ASKSAGE: Sage Q&A Forum - RSS feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Thu, 13 Dec 2012 04:04:43 +0100Normalization (integral closure)https://ask.sagemath.org/question/7876/normalization-integral-closure/I tried doing some calculations that involve integral closures, but I seem to run into problems:
Say I define a ring as a quotient ring: for example the quotient ring $\mathbb{C}[x,y]/(y^3-x^2)$. Then if I use the integral_closure option, he rejects me. As far as I can tell this is because he treats quotient rings as Commutative Rings rather than, say, integral domains, and so it doesn't have the option of integral closures.
I'm sure there's a way to do such things. What is it? Is the idea that we have to apply some functor so that that ring would be treated as an object in a different category?Sat, 15 Jan 2011 21:17:17 +0100https://ask.sagemath.org/question/7876/normalization-integral-closure/Answer by William Stein for <p>I tried doing some calculations that involve integral closures, but I seem to run into problems:</p>
<p>Say I define a ring as a quotient ring: for example the quotient ring $\mathbb{C}[x,y]/(y^3-x^2)$. Then if I use the integral_closure option, he rejects me. As far as I can tell this is because he treats quotient rings as Commutative Rings rather than, say, integral domains, and so it doesn't have the option of integral closures.</p>
<p>I'm sure there's a way to do such things. What is it? Is the idea that we have to apply some functor so that that ring would be treated as an object in a different category?</p>
https://ask.sagemath.org/question/7876/normalization-integral-closure/?answer=11985#post-id-11985I don't think there is any easy-to-use functionality in Sage for computing the integral closure of a polynomial quotient ring. What you would want would be for the following to work:
sage: R.<x,y> = QQ[]
sage: S = R.quotient(y^2 - x^3); S
Quotient of Multivariate Polynomial Ring in x, y over Rational Field by the ideal (-x^3 + y^2)
sage: S.integral_closure()
but of course it doesn't. The actual *hard work* to make this possible is already in Sage, via Singular, as [documented here](http://www.singular.uni-kl.de/Manual/3-1-0/sing_951.htm). Here is a complete example that shows actually computing the integral closure of S via singular:
sage: R.<x,y> = QQ[]
sage: I = singular(R.ideal([y^2 - x^3])); I
-x^3+y^2
sage: singular.load('normal.lib')
sage: I.normal()
[1]:
[1]:
// characteristic : 0
// number of vars : 3
// block 1 : ordering dp
// : names T(1)
// block 2 : ordering dp
// : names x y
// block 3 : ordering C
[2]:
[1]:
_[1]=x^2
_[2]=y
If you read the documentation I linked to above, you might see what the output means (after 3 minutes, I didn't). To make this a nice easy-to-use function in Sage, you would have to decide on exactly what the output should be in Sage, then implement a function for Sage that (under the hood) would just call the normal command in Singular. This would be a great project to get you into Sage development. Mon, 17 Jan 2011 13:50:53 +0100https://ask.sagemath.org/question/7876/normalization-integral-closure/?answer=11985#post-id-11985Answer by pkoprowski for <p>I tried doing some calculations that involve integral closures, but I seem to run into problems:</p>
<p>Say I define a ring as a quotient ring: for example the quotient ring $\mathbb{C}[x,y]/(y^3-x^2)$. Then if I use the integral_closure option, he rejects me. As far as I can tell this is because he treats quotient rings as Commutative Rings rather than, say, integral domains, and so it doesn't have the option of integral closures.</p>
<p>I'm sure there's a way to do such things. What is it? Is the idea that we have to apply some functor so that that ring would be treated as an object in a different category?</p>
https://ask.sagemath.org/question/7876/normalization-integral-closure/?answer=14368#post-id-14368I believe this can be easily computed with Macaulay-2, as documented here: http://www.math.uiuc.edu/Macaulay2/doc/Macaulay2-1.4/share/doc/Macaulay2/IntegralClosure/html/ (BTW, this is yet another reason to integrate Sage with Macaulay by default, not only as a optional package.)Thu, 13 Dec 2012 04:04:43 +0100https://ask.sagemath.org/question/7876/normalization-integral-closure/?answer=14368#post-id-14368