Puiseux series- This field is denoted by . Note that we have a double brace “[[ ]]” instead of “[]”. This implies that we have infinite series instead of finite ones (which would be polynomials). The Puiseux laurent series is denoted as
. This means that
is also allowed to have negative powers. Now
, which just means that
contains all rational powers of
now, and not just integral powers, as in the Laurent series. We seem to be generalizing in every successive step.
Now we define a valuation: , for
. So what we’ve essentially done is that we’ve written all rational powers (including integral ones) as fractions with denominator
. Clearly, amongst all the denominators in the rational powers of
,
has to be the largest denominator.
If the Puiseux series converges, then we have
Why is that? It seems to me that would give us the sum of all rational powers of
, which could possibly be infinite. Then why do we just get the lowest one?
Now for , set Trop
to be
. Take
for example. One should think about this variety as
instead, where
and
are power series in
(with rational powers). Then there are three possibilities:
i) and
.
ii) and
iii) .
These cases can be easily deduced to contain all possibilities. For instance, if , then
too. This is because
implies that
has negative powers of
. This implies that
too has to have negative powers of
, as
. When one of them contains strictly positive powers, the other has to contain
and no negative powers of
, which implies that if
, then
.