Package borderbasis

This article is about a function from ApCoCoA-2. If you are looking for the ApCoCoA-1 version of it, see Category:ApCoCoA-1:Package borderbasis.

This page describes the borderbasis package. The package contains various functions for computing with border bases of order ideals in a polynomial ring P=K[x_1,...,x_n] over a field K. We refer the book [M. Kreuzer and L. Robbiano, Computational Commutative Algebra 2, Springer-Verlag, Berlin, 2005] for more details about border bases. For a complete list of functions, see also Category:Package borderbasis.

The Global Alias of the package is BB.

List of the main functions

IsOrderIdeal

IsOrderIdeal(OO): checks whether OO is an order ideal.
    input: OO a non-empty set of terms in K[x[1..N]]
    output: boolean value for checking OO being an order ideal

Border

Border(OO): computes the border of an order ideal.
    input: list of terms
    output: list of terms in ascending order

Box

Box(P,D): computes the 'box' order ideal of type D=[D1,..,DN].
    input: list of integers D of length NumIndets(P), P=K[x[1..N]]
    output: list of terms (sorted w.r.t. current TO)

BBasisForOI

BBasisForOI(F,OO): computes the border basis of the ideal I=<F> with respect to the order ideal OO, 
          gives an error messages if no border basis exists, uses the O_sigma(I) border basis and the BB transformation.
    input: list of poly F, list of terms OO
    output: list of poly

BorderDivAlg

	
BorderDivAlg(P,F,OO,Prebasis): applies the Border Division Algorithm w.r.t. the order ideal OO and the border prebasis
          Prebasis to the polynomial F and returns a record with fields 
          Quotients and Remainder where Remainder is the normal OO-remainder of F.
    input: poly F, list of terms OO, list of poly Prebasis
    output: record with two fields Quotients and Remainder

Example for computations

See also: BB.Border