m,k,l is from {1,...,N} and I,J in {1,...,Len(OO)}. |
BBSGen.NonStandPoly | This function computes the non-standard polynomial generators of the vanishing ideal of border basis scheme with respect to the arrow grading. |
BBSGen.NonTriv | This function computes the non-trivial polynomials of the generating set of the vanishing ideal of a border basis scheme. |
BBSGen.PolDeg | This function computes the arrow degree of a given homogenous polynomial from the ring K[c](see BBSGen.WMat). |
BBSGen.TraceSyzFull | This function computes the trace polynomials. |
BBSGen.TraceSyzLin | : This function computes the K[c]-linear summand of trace polynomials.(see BBSGen.TraceSyzFull) |
BBSGen.TraceSyzLinStep | This function computes the K[c]-linear summand of the trace polynomial T_{Pi,X} with respect to a given term Pi and a variable from ring K[x_1,...,x_N].(see BBSGen.TraceSyzFull) |
BBSGen.TraceSyzStep | This function computes the trace polynomial T_{Pi,X} with respect to a given term Pi and a variable from ring K[x_1,...,x_N].(see BBSGen.TraceSyzFull) |
BBSGen.WMat | This function computes the Weight Matrix with respect to the arrow grading. |
Bertini.BCMSolve | Solves a zero dimensional non-homogeneous polynomial system of equations using multi-homogenization and user configurations. |
Bertini.BMSolve | Solves a zero dimensional non-homogeneous polynomial system using multi-homogenization and default configurations. |
Bertini.BPCSolve | Computes numerical irreducible decomposition by finding witness point supersets of a positive dimensional homogeneous or non-homogeneous polynomial systems of equations. |
Bertini.BPCSSolve | Sampling a component for a positive dimensional homogeneous or non-homogeneous polynomial system. |
Bertini.BPMCSolve | Membership testing for a positive dimensional homogeneous or non-homogeneous polynomial system. |
Bertini.BSolve | Solves a zero dimensional homogeneous or non-homogeneous polynomial system of equations with default configurations. |
Bertini.BUHSolve | Solves a zero dimensional non-homogeneous polynomial system of equations by user defined homotopy. |
Bertini.BZCSolve | Solves a zero dimensional homogeneous or non-homogeneous polynomial system of equations using configurations provided by the user. |
CharP.BBasisMutantStrategyF2 | Computes a Border Basis of a given ideal over F_2. |
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. |
CharP.IMBBasis | Computes a Border Basis of a given ideal over F_2. |
CharP.IMNLASolve | Computes the unique F_2-rational zero of a given polynomial system over F_2. |
CharP.IMXLSolve | Computes the unique F_2-rational zero of a given polynomial system over F_2. |
CharP.LAAlgorithm | Computes the unique F_2-rational zero of a given polynomial system over F_2. |
CharP.MBBasis | Computes a Border Basis of a given ideal over F_2. |
CharP.MNLASolve | Computes the unique F_2-rational zero of a given polynomial system over F_2. |
CharP.MXLSolve | Computes the unique F_2-rational zero of a given polynomial system over F_2. |
CharP.NLASolve | Computes the unique F_2-rational zero of a given polynomial system over F_2. |
CharP.XLSolve | Computes the unique F_2-rational zero of a given polynomial system over F_2. |
FGLM.FGLM | Performs a FGLM Groebner Basis conversion using ApCoCoAServer. |
GLPK.BPMax | Solving binary programmes by maximizing the objective function. |
GLPK.BPMin | Solving mixed integer linear programmes by minimizing the objective function. |
GLPK.IPCSolve | Solves a system of polynomial equations over F_2 for one solution in F_2^n. |
GLPK.L01PSolve | Solve a system of polynomial equations over F_2 for one solution in F_2^n. |
GLPK.LPMax | Solving linear programmes by maximizing the objective function. |
GLPK.LPMax | Solving mixed integer linear programmes by maximizing the objective function. |
GLPK.LPMin | Solving linear programmes by minimizing the objective function. |
GLPK.LPMin | Solving mixed integer linear programmes by minimizing the objective function. |
GLPK.LPSolve | Solving linear programmes. |
GLPK.MIPSolve | Solving linear programmes. |
GLPK.RIPCSolve | Solves a system of polynomial equations over F_2 for one solution in F_2^n. |
GLPK.RPCSolve | Solves a system of polynomial equations over F_2 for one solution in F_2^n. |
GLPK.RRPCSolve | Solves a system of polynomial equations over F_2 for one solution in F_2^n. |
Hom.HSolve | Solves a zero dimensional square homogeneous or non-homogeneous polynomial system of equations. |
Hom.LRSolve | Solves a non-square zero dimensional homogeneous or non-homogeneous polynomial system of equations. |
Hom.SRSolve | Solves a non-square zero dimensional homogeneous or non-homogeneous polynomial system of equations. |
IML.REF | Compute a row echelon form of a matrix. |
IML.Solve | Solves a system of linear equations. |
Latte.Count | Counts the lattice points of a polyhedral given by a number of linear constraints. |
Latte.Ehrhart | Computes the ehrhart series as a rational function for a polyhedral P given by a number of linear constraints. |
Latte.Maximize | Maximizes the objective function over a polyhedral P given by a number of linear constraints. |
Latte.Minimize | Minimizes the objective function over a polyhedral P given by a number of linear constraints. |
LinAlg.CharPoly | Computes the characteristic polynomial of a matrix. |
LinAlg.Det | Computes the determinant of a matrix. |
LinAlg.EF | Computes a row echelon form of a matrix over F_2 with record keeping. |
LinAlg.REF | Computes a row echelon form of a matrix. |
LinAlg.Solve | Solves a system of linear equations. |
LinBox.CharPoly | Computes the characteristic polynomial of a matrix. |
LinBox.Det | Computes the determinant of a matrix. |
LinBox.REF | Computes a row echelon form of a matrix. |
LinBox.Solve | Solves a system of linear equations. |
LinSyz.BettyNumber | Computes the N-th Betty number of a module generated by linear forms. |
LinSyz.BettyNumbers | Computes all Betty numbers of a module generated by linear forms. |
LinSyz.Resolution | Computes syzygy modules of a module generated by linear forms. |
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.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. |
Num.ABM | Computes the border basis of an almost vanishing ideal for a set of points using the ABM algorithm. |
Num.AVI | Computes a border basis of an almost vanishing ideal for a set of points. |
Num.BBABM | Computes the border basis of an almost vanishing ideal for a set of points using the BB ABM algorithm. |
Num.CABM | Computes the border basis of an almost vanishing ideal for a set of complex points. |
Num.CEXTABM | Computes the border basis of an almost vanishing ideal for a set of points. |
Num.ContainsLinearRelations | Checks if a given set of terms has some epsilon-linear dependencies with respect to a set of points. |
Num.DABM | In a differential setting, computes the border basis of an almost vanishing ideal for a set of points using the ABM algorithm. |
Num.EigenValues | Computes the eigenvalues of a matrix. |
Num.EigenValuesAndAllVectors | Computes eigenvalues and left and right eigenvectors of a matrix. |
Num.EigenValuesAndVectors | Computes the eigenvalues and eigenvectors of a matrix. |
Num.EXTABM | Computes the border basis of an almost vanishing ideal for a set of points. |
Num.IsAppBB | Checks if a given set of polynomials is an approximate border basis. |
Num.IsAVI | Checks if a given set of polynomials vanishes at a given set of points. |
Num.LeastSquaresQR | Computes the least squares solution of the linear system of equations Ax=b. |
Num.ProjectAVI | Computes the least squares solution of the general problem Ax=b, where x are coefficients of an order ideal. |
Num.QR | Computes the QR-decomposition of a matrix. |
Num.RatPoints | Computes the zero set of an exact zero dimensional border basis. The zeros are computed approximately using the eigenvalues of the transposed multiplication matrices. |
Num.SimDiag | Computes an approximate diagonalization of a set of matrices. |
Num.SingularValues | Computes the singular values of a matrix. |
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. |
Num.SVD | Computes the singular value decomposition of a matrix. |
PGBC.ParallelGBC | Computes a Gröbner Bases over a prime field using the degree reverse lexicographic term ordering in parallel. |
Slinalg.SEF | Computes the row echelon form of a sparse matrix over F2. |
Slinalg.SGEF | Performs specified steps of structured gaussian elimination on a sparse matrix over F2. |
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.Inw | Computes the initial form of a polynomial 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. |
Weyl.WNormalRemainder | Computes the normal remainder of a Weyl polynomial F with respect to a polynomial or a set of polynomials. |
Weyl.WNR | Computes the normal remainder of a Weyl polynomial F with respect to a polynomial or a list of Weyl polynomials using corresponding implementation in ApCoCoALib. |
Weyl.WRedGB | Computes reduced Groebner basis of a D-ideal in Weyl algebra A_n. |
Weyl.WRGB | Reduced Groebner basis of an ideal I in Weyl algebra A_n. |
Weyl.WRGBS | Convert a Groebner basis of an ideal in Weyl algebra A_n in to its reduced Groebner Basis using corresponding implementation in ApCoCoALib. |