I was reading the proof of Hölder's theorem on wikipedia that the gamma function $\Gamma(z)$ does not satisfy any differential equation of form $P(z;\Gamma(z),\Gamma'(z),...,\Gamma^{(n)}(z))=0$, where $P(X;Y_0,Y_1,...,Y_n)$ is a polynomial (I don't know why there is a semicolon, but see the last paragraph). One of the key idea is to define a total order on monomials, and therefore we may talk about the highest term or "degree" of a polynomial. However the article is kind of vague about that. According to this there are several orders on monomials. Which of them is being used here? More specifically, in the wiki article the following example is given:
$\deg \left(-3 X^{10} Y_{0}^{2} Y_{1}^{4} + i X^{2}Y_{2} \right) < \deg \left( 2 X Y_{0}^{3} - Y_{1}^{4} \right)$
Below is part of the proof.
Furthermore, if $X^{h} Y_{0}^{h_{0}} Y_{1}^{h_{1}} \cdots Y_{n}^{h_{n}}$ is the highest-degree monomial term in $P$, then the highest-degree monomial term in $Q\stackrel{\text{df}}{=} ~ P(X + 1;X Y_{0},X Y_{1} + Y_{0},X Y_{2} + 2 Y_{1},\ldots,X Y_{n} + n Y_{n - 1})$ is
$X^{h + h_{0} + h_{1} + \cdots + h_{n}}Y_{0}^{h_{0}}Y_{1}^{h_{1}} \cdots Y_{n}^{h_{n}}$.
Consequently, the polynomial
$Q - X^{h_{0} + h_{1} + \cdots + h_{n}} P$
has a smaller overall degree than $P$.
I think for the argument to work we should use the lexicographic order $Y_n<Y_n-1<...<Y_0<X$, i.e., given two monomials first compare the order of $Y_n$, and then $Y_{n-1}$, and finally $X$. In this case the example given above should be false. Am I correct?
Wikipedia gives two references, A Survey of Transcendentally Transcendental Functions by Lee A. Rubel and Irresistible Integrals by George Boros and Victor Moll. Both of their proofs are quite similar to wikipedia article, but the order they use seems to be more mysterious: they view $X$ as coefficient rather than indeterminate and the order does not involve $X$, so the "highest degree term" is something like $q(X)Y_{0}^{h_{0}}Y_{1}^{h_{1}} \cdots Y_{n}^{h_{n}}$. Then they use some "Euclidean algorithm", which I do not understand how to work for multivariate polynomials. Should the monomial order involve $X$ or not ?