how to find standard monomials of an Ideal?

AlgebraicGeometry

asked 11 years ago

Neda
75 ●6 ●10 ●20

Hello Is there any special command for contributing standard monomials of an Ideal I?

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answered 2 years ago

rburing

10964 ●4 ●80 ●219 https://www.rburing.nl/

A basis of the quotient ring $K[x_1,\ldots,x_n]/I$ 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).

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answered 11 years ago

slelievre
17799 ●22 ●163 ●352 http://carva.org/samue...

updated 2 years ago

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.

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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.

Neda ( 11 years ago )

By the way Do you know any English book with some good examples for learning and practicing sage? thank you so much

Neda ( 11 years ago )

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!

PeterMueller ( 2 years ago )

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