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6.2.14 ApCoCoAServer
The following are the functions using 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. 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.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. 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.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. 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.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.




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