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6.2.19 Border Basis
The following are commands and functions for Border Basis:
BB.ASgens Computes the generators of the vanishing ideal of a border basis scheme.
BB.ASneighbors Computes a list of across-the-street neighbors.
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.BBscheme Computes the defining equations of a border basis scheme.
BB.Border Computes the border of an order ideal.
BB.BorderDivAlg Applies the border division algorithm.
BB.Box Computes a box order ideal.
BB.CoeffOfBB Computes the coefficient matrix of a border basis.
BB.GenericBB Computes a generic border basis.
BB.GenericHomBB Computes a generic homogeneous border basis.
BB.GenHomMultMat Computes a generic homogeneous multiplication matrix.
BB.GenMultMat Computes a generic multiplication matrix.
BB.HomASgens Computes the generators of the vanishing ideal of a homogeneous border basis scheme.
BB.HomBBscheme Computes the defining equations 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.LiftHomAS Computes the homogeneous border basis scheme generators obtained from lifting of AS neighbours.
BB.LiftHomND Computes the homogeneous border basis scheme ideal generators obtained from lifting of next-door neighbors.
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.MultMat Computes the i-th multiplication matrix associated to a border basis.
BB.NDgens Computes the generators of the vanishing ideal of a border basis scheme.
BB.NDneighbors Computes a list of next-door neighbors.
BB.TransformBBIntoGB Transforms a border basis into a Groebner basis.
BB.TransformGBIntoBB Transforms a Groebner basis into a border basis.
BBF.Cancel Cancels the current border basis computation.
BBF.Finish Finishes a border basis computation.
BBF.Init Initializes a border basis computation.
BBF.Quit Terminates the ApCoCoABBFServer.
BBF.RetrieveBorderBasis Retrieves a computed border basis.
BBF.RetrieveBorderBasisLTs Retrieves leading terms of a computed border basis.
BBF.RetrieveBorderBasisPolyByLT Retrieves a computed border basis polynomial.
BBF.RetrieveOGenerators Retrieves generators of the order ideal.
BBF.RetrieveUGenerators Retrieves generators of the order ideal.
BBF.RetrieveV Retrieves polynomials from V.
BBF.RetrieveVLTs Retrieve leading terms of polynomials of V.
BBF.RetrieveVPolysWithTermInSupport Retrieves polynomials with specific support term from V.
BBF.RetrieveWPrime Retrieves polynomials from W'.
BBF.RetrieveWPrimeLTs Retrieves leading terms of polynomials of W'.
BBF.Status Retrieves status information from ApCoCoABBFServer.
BBF.Steps Executes border basis computation steps.
BBF.SwitchPort Changes the ApCoCoABBFServer communication port.
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.LinIndepGen Let OO be an order ideal and BO be its border. Let Mu:=Len(OO) and Nu:=Len(BO). This function computes the equivalent indeterminates from K[c_11,...,c_Mu Nu] modulo m^2, where m is the maximal ideal generated by the indeterminates {c_11,...,c_Mu Nu} from the coordinate ring of the border basis scheme. As out-put, it gives every equivalence class as a list.
BBSGen.NonStand This function computes the non-standard indeterminates from K[c] with respect to the arrow grading.
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.PurPow This function finds the pure power indeterminates in the ring K[c].
Num.IsAppBB Checks if a given set of polynomials is an approximate border basis.




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