[Axiom-math] Re: [Axiom-mail] Solving inequalities

[Martin Rubey]

Hello,
find all values of x such that f(x) > 1,
where f(x) is an `Expression Integer'
`solve' and `radicalSolve' are apparently not helpful here. Any idea on
how to achieve this?

For polynomial f this can be done using "cylindrical algebraic decompositions"
(CADs). Quite a bit of this is actually implemented by Renaud Rioboo, see
AxiomContributions on MathAction. Unfortunately, some pieces are missing in
order to solve inequalities.

If you were able to fill the gaps, this would be really great. The algorithm is
described in Renaud Rioboos thesis. Maybe Renaud could give some instructions
on how to proceed from his packages. Another way to help would be to provide
explanations for Renaud's stuff, although I guess for you it would make more
sense to read up on CAD and provide the missing pieces to make it into a
inequality solver. This would be a really nice thing to have for Axiom!!! Maybe
Renaud would be able to help you there?

By the way: an implementation exists in Mathematica (it's called
InequalitySolve there), and in fact, it was able to prove a theorem for me
once.

Thanks, and sorry for bothering you,

why exactly should you be sorry? If I wouldn't love to work on axiom/aldor, I
wouldn't do it.


Martin

[Ralf Hemmecke]

Post by Martin Rubey

find all values of x such that f(x) > 1,
where f(x) is an `Expression Integer'
`solve' and `radicalSolve' are apparently not helpful here. Any idea on
how to achieve this?

For polynomial f this can be done using "cylindrical algebraic decompositions"
(CADs). Quite a bit of this is actually implemented by Renaud Rioboo, see
AxiomContributions on MathAction. Unfortunately, some pieces are missing in
order to solve inequalities.
If you were able to fill the gaps, this would be really great.

(Sorry for the late reply.)
Actually, I'm not a mathematician and I'm afraid I wouldn't be able to
contribute to it without attending tens of hours of classes on that
topic. ;-)

The problem is, as Martin has already pointed out, that your problem is
undecidable if you do not restrict f to a certain class of functions
(like for example polynomial functions). So nobody will be able to write
a general solver for your problem.

Could you specify they type of functions more clearly, ie, replace
"Expression Integer" to some smaller class of functions?

Ralf

[Ludovic Courtès]

Hi,

Post by Ralf Hemmecke
The problem is, as Martin has already pointed out, that your problem
is undecidable if you do not restrict f to a certain class of
functions (like for example polynomial functions). So nobody will be
able to write a general solver for your problem.
Could you specify they type of functions more clearly, ie, replace
"Expression Integer" to some smaller class of functions?

Yes, I can restrict it to a `Fraction Polynomial Integer'.
Does it help?

Thanks,
Ludovic.