up previous next

QuotientBasis    --    vector space basis for zero-dimensional quotient rings

Syntax

QuotientBasis(I: IDEAL): LIST

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 (and consequently they form a factor-closed set).

The power-products in the result are sorted in increasing lex ordering. See QuotientBasisSorted for sorting them according to the term-ordering of the ring.

Example

/**/ use P ::= QQ[x,y,z]; /**/ I := intersection(ideal(x,y,z)^2, ideal(x-1, y+1, z)^2); /**/ QB := QuotientBasis(I); /**/ QB; -- power-products underneath the reduced GBasis of I [1, z, y, y*z, y^2, y^3, x, x*y]

See Also

• IdealOfPoints
• IsFactorClosed