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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