For one thing, I suppose that the Order(x) function is missing a power evaluation of the kind Order(x)^e -> Order(x^e). Does this sound right?
Thanks for looking into this. The substitution above is not correct if e<0. For example: x = Order(1/A) => |x| <= Const * 1/A but x = 1/Order(A) => |x| >= 1 / (Const * A) so Order(1/A) =/= 1/Order(A) It is correct for non-negative integer e though.
The attached patch adds this. It solves some of your problems but not all. It doesn't seem to introduce regressions. Can you test this, please?
I guess ginsh> series((x+x^2)^2,x,0); Order(x^(-2)) is preferable to (Order(1)^2)*x^(-2)+Order(1) but it is still incorrect and not that helpful: I'm asking for the expansion up to x^0, not x^-2.
But there's something else fishy going on in pseries::power_const(p, deg).
Seems so. * * * On a bit of a tangential topic: I think the fact that Order(x^0) is simplified to Order(1) is a bad design choice, because the variable in which the expansion was made is lost. This leads to all kinds of special cases if one wants to work with series; e.g. Order(x^n)*x can be simplified to Order(x^(n+1)), but Order(1)*x can no longer be Order(x). Similarly, Order(x) + Order(x^2) = Order(x), but Order(1) + Order(x) =/= Order(1), because Order(1) could have been Order(othervar^0). I'd much rather have Order() to have the variable and the exponent separately, i.e. Order(x, 0) instead of Order(1). This is a separate question though.