subrings

[Jessica Sidman]

Suppose that I have polynomial ring 

R = ZZ/32003[x,y,z]

and I want to generate the subring of R generated by x+y-z (and I really do want those coordinates).  Is there a way to do this in M2?  I can only see how to get a module generated by an element like that, and not the subring.

Thanks,

Jessica

[David Eisenbud]

Define a ring map from a polynomial ring S on 1 variable to R; compute the kernel I subset S of the map; then define the map from S/I to R.

--
David Eisenbud
Director, Mathematical Sciences Research Institute; and
Professor of Mathematics,University of California, Berkeley
www.msri.org/~de

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[Daniel R. Grayson]

Right.  Moreover, if f denotes the ring map, then "coimage f" will compute S/I for you.  (In Macaulay2 the notion of quotient ring

is supported, but not the notion of subring.)

On Friday, August 25, 2017 at 9:34:14 PM UTC-4, David Eisenbud wrote:

Define a ring map from a polynomial ring S on 1 variable to R; compute the kernel I subset S of the map; then define the map from S/I to R.

--
David Eisenbud
Director, Mathematical Sciences Research Institute; and
Professor of Mathematics,University of California, Berkeley
www.msri.org/~de

On Aug 25, 2017, at 8:23 AM, Jessica Sidman <jsid...@gmail.com> wrote:

Suppose that I have polynomial ring 

R = ZZ/32003[x,y,z]

and I want to generate the subring of R generated by x+y-z (and I really do want those coordinates).  Is there a way to do this in M2?  I can only see how to get a module generated by an element like that, and not the subring.

Thanks,

Jessica

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[Jessica Sidman]

Thanks, Dan and David.  Unfortunately I'm still confused.  What I think you're saying is to do something like this:

--Want the subring ZZ/32033[x+y] contained in R.

R = ZZ/32003[x,y,z]

--ring with one variable

S = ZZ/32003[w]

--map from S to R.  (But the kernel of f is (0), so coimage f is just S again.)

f = map(R,S, {x+y})

g = map(R, coimage f, {x+y})

Which doesn't get me what I want (I think?), which is a subring of R that I can intersect with an ideal of R.

Maybe I should describe what I really want to do.  I have an ideal, and I want to eliminate some variables.  Unfortunately, the generators of the ideal are bad (=not sparse) in the most natural coordinates, and the computations won't terminate.  I happen to know a change of coordinates that makes computations with this ideal much better (elimination of variables in the new coordinates is fast!).  But in the new coordinates I don't want to eliminate variables, I want to intersect with a subring generated by linear forms that are not monomials.

Is this not going to be possible because subrings are not supported in M2?

Thanks!

Jessica

[David Eisenbud]

How about the kernel of 

R \to S/I ?

David Eisenbud

Professor of Mathematics

UC Berkeley

Sent from my phone

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[zteitler]

Can you get what you want by using the 'preimage' command?

https://faculty.math.illinois.edu/Macaulay2/doc/Macaulay2-1.10/share/doc/Macaulay2/Macaulay2Doc/html/_preimage.html

Zach

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[Douglas Leonard]

For what it is worth, I often do something similar in a very naive way.

I may have something in the variables x,y,z and want to change to variables a,b,c.

This can be done in a ring of 3+3 variables by eliminating the first 3, as in the following concrete

example where x,y,z satisfy g and a,b,c will satisfy G_(0,0), with the rest of G giving

x,y,z in terms of a,b,c.

R=QQ[x,y,z,a,b,c,MonomialOrder=>{Eliminate 3}];
f1=x+y-z-a;
f2=y+2*z-b;
f3=3*x-y-c;
g=x^2+y^3*z-x*y*z;
I=ideal(f1,f2,f3,g);
G=gens gb I

---

From: maca...@googlegroups.com <maca...@googlegroups.com> on behalf of zteitler <ztei...@gmail.com>
Sent: Sunday, August 27, 2017 10:37 PM
To: Macaulay2
Subject: Re: [Macaulay2] subrings