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Re: Resolve[Exists[i,{-1,0,2,-3,4,0}>0],Integers] does not work

On 9 Feb 2008, at 10:19, janos wrote:

> Resolve[Exists[x, a x^2 + b x + c == 0 && x > 0], Reals]
> (taken from the Help) works fine. However,
> Resolve[Exists[i, Part[{-1, 0, 2, -3, 4, 0}, i] > 0], Integers]
> does not work.
> My guess is that Part is evaluated differently from Equal,
> but I do not know how to fix the problem.
> Thank you.
> Janos

The problem is that Quanitifer Elimination , which is the algorithm  
that is being used here, works only with polynomial equations and  
inequalities and only over the real or complex numbers. So obviously  
you can't use Part or any other programming (rather than algebraic)  
construct. That's the main reason why this sort of thing won't work at  
all. But in addition, no known general quantifier elimination  
algorithm works over the integers so only very simple special cases  
have been implmented in Mathematica. For example:

Resolve[Exists[i, Element[i, Integers], i^2 == 2]]

works, but the more complicated:

  Resolve[ForAll[i, Element[i, Integers],
      Exists[j, Element[j, Integers], j^2 == i]]]

ForAll[i, Element[i, Integers],
    Exists[j, Element[j, Integers], j^2 == i]]

doesn't, while over the reals there is no problem:

  Resolve[ForAll[i, Element[i, Reals], Exists[j, Element[j, Reals],
        j^2 == i]]]


  Resolve[ForAll[i, Element[i, Reals] && i >= 0,
      Exists[j, Element[j, Reals], j^2 == i]]]

and so on.

Andrzej Kozlowski

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