how to find standard monomials of an Ideal?
Hello Is there any special command for contributing standard monomials of an Ideal I?
Hello Is there any special command for contributing standard monomials of an Ideal I?
A basis of the quotient ring K[x1,…,xn]/I as a K-vector space is also called a normal basis of the ideal I.
sage: R.<x,y> = PolynomialRing(QQ, order='lex')
sage: I = R.ideal(x^2+y^2-1, 16*x^2*y^2-1)
sage: I.normal_basis()
[x*y^3, y^3, x*y^2, y^2, x*y, y, x, 1]
See the documentation of normal_basis
for more options (e.g., when I is not 0-dimensional).
Is groebner_basis
the command you need?
sage: R.<x,y> = PolynomialRing(QQ, order='lex')
sage: R.ideal(x^2+y^2-1, 16*x^2*y^2-1).groebner_basis()
[x^2 + y^2 - 1, y^4 - y^2 + 1/16]
This example is from section 9.3 of the book Calcul mathématique avec Sage / Computational mathematics with SageMath (available in French, English, German). The link is to a page where you can download or order the book.
Thank you, but I need to know is there any commands for standard monomial of an ideal i.e. for each ideal I with an special order, the monomials which not exist in <lt(i)> are standard monomials.
By the way Do you know any English book with some good examples for learning and practicing sage? thank you so much
I believe that this wasn't the OP's question, but rather this was intended to ask: Let I be a 0-dimensional ideal over a polynomial ring R over a field K. Then R/I is a finite dimensional K-algebra, and the requested set consists of monomials from R whose image in R/I is a K-vector space basis. In fact I came across this question because I too was wondering how to do that in SageMath. @rburing: Thank you for the quick answer!
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Last updated: Jul 07 '22