Thom Encoding Representation

Question

I'm currently doing some work which involves Thom Encodings. However I'm having trouble about their representation. This is the algorithm which uses them:

INPUT: Real algebraic numbers α and β given by their Thom encodings $σ_{P_1}(α)$ and $σ_{P_2}(β)$ with respect to polynomials $P_1, P_2 ∈ Z[X]$ such that $deg(P_1), deg(P_2) ≤ d (d ≥ 2)$ and $H(P_1), H(P_2) ≤ H$. Where H is the height of a polynomial.

OUTPUT: The sign $s := sign(e^β − α)$.

1. Let $c := (2^{d+1}(d + 1)H)^{−2^{41}d^6(5d+4⌈log(H)⌉)}$.
2. Compute $w ∈Q$ such that $|e^β −w|(a) Compute $w_1 ∈Q$ such that $|β−w_1|< (\frac{c}{(2.3^{H+2}})$
   (b) Compute $w ∈Q$ such that $|e^{w_1} −w|< (\frac{c}{2})$
3. Compute $s = sign(w − α)$.

What I'm wondering is what is α and β? I believe that a Thom Encoding is a tuple of the polynomial and the signs when calculated at their derivatives. If this is true, how could α and β, a tuple, be used in the algorithm? If possible, could you provide an example please?
Thanks, help appreciated.

Answer 1

in the above case, $\alpha$ and $\beta$ represent real algebraic numbers--specific roots of a monic polynomial in $\mathbb{Q}[x]$.

The tuple you are referring to--the Thom encoding--is a representation of those numbers as the combination of their unique minimal polynomials and a sequence of signs representing the sign of the derivatives of that polynomial at the given root $\alpha$ or $\beta$. The sign sequence is what identifies the specific root uniquely, and is one way to canonically represent a real algebraic number. (The other, probably more common way, is to use the polynomial in combination with an isolating interval that contains the root in question and excludes all other roots.) To be pedantic, the Thom encoding is the sign sequence itself, rather than the aforementioned tuple; i.e., polynomial + Thom encoding = unique representation.

So the $\alpha$ and $\beta$ in the algorithm are specific real values that can be calculated with. The Thom encoding is just (part) of a representation of these values.