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

Posted 10 years ago
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 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 useResolve[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)] > 0that 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 commandResolve[Exists[{x1,x2,...,xL},inq1>0&&inq2>0&&...&&inqM>0&&eq1==0&&eq2==0&&...&&eqN==0]]updated toResolve[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