Suppose that $a$ and $b$ are two algebraic numbers with $0<|a-b|\approx 10^{-50}.$ Suppose further that a calculator can evaluate $a$ and $b$ up to say 12 digits. Are there some general transformation I can do such that my calculator will output if $a<b$ or not?
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Suppose that $f(x)\in \mathbb{Q}[x]$ has roots $\{\alpha_1,\ldots,\alpha_k\}$ and $g(x)\in \mathbb{Q}[x]$ has roots $\{\beta_1,\ldots,\beta_m\}$. Let $m(x)$ denote the minimal polynomial for $\alpha_1-\beta_1$ (e.g.) over $\mathbb{Q}[x]$, which can be found as a factor of the polynomial $$F(x)=\prod_{i,j} (x-\alpha_i+\beta_j).$$ Thus $m(x)$ has a root very near $0$, the sign of which can be determined (heuristically) by looking at $m(0)$ and the sign of $m'(0)$. To make the answer obtained here precise, we just need to bound the other roots of $m$ away from zero, which is not too hard (the bound comes from the discreteness of the roots of $f$ and $g$.
awwalker
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Assuming the calculator is only deficient in its ability to display digits, but not store them, you could consider the output of $10^{50}(a-b)$.
Jared
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That would be a very weird calculator, don't you think? I don't think you can do anything on this calculator to distinguish these two numbers. The fact that are algebraic is irrelevant. – Heberto del Rio May 07 '13 at 21:20