70
G. Gubbiotti
Then we consider the following limit:
ε = lim
k→∞
1
k
log d k ,
(9)
called the algebraic entropy. If the growth of the map ϕ is sub-exponential, then the
algebraic entropy (9) vanishes and we say the map ϕ is integrable in the sense of
the algebraic entropy [5, 13, 42]. As a particular case, when the growth of a map is
linear the map is believed to be linearizable [25].
Algebraic entropy is an invariant of bi-rational maps, meaning that its value is
unchanged up to bi-rational equivalence. Moreover, its value is determined by the
singularity structure of a map [5, 34, 43].
To compute the algebraic entropy from (9) we need to know the asymptotic
behaviour of the sequence d n . For the majority of applications such behaviour can
be inferred by using a generating function [27], that is, a function g = g (z) such
that:
g (z) =
∞
n=0
d k z
k .
(10)
A generating function is a predictive tool which can be used to test the successive
members of a finite sequence. It follows that the algebraic entropy is given by the
logarithm of the smallest pole of the generating function, see [17, 18].
Remark 1 Finding invariants is a hard task. Here we recall briefly a method for
finding invariants of bi-rational maps presented first in [13] and recently reprised in
[8]. If the ratio P /Q is an invariant of a map ϕ, then the pullback of ϕ on P /Q is
invariant: ϕ ∗ (P /Q) = P /Q. This implies
ϕ
∗ (P ) = aP and ϕ
∗ (Q) = aQ
(11)
for some polynomial factor a. Using the fact that ψ ◦ ϕ = κ Id where κ is a
polynomial one gets that a must contain some of the factors dividing κ. Hence one
can search for invariants imposing the form of P , then searching for the appropriate
factors. We get an invariant when we obtain more than one solution for the same a.
By taking ratios of the solutions we obtain the invariants.
The problem with this method is that it is not bounded as we do not know a priori
the degree of P . However, in practice this method is quite useful for the explicit
computation of the invariants, since the conditions in (11) are linear, even though
their number can become huge as deg(P ) grows.
G. Gubbiotti
Then we consider the following limit:
ε = lim
k→∞
1
k
log d k ,
(9)
called the algebraic entropy. If the growth of the map ϕ is sub-exponential, then the
algebraic entropy (9) vanishes and we say the map ϕ is integrable in the sense of
the algebraic entropy [5, 13, 42]. As a particular case, when the growth of a map is
linear the map is believed to be linearizable [25].
Algebraic entropy is an invariant of bi-rational maps, meaning that its value is
unchanged up to bi-rational equivalence. Moreover, its value is determined by the
singularity structure of a map [5, 34, 43].
To compute the algebraic entropy from (9) we need to know the asymptotic
behaviour of the sequence d n . For the majority of applications such behaviour can
be inferred by using a generating function [27], that is, a function g = g (z) such
that:
g (z) =
∞
n=0
d k z
k .
(10)
A generating function is a predictive tool which can be used to test the successive
members of a finite sequence. It follows that the algebraic entropy is given by the
logarithm of the smallest pole of the generating function, see [17, 18].
Remark 1 Finding invariants is a hard task. Here we recall briefly a method for
finding invariants of bi-rational maps presented first in [13] and recently reprised in
[8]. If the ratio P /Q is an invariant of a map ϕ, then the pullback of ϕ on P /Q is
invariant: ϕ ∗ (P /Q) = P /Q. This implies
ϕ
∗ (P ) = aP and ϕ
∗ (Q) = aQ
(11)
for some polynomial factor a. Using the fact that ψ ◦ ϕ = κ Id where κ is a
polynomial one gets that a must contain some of the factors dividing κ. Hence one
can search for invariants imposing the form of P , then searching for the appropriate
factors. We get an invariant when we obtain more than one solution for the same a.
By taking ratios of the solutions we obtain the invariants.
The problem with this method is that it is not bounded as we do not know a priori
the degree of P . However, in practice this method is quite useful for the explicit
computation of the invariants, since the conditions in (11) are linear, even though
their number can become huge as deg(P ) grows.