What is an "algebraic point" in differential equations?

Question

This wikipedia article https://en.wikipedia.org/wiki/Hypertranscendental_number mentions an "algebraic point" when describing the definition of a hypertranscendental number. However, there's no wiki entry for an algebraic point and google doesn't give me any results because the terms are too ambiguous.

Answer 1

Transcendental numbers are those that do not appear as roots of polynomials over the integers. That is, there is no finite encoding of exactly that number using a rationally bounded interval (or complex square) to contain this number as the exactly one root of a polynomial with integer coefficients.

Now one can use other concepts like ODE to find a finite (in bit length) description of some of those algebraically transcendental numbers. Thus the idea to take as right side of an ODE a polynomial, which has a finite number of parameters, make these parameters finitely encoded, that is algebraic numbers, use algebraic initial conditions and an algebraic time. This gives a definite result, as long as the time is inside the maximal domain of the solution, and a finite description, regardless of how complex the structure. The given example is $$ e=y(1)\text{ where }y'=y, ~y(0)=1. $$

Thus by the definition in the link, hypertranscendental numbers are those that do not appear as components of function values $y(t)$ at rational or algebraic (over $\Bbb Z$) times $t\in\bar {\Bbb Z}$ (algebraic numbers) of any IVP

$y'=p(t,y)$ with $y(0)\in\bar {\Bbb Z}$ and $p\in\Bbb Z[t,x]$.

I'm not sure if the original sources also include implicit ODE, so that $p$ could also be algebraic over $\Bbb Z[t,x]$. Also, the coefficients could also be algebraic, wikipedia is not the most faithful source. Other points of consideration are how arithmetically and algebraically closed that set of numbers is, or if one can sacrifice dimensional compactness to get simpler data, that is, increase the dimension of the ODE system to get a system where some or all parameters are rational or integer,...

As you can see, "algebraic point" means exactly that, an algebraic number for the time $t$. I'm not sure why there has to be a (dead) link.