Inequalities in InfinitePolynomialRing

[Nathann Cohen]

Hello everybody !!!!

Could you tell me why there does not seem to be any support for inequalities in InfinitePolynomialRing ?

I would like to express general inequalities of the form x[1] + x[2] < x[3], but each time I evaluate it I get a boolean, which is not quite what I would like to keep of the expressopn :-/

Besides, what is the meaning of this boolean ?

Thank you !!!

Nathann

[Simon King]

Hi!

On Sep 1, 8:31 am, Nathann Cohen <nathann.co...@gmail.com> wrote:
> Could you tell me why there does not seem to be any support for inequalities
> in InfinitePolynomialRing ?

Because what you call "inequalities" are symbolic expressions.

InfinitePolynomialRing has nothing to do with symbolics. It is about
some very recent development in commutative algebra, namely
1. polynomials over a countably infinite set of variables x[1],x[2],x
[3],...
2. "symmetric ideals", i.e., ideals that are invariant under variable
permutations.

The amazing fact is that an InfinitePolynomialRing R, though not
noetherian as a ring, is noetherian as an RG_\infty-module, where G_
\infty is the infinite permutation group acting on the indices of the
variables. In other words, every "symmetric ideal" can be finitely
generated up to sums, multiplication by elements of R, and variable
permutation.

And using Aschenbrenner's and Hillar's version of Groebner theory, one
can solve "symmetric ideal membership" and "symmetric ideal equality"
problems. Those things are explained in the Sage documentation.

> I would like to express general inequalities of the form x[1] + x[2] < x[3],
> but each time I evaluate it I get a boolean, which is not quite what I would
> like to keep of the expressopn :-/

x[1] + x[2] < x[3] is not an expression. It is simply a test which of
the two polynomials is smaller, in the monomial ordering of the ring.

> Besides, what is the meaning of this boolean ?

x[1] + x[2] < x[3] returns "True", and this simply means that the
leading monomial of x[1] + x[2] is smaller in the monomial ordering of
the ring than x[3]. By the way, in an InfinitePolynomialRing one will
always have x[1]<x[2]<x[3]< ...

However, if you want to use InfinitePolynomialRing as a comfortable
way to create symbolic expressions, you can easily do so:

sage: SR(x[1]+x[2])<SR(x[3])
x1 + x2 < x3

Now, SR(x[1]+x[2])<SR(x[3]) is a symbolic expression in the symbolic
variables x1,x2,x3.

I have no idea how one can solve inequalities, though. That's not my
field of expertise.

Best regards,
Simon

[Minh Nguyen]

Hi Simon,

On Tue, Sep 1, 2009 at 6:34 PM, Simon King<simon...@nuigalway.ie> wrote:

<SNIP>

> I have no idea how one can solve inequalities, though. That's not my
> field of expertise.

I'm based in a department with a research group in mathematical
inequalities and applications, but ironically I don't know anything
about solving inequalities apart from techniques from basic algebra.

--
Regards
Minh Van Nguyen

[William Stein]

I think Cylindrical Algebraic  Decomposition (something Carl Witty was working on) is relevant.

William

[Jason Grout]

You might look at the qepcad (which does CAD) interface Carl wrote too.

sage: qepcad?
sage: sage.interfaces.qepcad?

Jason

--
Jason Grout