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How to Reduce a system of algebraic inequalities in two variables?

Posted 3 months ago
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I want to prove that

expression1[u,v] <= 0 for all real u, v satisfying 0<=u<=2 and v^2<=u^3

where the formula for expression1 is algebraic but very large and messy. Based on numerical experiments, I strongly believe this statement is true.

In other words, I want to compute

Reduce[expression1[u,v]<=0 && 0<=u<=2 && v^2<=u^3, {u,v}]

However, I am unable to compute this on my laptop (not enough computational resources I suppose, Reduce keeps running even after several hours). So I have two questions:

(1) Is there a way to monitor the progress of Reduce[]?

(2) I have access to powerful computing clusters as well as AWS. Is there a way to use these resources to solve this problem in a reasonable amount of time?

I have attached the explicit formula for expression1.

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Posted 3 months ago

Your conjecture appears to be true. Here's a visual approach:

Show[Plot[{-Sqrt[u^3], Sqrt[u^3]}, {u, 0, 2}, PlotStyle -> Red,
   Filling -> {1 -> {{2}, {LightRed, Transparent}}}, Frame -> True], 
   ContourPlot[expression1[u, v], {u, 0, 2}, {v, -Sqrt[8], Sqrt[8]}, PlotRange -> All,
    Contours -> 20, ContourLabels -> True, ContourShading -> None]
 ]

Contour plot with overlay of a region

POSTED BY: Jim Baldwin
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