CoCoA:QuotientBasis

QuotientBasis

vector space basis for zero-dimensional quotient rings

Description

This function determines a vector space basis (of power products)

for the quotient space associated to a zero-dimensional ideal.

That is, if R is a polynomial ring with field of coefficients k, and I is a zero-dimensional ideal in R then QuotientBasis(I) is a set of power products forming a k-vector space basis of R/I.

The actual set of power products chosen depends on the term ordering

in the ring R: the power products chosen are those not divisible by

the leading term of any member of the reduced Groebner basis of I.

Example

  Points := [[Rand(-9,9) | N In 1..3] | S In 1..25];
  Use Q[x,y,z];
  I := IdealOfPoints(Points);
  QuotientBasis(I);     -- power products underneath the DegRevLex reduced GBasis
[1, z, z^2, z^3, z^4, y, yz, yz^2, yz^3, y^2, y^2z, y^2z^2, y^3, x,
xz, xz^2, xz^3, xy, xyz, xyz^2, xy^2, x^2, x^2z, x^2y, x^3]
-------------------------------
  Use Q[x,y,z],Lex;
  I := IdealOfPoints(Points);
  QuotientBasis(I);     -- power products underneath the Lex reduced GBasis
[1, z, z^2, z^3, z^4, z^5, z^6, z^7, z^8, z^9, z^10, z^11, z^12, z^13,
y, yz, yz^2, yz^3, yz^4, yz^5, yz^6, y^2, y^2z, y^2z^2, y^2z^3]
-------------------------------

Syntax

QuotientBasis(I:IDEAL):LIST

IdealOfPoints

   <type>ideal</type>