# how to find standard monomials of an Ideal?

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

how to find standard monomials of an Ideal?

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

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1

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).

0

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 mathÃ©matique 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.

1

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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Asked: ** 2013-06-07 11:17:14 +0200 **

Seen: **442 times**

Last updated: **Jul 07 '22**

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