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Understanding Resolve[Exists[...]] type command

Posted 11 years ago
Hi everyone,
I'd be most thankful for any and all inputs into understanding what I describe below. I thank you for your patience and time in hearing out my question.
I frequently use
Resolve[Exists[{x1,x2,...,xL},inq1>0&&inq2>0&&...&&inqM>0&&eq1==0&&eq2==0&&...&&eqN==0]]
where inq1,...,inqM and eq1,...,eqN are all real multivariable polynomials in x1,...,xL (and x1,..,xL are all positive and so real).
I use the command above to check for the existence of a solution satisfying all the inequalities and equalities as represented by the expressions inq1,...,inqM>0 and eq1==...=eqN==0. The command works perfectly fine.
But now there is one additional inequality
(Polynomial 1 in {x1,...,xL}) - (Polynomial 2 in {x1,...,xL})*Log[(1+x1)/(1-x1*x2)] > 0
that I'd like to introduce into the command above, call it INQ.
In generally, introducing INQ into the above appears to bottleneck Mathematica. In fact, leaving the conditions inq1,...,inqM>0 and eq1==...=eqM==0 aside,
Resolve[Exists[{x1,...,xL},INQ>0]]
which is obviously true in my case does not terminate. My first question is, is there any reason as to why the commands Resolve[] and Log[] are not compatible?
I tried Resolve[Exists[...],WorkingPrecision->number (say, 10)] to potentially quicken the calculation.
I am not sure why, but regardless of introducing INQ>0, the original command
Resolve[Exists[{x1,x2,...,xL},inq1>0&&inq2>0&&...&&inqM>0&&eq1==0&&eq2==0&&...&&eqN==0]]
updated to
Resolve[Exists[{x1,x2,...,xL},inq1>0&&inq2>0&&...&&inqM>0&&eq1==0&&eq2==0&&...&&eqN==0],WorkingPrecision->10]
does not converge. For example, the original command takes less than 10 seconds, but the new command simply doesn't converge.
My second question is, isn't WorkingPrecision->number supposed to fasten the calculations? Why is this happening?
Finally, my third question is, is there any way of quickening Resolve[Exists[...]] type of command?

Thank you so much in advance for all your help!!

Best,
John
POSTED BY: J K
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