I am reading Bochnak, Coste, Roy's real algebraic geometry and the definition of ordering on $R(X)$ is not clear to me why/how can the authors write the polynomial $P(X)=a_nX^n+a_{n-1}X^{n-1}+\cdots+a_kX^k$ this way (with $a_k \neq0$ and then proceed to define the ordering)? Is the leading coefficient $a_n$ or $a_k$?
Edit: the definition proceeds as follow. With respect to the above form, $P(X)>0$ if $a_k>0$. Can someone explain why this is well defined? I dont understand the meaning/significance of $a_k$ as it is not uniquely defined.