I know this Lemma appears more in trascendental number theory, but I saw a proof of the irrationality of $e^\pi$ it's a proof by contradiction, finding a integer between $0$ and $1$, but to that it's uses a few ideas, like the Lemma and other things, that I don't understand. I'm more interested in the irrational theory rather than the transcedental part (I'm not there yet), but it's fine to include anything if it's needed.
From wikipedia: "Siegel's lemma refers to bounds on the solutions of linear equations obtained by the construction of auxiliary functions"
In the proof that I read for $e^\pi$ the auxiliary function came from a two variable polynomial.
Also from wikipedia, these auxiliary functions must have some properties "such as being $0$ for many arguments, or having a zero of high order at some point"
Why are the properties above needed in a irrationality proof?
How finding bounds to the solutions of these auxiliary functions helps in the proof?