Risch differential equation algorithm by Bronstein

Question

I was implementing the algorithm by Manuel Bronstein for solving the Risch differential equation.

My question is:

What does Bronstein mean by "Order" in the algorithm poly_DE (exponential case, page 56)?

Thanks everyone for helping

Answer 1

My guess would be that if you expand $Q$ as a Laurent series about $0$:

$$ Q(x)=\sum_{n=-\infty}^\infty \frac{a_n}{n!}x^n $$

then "a lower bound on the order at $0$ of $Q$", $b$, is defined such that there is guaranteed to be a $c\geq b$ such that either $a_c\neq 0$ or $a_{-c}\neq 0$ or both.

So it is, in a sense, the opposite of the "order of vanishing" of a function.