up previous next
6.2.9 Polynomials
The following are commands and functions for polynomials:
BB.BorderDivAlg Applies the border division algorithm.
BB.TransformBBIntoGB Transforms a border basis into a Groebner basis.
BB.TransformGBIntoBB Transforms a Groebner basis into a border basis.
BBF.RetrieveBorderBasisPolyByLT Retrieves a computed border basis polynomial.
BBF.RetrieveVPolysWithTermInSupport Retrieves polynomials with specific support term from V.
DA.Class Returns the class of a given derivative.
DA.DiffAutoReduce Computes a reduced list of differential polynomials.
DA.Differentiate Computes the derivation of a differential polynomial.
DA.DiffGB Computes a differential Groebner basis.
DA.DiffOrd Computes the differential order of a differential polynomial.
DA.DiffReduce Computes a differential normal form of a differential polynomial wrt. a list of differential polynomials.
DA.DiffSupp Computes the differential support of a differential polynomial.
DA.Initial Computes the initial of a differential polynomial.
DA.InitialOfDer Computes the initial wrt. a given derivative.
DA.LD Computes the leading derivative of a differential polynomial.
DA.LPot Computes the leading power of a differential polynomial.
DA.NthDerivation Computes the N-th derivation of a differential polynomial.
DA.PseudoAutoReduce Computes a pseudo reduced list of differential polynomials.
DA.PseudoReduce Computes a pseudo normal form of a differential polynomial.
DA.Sep Computes the separand of a differential polynomial.
DA.Weight Computes the weight of a differential polynomial.
Dec Pretty Printing of Objects.
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.GDeg (Standard) degree of a polynomial in a free monoid ring over 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.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.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.GSubtract Subtraction of two polynomials in a free monoid ring over F2.
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.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.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.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.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.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.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.Subtract Subtraction of two polynomials in a free monoid ring.
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.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.
PGBC.ParallelGBC Computes a Gröbner Bases over a prime field using the degree reverse lexicographic term ordering in parallel.
SAT.ConvertToCNF Converts a given quadratic (cubic) system of polynomial equations (SPE) over GF(2) to CNF. Writes the CNF to the file sat.cnf
SAT.ConvertToXOR Converts a given quadratic (cubic) system of polynomial equations (SPE) over GF(2) to XOR-CNF. Writes the XOR-CNF to the file sat_xor.cnf.
SAT.FixBits Allows to fix chosen bits in advance within a SAT input file. The new input file is sat_fixed.cnf.
SAT.GetResult Looks up the configuration of the indeterminates in the file output produced by a SAT-Solver. The file output must be in the folder ApCoCoA-directory/sat/bin.
SAT.LaunchCryptoMiniSat Launches CryptoMiniSat via the ApCoCoAServer on a given input file located in ApCoCoA-directory/sat/bin. The result is written to the file output.
SAT.LaunchMiniSat Launches MiniSat via the ApCoCoAServer on a given input file located in ApCoCoA-directory/sat/bin. The result is written to the file output.
Weyl.WeylMul Computes the product F*G of two Weyl polynomials, F and G, in normal form.
Weyl.WMul Computes the product F*G of Weyl polynomial F and G in normal form.
Weyl.WMulByMonom Computes the product M*F of a Weyl monomial M and a Weyl polynomial F in normal form.
Weyl.WMult Computes the product F*G of two Weyl polynomials, F and G, in normal form using corresponding implementation in ApCoCoALib.
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.WPower Computes the N-th power of a Weyl polynomial.
Weyl.WSPoly Computes the S-polynomial of two Weyl polynomials.
Weyl.WStandardForm Computes the Standard form of a Weyl polynomial.




For details look up each item by name. Online, try ?ItemName or H.Syntax("ItemName").