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Suppose two elements of the ring of periods are given by their systems of polynomial inequalities with rational coefficients. Is there a known algorithm deciding their equality? Is it known if their equality is algorithmically decidable at all?

It is quite obvious that if they are not equal, then it can be shown after finite number of steps just by evaluating them with big enough precision. The difficult part is when they are equal.

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There is a deep conjecture here which is explained nicely in the paper of Kontsevich-Zagier. – Moosbrugger Dec 8 '11 at 22:05
This is Problem 1 of Kontsevich-Zagier. They say: "Problem 1 therefore looks completely intratable now and may remain so for many years" (page 7). – François G. Dorais Dec 9 '11 at 4:02
up vote 7 down vote accepted

There is no known algorithm for that, and it is unknown if such an algorithm exists.

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