Symmetry, Integrability and Geometry: Methods and Applications SIGMA 11 (2015), 056, 36 pages
From Polygons to Ultradiscrete Painleve´ Equations
Christopher Michael ORMEROD † and Yasuhiko YAMADA ‡
† Department of Mathematics, California Institute of Technology,
1200 E California Blvd, Pasadena, CA, 91125, USA
E-mail: cormerod@caltech.edu
‡ Department of Mathematics, Kobe University, Rokko, 657–8501, Japan
E-mail: yamaday@math.kobe-u.ac.jp
Received January 29, 2015, in final form July 10, 2015; Published online July 23, 2015
http://dx.doi.org/10.3842/SIGMA.2015.056
Abstract. The rays of tropical genus one curves are constrained in a way that defines
a bounded polygon. When we relax this constraint, the resulting curves do not close,
giving rise to a system of spiraling polygons. The piecewise linear transformations that
preserve the forms of those rays form tropical rational presentations of groups of affine
Weyl type. We present a selection of spiraling polygons with three to eleven sides whose
groups of piecewise linear transformations coincide with the Ba¨cklund transformations and
the evolution equations for the ultradiscrete Painleve´ equations.
Key words: ultradiscrete; tropical; Painleve´; QRT; Cremona
2010 Mathematics Subject Classification: 14T05; 14H70; 39A13
1 Introduction
A significant contribution to our understanding of the Painleve´ equations, both discrete and
continuous, has been their characterization in terms of their rational surfaces of initial conditions
[33, 47]. These works related the symmetries of the Painleve´ equations to Cremona isometries
of rational surfaces [24, 27, 28], which are groups of affine Weyl type [6, 7, 17]. This provided
a geometric setting for many previous studies that were based purely on the symmetries of the
Painleve´ equations [19, 20, 31]. In the autonomous limit, the Painleve´ equations degenerate to
elliptic equations or QRT maps [40, 41] and their associated surfaces of initial conditions are
rational elliptic surfaces [8, 53].
Given a subtraction free discrete Painleve´ equation, one may obtain an ultradiscrete Painleve´
equation by applying the ultradiscretization procedure [52]. The ultradiscretization procedure
famously related integrable difference equations with integrable cellular automata [49, 51, 52],
hence, the process is thought to preserve integrability [21, 43]. The ultradiscrete Painleve´
equations are second order non-linear difference equations defined over the max-plus semifield
that are integrable in the sense that they possess many of same properties of the continuous
and discrete Painleve´ equations that are associated with integrability, albeit, in some tropical
form. These properties include tropical Lax representations [15, 35] and tropical singularity
confinement [14, 36]. They also admit symmetry groups of affine Weyl type [18, 19] and special
solutions of rational and hypergeometric type [26, 34, 50]. The ultradiscrete QRT maps may
also be obtained as autonomous limits of the ultradiscrete Painleve´ equations [29, 39].
The ultradiscrete QRT maps preserve a pencil of curves arising as the level sets of tropical
biquadratic functions [29, 39]. Since every non-degenerate level set of a tropical biquadratic
function is a tropical genus one curve, one may say that the ultradiscrete QRT maps can be
lifted to automorphisms of tropical elliptic surfaces. Given the geometric interpretation of tro-
pical singularity confinement [36], the positions of the rays in any pencil of tropical genus one
curves play the same role as the positions of the base points in a pencil of genus one curves. In
2 C.M. Ormerod and Y. Yamada
Figure 1. A fibration of closed tropical curves (left) corresponds to ultradiscrete QRT maps. Breaking
this closure condition results in spiraling polygons (right), which corresponds to ultradiscrete Painleve´
equations.
this way, there is an analogous constraint on the positions of the rays of any pencil of tropical
genus one curves, which when removed, results in curves that are no longer closed. We refer to
the resulting set of piecewise linear curves as spiraling polygons, which are depicted in Fig. 1.
This situation mimics the generalization of elliptic surfaces to surfaces of initial conditions for
discrete Painleve´ equations.
This article is concerned with groups of piecewise linear transformations of the plane which
preserve the forms of the spiraling polygons. We specify a selection spiraling polygons with be-
tween three and eleven sides whose groups of transformations form representations of affine Weyl
groups with types that coincide with those of the Ba¨cklund transformations for the multiplica-
tive Painleve´ equations [47]. The piecewise linear transformations corresponding to translations
in the affine Weyl group are shown to be ultradiscrete Painleve´ equations. A list of the cor-
respondences between polygons, symmetry groups and ultradiscrete Painleve´ equations, along
with where these systems first appeared, is provided in Table 1. This work provides a geometric
interpretation for the group of Ba¨cklund transformations of the ultradiscrete QRT maps and
ultradiscrete Painleve´ equations.
Our construction replicates the ultradiscretization of known subtraction-free affine Weyl
representations in the unpublished work of Kajiwara et al. [18], however, our derivation does
not use or require the ultradiscretization procedure. Finding generators for the representations
is reduced to combinatorial properties of the underlying polygons. By considering genus one
tropical plane cubic, quartic and sextic curves, we treat polygons with up to eleven sides. We
mention that the case of octagons arising as level sets of tropical biquadratic functions also
appeared in this context in the work of Rojas [46], Nobe [29] and Scully [48], as do a very small
collection of the symmetries we list in [46].
We set out this paper as follows: we first briefly review a geometric setting for QRT maps and
the discrete Painleve´ equations in Section 2, then we review the ultradiscretization procedure
with some relevant tools from tropical geometry in Section 3. A description of the canonical
classes of transformations that preserve given spiral structures is presented in Section 4, which
we use in Section 5 to give explicit presentations of the piecewise linear transformations that
may be used to construct the ultradiscrete Painleve´ equations. We have a brief discussion of the
difficulties in extending this to polygons with greater than eleven sides in Section 6.
From Polygons to Ultradiscrete Painleve´ Equations 3
Table 1. A labelling of the various polygons and the affine Weyl groups of symmetries they possess.
The references refer to the first known appearence of the ultradiscrete Painleve´ equation in the literature.
Sides Polygon Affine Weyl group Painleve´ equation
3 Triangle A(1)0
4 Quadrilateral A(1)1 , A
(1)
1 +D8 u-PI, u-P
′
I [43]
5 Pentagon (A1 +A1)
(1) u-PII [43]
6 Hexagon (A2 +A1)
(1) u-PIII/ u-PIV [19]
7 Heptagon A(1)4 u-PV [43]
8 Octagon D(1)5 u-PVI [43]
9 Enneagon E(1)6 u-P
(
A(1)2
)
[18]
10 Decagon E(1)7 u-P
(
A(1)1
)
[18]
11 Undecagon E(1)8 u-P
(
A(1)∗0
)
[18]
2 The geometry of QRT maps and discrete Painleve´ equations
The QRT maps are integrable second order autonomous difference equations [40, 41]. They are
Lax integrable, measure preserving and possess the singularity confinement property. The QRT
maps may broadly be considered discrete analogue of elliptic equations [53]. To construct a QRT
map, one takes two linearly independent biquadratics, h0(x, y) and h1(x, y), and a generic point,
p = (x, y), to which we associate an element, z = [z0 : z1] ∈ P1, by the relation
z0h0(x, y) + z1h1(x, y) = 0. (2.1)
That is to say that h0(x, y) and h1(x, y) define a pencil of biquadratic curves. If we let h(x, y) =
h0(x, y)/h1(x, y), then the QRT map, φ : (x, y)→ (x˜, y˜), is defined by the condition that x˜ and y˜
are related to x and y by
h(x, y) = h(x, y˜), (2.2a)
h(x, y˜) = h(x˜, y˜), (2.2b)
where the trivial solutions, x = x˜ and y = y˜, are discarded [40, 41]. In this way, the map is an
endomorphism of the curve defined by (2.1) for each value of z.
If we take a point in the intersection of the curves h0(x, y) = 0 and h1(x, y) = 0, then z0 and z1
may be chosen arbitrarily, hence, an entire pencil of curves intersect at these points. These points
are called base-points and the number of base points for any pencil of biquadratics is 8, counting
multiplicities. A case in which there are eight distinct base points in R2 is depicted in Fig. 2.
By blowing up these base points, possibly multiple times in the case of higher multiplicities,
we obtain a surface admitting a fibration by smooth biquadratic curves (i.e., elliptic curves).
Lifting the QRT map to this surface gives an automorphism of an elliptic surface [8, 53].
A classic example is the QRT map defined by the invariant
h(x, y) =
y
a3
+
y
a4
+
(a1 + a2)b1b2
ya1a2
+
(y + b1)(y + b2)
xy
+
x(y + b3)(y + b4)
ya3a4
, (2.3)
where we require the condition
a1a2b3b4 = b1b2a3a4. (2.4)
The map, (x, y)→ (x˜, y˜), is specified by relations
x˜x =
a3a4(y˜ + b1)(y˜ + b2)
(y˜ + b3)(y˜ + b4)
, (2.5a)
4 C.M. Ormerod and Y. Yamada
Figure 2. A collection of elements of the pencil of biquadratic curves with eight distinct base-points
in R2.
y˜y =
b3b4(x+ a1)(x+ a2)
(x+ a3)(x+ a4)
. (2.5b)
The base points of (2.5) lie on the lines x, y = 0,∞ in P21. The blow-up at these points, with (2.4)
as a constraint, is an elliptic surface [8].
The discrete Painleve´ equations are integrable second order difference equations that admit
the continuous Painleve´ equations as a continuum limit [42] and QRT maps in an autonomous
limit. The discrete Painleve´ equations and QRT maps are integrable by many of the same crite-
ria; Lax integrability [13, 37], vanishing algebraic entropy [2] and singularity confinement [42].
One way to obtain a non-autonomous second order difference equation from a QRT map
is by assuming the parameters vary in a manner that preserves the singularity confinement
property [42]. Given the autonomous system defined by (2.5), we may deautonomize to the
system to obtain the nonlinear q-difference equation
y˜y =
b3b4(x+ a1t)(x+ a2t)
(x+ a3)(x+ a4)
, (2.6a)
x˜x =
a3a4(y˜ + qb1t)(y˜ + qb2t)
(y˜ + b3)(y˜ + b4)
, (2.6b)
where x = x(t), y = y(t), x˜ = x(qt) and y˜ = y(qt). If we think of this as a difference equation
for y = yn and x = xn, with independent parameter, n, this is equivalent to n appearing in an
exponent as t = t0qn. The parameter q ∈ C \ {0} is a constant defined by the relation
q =
a1a2b3b4
b1b2a3a4
. (2.7)
This system was first derived as a connection preserving deformation [13]. While these are often
thought of as nonlinear q-difference equations in t, from the viewpoint of symmetries, it is more
conducive to think of (2.6) as a map
φ :
(
a1, a2, a3, a4
b1, b2, b3, b4
;x, y
)
→
(
qa1, qa2, a3, a4
qb1, qb2, b3, b4
; x˜, y˜
)
, (2.8)
From Polygons to Ultradiscrete Painleve´ Equations 5
y = 0
y =∞
x = 0 x =∞
Figure 3. The positions of the blow-up points for (2.5) and (2.6).
where x˜ and y˜ are related by (2.6) and we absorb t into the definitions of a1, a2, b1 and b2
(equivalent to setting t = 1 in (2.6)). When we blow up the eight points, P = {p1, . . . , p8} ⊂ P21,
given by
p1 = (−a1, 0), p2 = (−a2, 0), p3 = (−a3,∞), p4 = (−a4,∞),
p5 = (0,−b1), p6 = (0,−b2), p7 = (∞,−b3), p8 = (∞,−b4),
the resulting surface, XP , has been called a generalized Halphen surface [47]. Lifting the map
defined by (2.6) is not an automorphism of XP , but rather an isomorphism, ϕ : XP → XP˜ ,
where P˜ is the set of points defined by the image of (2.8). This map is bijective for the same
reasons as for the QRT case. In the autonomous limit as q = 1, (2.7) coincides with (2.4), P = P˜
and ϕ is an automorphism of an elliptic surface that coincides with the lift of (2.5).
In the same way as (2.5), the blow-up points for (2.6) lie on the lines x, y = 0,∞, as shown
in Fig. 3. We can identify the affine coordinates, x and y, with projective coordinates, [x0 : x1]
and [y0 : y1], via the relations x = x1/x0 and y = y1/y0 in which the points, P , lie on the
decomposable curve defined by x0x1y0y1 = 0.
If we were to follow up the construction of the surface, one notices that if we were to inter-
change the blow-up points, we obtain a surface that is isomorphic. We notice that the blow-up
co-ordinates, (z10 : z
1
1) and (z
3
0 : z
3
1), for the points, p1 and p3 respectively, satisfy the relations
z11(x+ a1) = z
1
0y, z
3
1(x+ a3) =
z30
y
,
then if we define the transformation (x, y)→ (xˆ, yˆ), by
xˆ = x, yˆ = y
x+ a3
x+ a1
,
then the blow-up co-ordinates in xˆ and yˆ satisfy the relations
z11(xˆ+ a3) = z
1
0 yˆ, z
3
1(xˆ+ a1) =
z30
yˆ
.
This transformation also has a scaling effect on the positions of p5 and p6.
(
a1, a2, a3, a4
b1, b2, b3, b4
;x, y
)
→
(
a3, a2, a1, a4
b1
a3
a1
, b2
a3
a1
, b3, b4
; xˆ, yˆ
)
. (2.9)
Both the constraint, (2.4), and the variable q, defined by (2.7), remain valid on the new surface,
hence, the transformation (x, y)→ (xˆ, yˆ) may be lifted to an isomorphism of surfaces.
6 C.M. Ormerod and Y. Yamada
Let σi,j denote the isomorphism identifying the surfaces in which the blowups at points pi
and pj are interchanged, then we have a natural set of elements, w0 = σ7,8, w1 = σ5,6, w4 = σ1,2
and w5 = σ3,4. We label the transformation from (2.9) by w3 and the corresponding operation
using points p5 and p7 by w2. These transformations and two natural symmetries, ρ1 and ρ2,
form a representation of an affine Weyl group of type D(1)5 (see [47, Section 2] for more details).
Furthermore, as an infinite order isomorphism, both (2.5) and (2.6) may be represented as
a product of these involutions as
T = ρ2 ◦ w2 ◦ w0 ◦ w1 ◦ w2 ◦ ρ1 ◦ w3 ◦ w5 ◦ w4 ◦ w3.
In many cases, such birational representations were studied independently.
While we have been considering biquadratics over P21, we may extend these arguments to
plane curves in P2 via the birational map, pi : P21 → P2, defined by
pi : ([x0 : x1], [y0 : y1]) = [x0y0 : x1y0 : x0y1],
which is not defined when x0 = y0 = 0 (corresponding to (∞,∞)). The inverse,
pi−1([u0 : u1 : u2]) = ([u0 : u1], [u0 : u2]),
is not defined at [0 : 0 : 1] and [0 : 1 : 0]. These maps are isomorphisms when restricted to the
copies of C2 defined by x0 = y0 = 1 and u0 = 1 respectively (or more precisely, x0, y0 and u0
are not 0). Any biquadratic curve,
b(x, y) =
∑
bi,jx
i
0x
2−i
1 y
j
0y
2−j
1 = 0,
going through (∞,∞) (i.e., b2,2 = 0) is mapped, via pi, to a cubic plane curve
c(u) =
∑
0≤i,j≤2,i+j>0
ci,ju
i+j−1
0 u
2−i
1 u
2−j
2 ,
which goes through [0 : 0 : 1] and [0 : 1 : 0]. In this way, our two generating biquadratics, h0
and h1 from (2.1), map to two cubic planar curves which generally intersect at 9 points (also
constrained). In this way, we can naturally pass from a pencil of biquadratics on P21, which is
resolved by blowing up eight points to a pencil of cubic plane curves, and a surface obtained by
blowing up P2 at nine points.
In passing from the QRT maps to discrete Painleve´ equations via singularity confinement,
where the base points are allowed to move, the resulting systems are one of three types of
nonautonomous difference equations; h-difference, q-difference or elliptic difference equations.
The points can still lie in non-generic positions, but the additional constraint associated with
the QRT maps is relaxed. The positions and multiplicities of these nine points determine the
symmetries of the surface and of the equation. All the equations admitting ultradiscretization
(or tropicalization) are special cases of q-difference equations, where all the parameters are
assumed to be positive. The class of surfaces giving rise to q-difference equations was studied
by Looijenga [24].
When the nine points are in any non-generic position and appear with different multiplicities,
one can not interchange blow-up points in any ad-hoc manner. For example, in the case of (2.5),
the points lie on four distinct lines with an intersection form of type A(1)3 , and the positions of
those points are subject to the constraint (2.4). The type of surface is characterized by this
intersection form, and we may only interchange blow-up points in a way that preserves the
intersection form. In this way we obtain two root systems, one describing the symmetry group
of the equation, the other describing the surface type. A degeneration diagram which lists the
surface type and the symmetries of the corresponding q-Painleve´ equations is given in Fig. 4.
From Polygons to Ultradiscrete Painleve´ Equations 7
E(1)8
A(1)0
E(1)7
A(1)1
E(1)6
A(1)2
D(1)5
A(1)3
A(1)4
A(1)4
(A2 +A1)(1)
A(1)5
(A1 +A1)(1)
A(1)6
A(1)1
A(1)7
A(1)0
A(1)8
A˜(1)1
A(1)7
Figure 4. The coalescence diagram for q/u-Painleve´ equations. The symmetry of the equation appears
on top and the surface type appears below.
By identifying the Picard lattices of isomorphic surfaces, we have an alternative interpretation
of these maps and their symmetries [47]. From the theory of rational surfaces (as blow-ups of
the minimal surfaces Σ0 = P21 or Σ1 = P2), we have the isomorphism Pic(X) = H
1(X,O∗) ∼=
H2(X,Z), with an endowed intersection form [27, 28]. The interchange of blow-up and blow-
down structures [1] preserves this intersection form and leaves the canonical class fixed [24], so
we may interpret these as reflections in Pic(X). This defines a group of Cremona isometries,
which are of affine Weyl type. The work of Sakai extended [24] and realized the action of the
translational Cremona isometries as discrete Painleve´ equations [47].
3 Tropicalization
Tropicalization can be thought of as the pointwise application of a nonarchimedean valuation
to geometric structures. Tropicalization sends curves to lines, surfaces to polygons and more
generally, smooth structures to piecewise linear ones [3, 45]. In the integrable community a non-
analytic limit known as ultradiscretization is used as a way of obtaining new and interesting
piecewise linear integrable systems [52]. Relating tropicalization with ultradiscretization gives
us a way of understanding the geometry of ultradiscrete systems [36].
Let us first consider the ultradiscretization procedure as it was originally considered in [52].
Given a subtraction free rational function in a number of strictly positive variables, f(x1, . . . , xn),
we introduce ultradiscrete variables, X1, . . . , Xn, related by xi = eXi/. The ultradiscretization
of f , denoted F , is obtained by the limit
F (X1, . . . , Xn) := lim
→0+
 ln f(x1, . . . , xn). (3.1)
The subtraction free nature of the function is required so that we need not consider the logarithm
of a negative number. Roughly speaking, the ultradiscretization procedure replaces variables
and binary operations as follows:
x1x2 → X1 +X2, x1 + x2 → max(X1, X2), x1/x2 → X1 −X2,
where there is no (natural) replacement of subtraction.
Given a difference equation, such as (2.6), we may apply the ultradiscretization procedure to
obtain a system known as u-PVI [43], given by
X + X˜ = A3 +A4 + max(Q+ T +B1, Y˜ ) + max(Q+ T +B2, Y˜ )
−max(B3, Y˜ )−max(B4, Y˜ ), (3.2a)
Y + Y˜ = B3 +B4 + max(A1 + T, X˜) + max(A2 + T,X)
−max(B3, X)−max(B4, X), (3.2b)
8 C.M. Ormerod and Y. Yamada
where the variable Q is specified by the relation
Q = A1 +A2 −A3 −A4 −B1 −B2 +B3 +B4.
A special case of this system was shown to arise as an ultradiscrete connection preserving
deformation [35]. In the same way as (2.6), we may think of this as a map
Φ:
(
A1, A2, A3, A4
B1, B2, B3, B4
;X,Y
)
→
(
Q+A1, Q+A2, A3, A4
Q+B1, Q+B2, B3, B4
; X˜, Y˜
)
.
In the autonomous limit, when we let Q = 0, the above ultradiscrete Painleve´ equation becomes
an ultradiscrete QRT map (i.e., the ultradiscretization of (2.5)), which was introduced in [39]
and studied from a tropical geometric viewpoint by Nobe [29]. The ultradiscretization of (2.3)
gives the following piecewise linear function
H(X,Y ) = max
(
Y −A3, Y −A4, B1 +B2 max(−A1,−A2)− Y,
max(Y,B1) + max(Y,B2)−X − Y,
X − Y + max(Y,B3) + max(Y,B4)−A3 −A4
)
, (3.3)
which is also an invariant of the ultradiscrete QRT map, i.e., H(X,Y ) = H(X˜, Y˜ ) [29]. Further-
more, the evolution of the ultradiscrete QRT map defines a linear evolution on the Jacobian of
the invariant, hence, the ultradiscrete QRT map may be expressed in terms of the addition law
on a tropical elliptic curve [5, 29].
While we may be able to solve (2.2) in a subtraction free manner, given an invariant such
as (3.3), the equation H(X,Y ) = H(X˜, Y˜ ) involves a max on both the left and right, hence,
cannot generally be solved within the limited framework of tropical arithmetic. Our approach
is different in that we only consider transformations that preserve the structure of the tropical
curves of the form H(X,Y ) = H0 where H0 is some constant. Any automorphism of tropical
curves of this form can be expressed in terms of compositions of more fundamental operations.
We need to consider these curves more carefully, hence, we will briefly review some tropical
geometry [45].
The discrete dynamical system, (3.2), is most naturally defined over a tropical semifield [38],
more precisely, the max-plus semifield, which is the set T = R∪{−∞}, equipped with the binary
operations
X1 ⊕X2 := max(X1, X2), X1 ⊗X2 := X1 +X2,
which are known as tropical addition and tropical multiplication respectively. The element −∞
plays the role of the tropical additive identity and 0 plays the role of the tropical multiplicative
identity [38].
The geometry of objects over the tropical semifields is the subject of tropical geometry [45].
A tropical polynomial, F ∈ T[X1, . . . , Xn] defines a piecewise linear function from Tn → T,
given by
F (X1, . . . , Xn) = max
j
(Cj +Aj,1X1 + · · ·+Aj,nXn), (3.4)
where {Aj,i} is a set of integers and {Cj} is a set of elements of T. The tropical variety associated
with F ∈ T[X1, . . . , Xn], denoted V(F ), is defined to be
V(F ) =
{
X = (X1, . . . , Xn) ∈ Tn such that F is not differentiable at X
}
,
which occurs precisely when one argument of the max-expression becomes dominant over another
argument [45].
From Polygons to Ultradiscrete Painleve´ Equations 9
Another equivalent algebraic characterization of tropical varieties relies on nonarchimedean
valuations. Every non-zero algebraic function, f ∈ C(t), admits a representation as a Puiseux
series,
f(t) = c1t
q1 + c2t
q2 + · · · ,
where c1 6= 0 and {qi} are rational and ordered such that qi < qi+1. The function, ν : C(t)→ T,
given by
ν(f) = −q1,
is a nonarchimedean valuation. This may be extended to an algebraically and topologically closed
field with a valuation ring of R, which we simply denote K = C(t) [25]. If I ⊂ K[x±11 , . . . , x
±1
n ]
is an ideal, then we define V (I) ⊂ Kn as
V (I) = {(x1, . . . , xn) : f(x1, . . . , xn) = 0 for all f ∈ I}.
The tropical variety associated with I is the topological closure of the point-wise application of ν
to V (I), i.e., V(I) = ν(V (I)) ⊂ Tn. For every tropical variety V(F ), there exists a function, f ,
such that V(F ) = V(〈f〉) where 〈f〉 ⊂ K[x±11 , . . . , x
±1
n ] denotes the ideal generated by f . This
means that we may define a tropical variety in terms of either piecewise linear functions or
ideals of K[x±11 , . . . , x
±1
n ]. The equivalence of the set of points of non-differentiability and the
image of the valuations is outlined in [45]. Each tropical curve is a collection vertices, finite line
segments, called edges, and a collection of semi-infinite line segments, called rays.
In the same way as affine n-space may be considered to be embedded in projective space,
we may naturally consider Tn as being embedded in tropical projective space. Define the
equivalence relation, ∼, on Tn+1 so that
V ∼ U if and only if V = U + λ(1, 1, . . . , 1),
for some λ, then tropical projective n-space is the set
TPn = Tn+1/ ∼ .
A tropical function of the form (3.4) is said to be homogeneous if there exists a d such that for
every j
∑
i
Aj,i = d.
The set of non-differentiable points of a tropically homogeneous polynomial defines a tropical
projective variety.
Given a rational function in a number of variables, f(x1, . . . , xn), we can lift the function up
to the field of algebraic functions by letting xi = tXi for some Xi, then the ultradiscretization
procedure is known to coincide with
F (X1, . . . , Xn) = ν(f(x1, . . . , xn)), (3.5)
for all subtraction free functions [34, 36]. The above extension, given by (3.5), is one of
a number of ways to incorporate a version of subtraction into the ultradiscretization procedure
[12, 22, 23, 32].
The most immediate consequence from the viewpoint of the geometry is that singularities of
a map manifest themselves as points of non-differentiability [3, 36, 45]. This interpretation was
10 C.M. Ormerod and Y. Yamada
Figure 5. A tropical biquadratic with the rays labeled in red.
also present in the work of Joshi and Lafortune who elucidated what the analogue of singularity
confinement should be for tropical integrable difference equations [14].
One of the characteristic features of the QRT map is that the invariant curves all intersect at
the base points. From looking at the invariant curves of (2.5), depicted in Fig. 1, this feature is
not apparent in the tropical setting. When we consider the extension of the ultradiscretization
via (3.5), another way of looking at the invariant is that the level set is a subset of the tropical
variety associated the ideal
IH0 =
〈
h(x, y)− tH0
〉
,
in K[x, y], which is the set
V(IH0) := ν(V (IH0)). (3.6)
For each x = tX where X ∈ Q, the equation
h
(
tX , y
)
− tH0 = 0,
is quadratic in y, and as K is algebraically closed, we have two algebraic solutions, y1 and y2
over K. That is for each X, we obtain values Y1 = ν(y1) and Y2 = ν(y2) in T, which form
infinite rays (also called tentacles in [5, 29]). These form points of V(IH0) that do not appear
in the level set of H(X,Y ). Notice that each of the rays intersect on the lines at X = ±∞ and
Y = ±∞, and positions of the rays define where on that line they intersect. The inclusion of the
rays to the level sets, as seen in Fig. 5, makes them smooth tropical curves in the sense of [45].
We may extend these tropical biquadratics to TP21 by using homogeneous co-ordinates X =
[X0 : X1] and Y = [Y0 : Y1]. The maps pi and pi−1 possess tropical analogues, Π: TP21 → TP2
and Π−1 : TP2 → TP21, given by
Π: ([X0, X1], [Y0, Y1])→ [X0 + Y0 : X1 + Y0 : X0 + Y1],
Π−1 : [U0 : U1 : U2]→ ([U0 : U1], [U0 : U2]).
These are isomorphisms between the copies of T2 specified by X0 = Y0 = 0 and U0 = 0
respectively. The map Π is not defined when X0 = Y0 = −∞ and the inverse is not defined at
[−∞ : 0 : −∞] and [−∞ : −∞ : 0]. The level set of a tropical biquadratic function
H(X,Y ) = max
i,j=0,1,2
(
Bi,j + iX0 + (2− i)X1 + jY0 + (2− j)Y1
)
,
From Polygons to Ultradiscrete Painleve´ Equations 11
Figure 6. A tropical cubic plane curve with rays labeled in red.
in which B2,2 = −∞ maps via Π to a tropical cubic plane curve, specified by the level set of
some cubic,
H(U) = max
0≤i,j≤2, i+j>0
(
Ci,j + (i+ j − 1)U0 + (2− i)U1 + (2− j)U2
)
.
Since Π maps the rays and edges over TP21 to rays and edges in TP2, we expect the image of the
level set of a biquadratic to be at most an octagon, however, the most general cubic plane curve
is an enneagon. If one considers the enneagon as the image of the variety over K[x, y, z], one
recovers nine rays counting multiplicities. The case of nine distinct rays is depicted in Fig. 6.
In this way, the information we have on rays in P21 applies equally well to the rays in TP2.
As the rays define the positions of the vertices of each polygon, they will play an important
role in the description of the symmetries. In Figs. 5 and 6, all the rays are asymptotic to one of
three forms;
Li : X −Ai = 0, Lj : Y −Aj = 0, Lk : Y −X −Ak = 0.
Since the rays in Figs. 5 and 6 are part of every variety of the form (3.6), this is equivalent to
each variety intersecting in TP2 at points
[Ai : −∞ : 0], [−∞ : Aj : 0], [0 : −∞ : Ak],
respectively. For the level set to close, there is a constraint on the positions of the rays, which
when relaxed gives a spiral diagram. For smooth biquadratics, we obtain spiraling octagons (see
Fig. 1). In the smooth cubic case we obtain spiraling enneagons (see Fig. 19). Given a polygon
arising as a tropical curve, there are two types of degenerations:
• We may make two parallel rays coincide.
• We may take two rays that are not parallel and merge them.
The latter corresponds to setting a coefficient of H(X,Y ) to −∞.
This construction may be generalized to tropical genus one curves of higher degrees, which
allows us to consider decagons and undecagons as level sets of tropical quartic and tropical
sextic plane curves respectively. In these cases, one finds twelve and thirteen rays, counting
multiplicities (when rays coincide). The decagon used will be a tropical quartic with four rays
of order one of the form Li : X −Ai, four rays of order one of the form Lj : Y −Aj and two rays
of order two of the form Lk : Y −X −Ak = 0. This would be the ultradiscretization of a curve
12 C.M. Ormerod and Y. Yamada
of degree four with eight singularities of order one and two of order two, which gives a genus of
one curve by the degree-genus formula,
g =
(d− 1)(d− 2)
2
−
∑
k
rk(rk − 1)
2
, (3.7)
where d is the degree of the curve and the ri is the order of the k-th singularity. In a similar
way, our undecagon is a the ultradiscretization of a genus one curve of degree six curve with
six rays of order one, two of order three and three of order two. This formula remains valid for
tropical varieties [9].
4 Piecewise linear transformations of polygons and spirals
Cremona transformations of the plane, and their subgroups, are a topic of classical and modern
interest [6, 10, 17]. The classical result of Noether [30] (see also [10]) states that Cremona
transformations are generated by the quadratic transformations, the simplest being the standard
Cremona transformation
τ : [x : y : z]→ [yz : xz : xy],
which may be interpreted as the blow-up of the points [1 : 0 : 0], [0 : 1 : 0] and [0 : 0 : 1]
combined with a blow-down on the co-ordinate lines given by xyz = 0. In a similar vein, our
aim is to specify a generating set of tropical Cremona transformations from which all the other
transformations may be obtained. Our aim is to specify subgroups of these that preserve a given
spiral diagram.
To specify any spiral diagram, we begin with a parameterization of the asymptotic form of
the rays,
X = {Li where Li : aiX + biY + ci = 0, and ai, bi, ci ∈ Z}.
The shape of the spirals are determined by the invariants obtained in the autonomous limit. We
seek a group of transformations that preserve the forms of these rays, more specifically, we seek
transformations, σ, such that
1) σ is a bijection of the plane;
2) for every ray, Lj , there is a ray, Li, such that σ : Li = L˜j , where L˜j differs only by some
translation.
These may be thought of as tropical Cremona isometries, as these conditions replicate conditions
that require the canonical class and intersection form of the surface be fixed.
Since the Cremona isometries are products of the interchange of blow-up and blow-down
structures [1], and the positions of these blow-up points are encoded in the positions of the rays,
it is sufficient to consider the shearing transformations that create and smooth out polygons
whose vertices lie along these rays. Analagously to the results of Noether [10], we propose the
following two generators:
ιA : (X,Y )→ (X,Y + max(0, X −A)), (4.1)
Ξ: (X,Y )→ (aX + cY, bX + dY ), (4.2)
where |ad− bc| = 1 and A ∈ T. The action of ιA can be seen as an analogous to the interchange
of blow-ups in the following way: if the vertices of the level sets of a polygon trace out the
rays, then ιA can smooth out all the vertices along a ray asymptotic to, L : X = A, while
From Polygons to Ultradiscrete Painleve´ Equations 13
Figure 7. Assuming B < A, the effect of ιB is depicted on the left, and σ = ι
−1
A ◦ ιB on the right.
simultaneously creating a kink along all the level sets along a ray of the same form, but in the
opposite direction. This means that if all the rays intersected at a point P = (A,−∞), the
transformed polygon has rays that intersect at (A,∞), or vise versa.
Let us use ιA to interchange rays that are of the same form in asymptotically opposite
directions. Suppose we have two rays, Li and Lj , which satisfy
Li : X −A = 0 and Lj : X −B = 0,
as Y → −∞ and Y → ∞ respectively. In the simplest case, these rays are order one, in that
the change in derivative is just one, in which case the transformation
σ = ι−1B ◦ ιA : (X,Y )→ (X,Y + max(0, X −A)−max(0, X −B)),
has the effect of creating a ray along the line X −A as Y →∞ and smooting out a set of kinks
along Li, and conservely doing the same for Lj . If we think of the surface as being parameterized
by A and B, then this action has the effect of swapping A and B. The overall shape of the
resulting polygon does not change by this transformation and the action is an isomorphism of
polygons. The action of ιA and σ on the plane is depicted in Fig. 7 and the action on the level
set of the form in Fig. 5 is depicted in Fig. 8.
Let us now consider how to swap rays given by
Li : X −A = 0, Lj : Y −B = 0,
as Y → −∞ and X → −∞ respectively. To describe this transformation, let us consider the
transformation, ρ : T2 → T2, given by
ρ : (X,Y )→ (X −max(0, Y ), X −max(0,−Y )),
whose inverse is given by
ρ−1 : (X,Y )→ (max(X,Y ), Y −X).
This transformation can be expressed as a composition of transformations of the form (4.1)
and (4.2) as
ρ : (X,Y )
Ξ
−→ (Y,X)
ι0−→ (Y,X −max(0, Y ))
Ξ
−→ (X −max(0, Y ), Y )
Ξ
−→ (X −max(0, Y ), Y +X −max(0, Y )) = (X −max(0, Y ), X −max(0,−Y )).
14 C.M. Ormerod and Y. Yamada
L1
L2
Figure 8. A depiction of action of σ, described above on an octagon with rays L1 and L2. The blue
octagon is the preimage and the green octagon is the image.
Roughly speaking, this sends every straight line of the form X − A = 0 to one that is bent 90
degree along the line Y = X. The conjugation of ιA by ρ, which we label ηA = ρ ◦ ιA ◦ ρ−1, is
given by the expression
ηA(X,Y ) = (X −max(0, Y −A), Y + max(A,X, Y )−max(A, Y )).
It should be clear that this has the same effect as ιA below the line Y = X, however, the effect
of ιA around Y = ∞ now occurs at X = −∞. We may now state that the transformation
swapping Li and Lj is given by
σ = ηB ◦ η
−1
A ,
whose max-plus expression may be simplified to
σ(X,Y ) =
(
B +X + max(A,X, Y )−max(A+B,B +X,A+ Y ),
A+ Y + max(B,X, Y )−max(A+B,B +X,A+ Y )
)
, (4.3)
or equivalently, this is the tropical projective transformation
σ([X : Y : Z]) =
[
B +X + max(A+ Z,X, Y ) : A+ Y + max(B + Z,X, Y ) :
Z + max(A+B + Z,B +X,A+ Y )
]
.
The effect of ηB is shown in Fig. 9 and the effect on a cubic plane curve with these rays is
depicted in Fig. 10.
Lastly, for bookkeeping reasons, we include a set of transformations simply permute the
roles of two rays that are of the same type. For example, if we have a tentacle, described by
Li : X − A = 0 as Y → −∞ and another, described by Lj : X − B = 0 as Y → −∞, then one
transformation simply swaps the roles of A and B, which swaps Li and Lj . In this case, σ acts
as the identity map on the plane and as a simple transformation of the parameter space. This
transformation can always be applied when there are two rays of the same form.
Each of these transformations is an isomorphism between either a collection of polygons
or between some spiral diagrams of the same form. We can now specify that each of the
From Polygons to Ultradiscrete Painleve´ Equations 15
Figure 9. The action of ηB and σ from (4.3) on TP2.
Figure 10. The effect of the σ from (4.3) on a tropical cubic plane curve with rays Li : X −A = 0 and
Lj : Y −B = 0.
ultradiscrete QRT maps and ultradiscrete Painleve´ equations are infinite order elements of the
group of transformations that preserve a pencil of polygons defined by tropical genus one curves
or their corresponding spiral diagrams respectively. This means they may be expressed in terms
of the simple transformations above. As an example, we consider (2.5) and (2.6). We start by
parameterizing the rays as follows:
L1 : X −A1 = 0, L2 : X −A2 = 0, L3 : X −A3 = 0, L4 : X −A4 = 0,
L5 : Y −B1 = 0, L6 : Y −B2 = 0, L7 : Y −B3 = 0, L8 : Y −B4 = 0,
where L1 and L2 extend downwords, L3 and L4 extend upwards, L5 and L6 extend to the left
and L7 and L8 extend to the right. We now have a group of type W (D
(1)
5 ) = 〈s0, . . . , s5〉 where
s0 = σ7,8, s1 = σ5,6, s2 = σ5,7,
s3 = σ1,3, s4 = σ1,2, s3 = σ3,4,
with two additional symmetries, p1 and p2, which are reflections through the line Y = (B3 +
B4)/2 and X = (A3 + A4)/2 respectively. We can now write the ultradiscrete QRT map and
the ultradiscrete Painleve´ equation as the composition
T = p2 ◦ s2 ◦ s0 ◦ s1 ◦ s2 ◦ s1 ◦ p1 ◦ s3 ◦ s5 ◦ s4 ◦ s3. (4.4)
16 C.M. Ormerod and Y. Yamada
s3 s5 ◦ s4 ◦ s3 p1
s2 s2 ◦ s1 ◦ s0 p2
Figure 11. Starting with a single spiral, we show each significant step in the sequence (4.4). In blue,
we show the result of previous transformations, in green is the result of the transformations listed below.
In the last step, we also show the original spiral (in red).
To show that each step is an isomorphism of spiral diagrams, we have depicted the nontrivial
steps in T on a typical spiral in Fig. 11.
We can present the nontrivial actions of these transformations as
s2 : X → X + max(Y,B3)−max(Y,B1),
s2 : A1 → A1 +B3 −B1, s2 : A2 → A2 +B3 −B1,
s3 : Y → Y + max(X,A3)−max(X,A1),
s3 : B1 → B1 +A3 −A1, s3 : B2 → B2 +A3 −A1,
p1 : Y → B3 +B4 − Y, p2 : X → A3 +A4 −X.
The composition in (4.4) gives (3.2).
Remark 4.1. The above constitutes the action on a tropical biquadratic that does not satisfy
the requirement that the image under Π is a tropical cubic plane curve. A cubic plane curve may
be obtained by applying ιA4 , which has the effect of removing the ray given by L4 and adding
a ray given by the same formula, but pointing downward instead of upwards. Up to translational
invariance, this is equivalent to the polygon considered in Section 5.7. In particular, their groups
of transformations are of the same affine Weyl type.
An aspect of defining the group of transformations for a polygon or spiral diagram that
we have not introduced in the above example is that we may always remove two parameters by
taking into account uniqueness of a group of transformations up to translational equivalence. We
can take this into account by insisting that two rays, of different asymptotic forms, pass through
the origin. This means that we will often compose one of the above types of transformation
with a translation so that any ray which is supposed to pass through the origin does so after
From Polygons to Ultradiscrete Painleve´ Equations 17
L1
L3
L2
Figure 12. The spiral diagram for the system with affine Weyl symmetry of type A(1)0 .
the transformation. This fixes a representation based on which rays we choose to pass through
the origin.
5 Tropical representations of affine Weyl groups
While the task of finding subtraction free versions of the Cremona transformations in Sakai’s list
was presented (but not published) by Kajiwara et al. [18], what we wish to present is a different
perspective. The derivation of the following list of affine Weyl representations will sometimes be
a slightly different parameterization of the transformations of [18] due to the manner in which
they were derived. We will also provide some of the geometric motivation behind our choices of
generators. To this end, we shall display a spiral diagram and a nontrivial translation for each
of the cases in Table 1. When the Newton polygon is known, this will also accompany the spiral
diagram on the right.
5.1 Triangles
At the bottom of the hierarchy of multiplicative surfaces in [47] is the system with a symmetry
of the dihedral group of order 6, which admits the presentation
D6 =
〈
p1, p2 : p
3
1 = p
2
2 = (p2p1)
2 = 1
〉
.
This is the group permuting the three rays in Fig. 12. The rays may be parameterized by the
equations
L1 : Y −X −A = 0, L2 : 2X + Y −B = 0, L3 : X + 2Y − C = 0.
By exploiting scaling (i.e., X → X + λ and Y → Y + µ), we can reduce this to the case where
we fix B = C = 0.
In this way, let p1 permute the lines so that p1 : (L1, L2, L3) → (L3, L1, L2). Similarly, p2 is
the transformation that swaps L2 and L3 via a reflection around the line Y = X. These are
explicitly given by the piecewise linear transformations
p1 : X → −X − Y −
A
3
, p2 : Y → X +
2A
3
, p2 : A→ A,
p2 : X → Y, p2 : X → Y, p2 : A→ −A.
18 C.M. Ormerod and Y. Yamada
L4
L3L1
L2
Figure 13. The spiral diagram for the system with affine Weyl symmetry of type A(1)1 with an additional
dihedral symmetry.
As a very degenerate case, the transformations are simple given by (up to translations) a sub-
group of actions of the type (4.2).
It is natural to see that Q = A. The limit which gives a fibration by tropical biquadratics is
the limit as A = Q = 0. The resulting polygons arise as level sets of
H(X,Y ) = max(−X − Y,X, Y ).
As the dihedral group, D6, contains no elements of infinite order, there is no difference equation
associated with this group.
5.2 Rectangles
We consider a spiral diagram of quadralaterals which gives an affine Weyl group of type W (A(1)1 )
with an additional D8 symmetry. In the same way as above, we may exploit scaling so that the
rays extending towards X = −∞ pass through the origin. We paramaterize our rays as follows:
L1 : Y +X = 0, L3 : Y −X −A = 0,
L2 : Y −X = 0, L4 : Y +X −B = 0.
This is depicted in Fig. 13.
The symmetry group for this system is the semidirect product of D8 = 〈p1, p2〉 and W
(
A(1)1
)
=
〈s0, s1〉. A presentation is given by
D8 nW
(
A(1)1
)
=
〈
p1, p2, s0, s1 : p
4
1 = p
2
2 = (p1p2)
2 = s20 = s
2
1 = s0p2s1p2 = 1
〉
,
where the action of D8 is specified up to translation by a clockwise rotation of the four defining
lines, p1, whereas p2 swaps L3 and L4. We write these transformations as
p1 : X → Y +
B
2
, p1 : Y → −X −
B
2
, p1 : A→ B, p1 : B → A,
p2 : X → X, p2 : Y → −Y, p2 : A→ −B, p2 : B → −A.
Let s0 be the conjugation of the transformation depicted in Fig. 7 with the piecewise linear
transformation that makes L2 and L3 parrellel to the y-axis (and L1 and L4 to the x-axis). The
transformation s1 may be obtained in a similar manner with L1 and L4, giving
s0 : X → X + max(0, Y −X +A)−max(0, Y −X)−
A
2
,
From Polygons to Ultradiscrete Painleve´ Equations 19
L4L1
L2
L3
Figure 14. The spiral diagram for the system with affine Weyl symmetry A(1)1 .
s0 : Y → Y + max(0, Y −X +A)−max(0, Y −X) +
A
2
,
s0 : A→ −A, s0 : B → B − 2A,
and s1 = p2 ◦ s0 ◦ p2, which we write as
s1 : X → X + max(0,−X − Y −B)−max(0,−X − Y ) +
B
2
,
s1 : Y → Y + max(0,−X − Y )−max(0,−X − Y −B) +
B
2
,
s1 : A→ A− 2B, s1 : B → −B.
We find that Q = A−B by tracing around the spiral. When A = B, we obtain the invariant
H(X,Y ) = max(−X,−Y, Y,X −A).
For the element T = s1 ◦ s0, we resort to co-ordinates U and V , where X = (U + V )/2 and
Y = (U − V )/2. The dynamical system in these variables is
U˜ − U = B + 2 max(A,A+ V )− 2 max(0, V ),
V˜ − V = 3A+ 2 max(0, U˜)− 2 max(B, 2A+ U˜),
T : A→ A+ 2Q, T : A→ A− 2Q,
where U˜ = T (U) and V˜ = T (V ).
5.3 Quadralaterals
We have another quadralateral that does not possess an additional dihedral symmetry. We break
the dihedral symmetry by fixing the parameterization of the four rays in the following manner:
L1 : Y +X = 0, L3 : Y + 2X −A = 0,
L2 : Y = 0, L4 : Y −X −B = 0,
as depicted in Fig. 14.
The group of transformations that preserves this spiral diagram is of type
W (A(1)1 ) =
〈
s0, s1 : (s0)
2 = (s1)
2〉,
20 C.M. Ormerod and Y. Yamada
where s1 ◦ s0 is the element of infinite order. The first transformation, s0, swaps the roles of L1
with L2 and L3 with L4, which we write as
s0 : Y → −Y −X, s0 : A→ −B,
s0 : X → X, s0 : B → −A.
The other involution, s1, is a reflection around X = B/2 above Y = 0 (so that L4 is sent to L1)
and a skewed reflection below Y = 0, given by X → −Y − X − B/2, which simplifies to the
following tropically rational transformation
s1 : X → max(0,−Y )−X −B, s1 : A→ −2B −A,
s1 : Y → Y, s1 : B → B.
This is simply the conjugation of ι0 with a swap of X and Y . We find Q = A + B by tracing
around one spiral. When A = −B, we obtain the invariant
H(X,Y ) = max(−X − Y,−X,Y,X −A).
The composition, T = s1 ◦ s0, gives the evolution equations
X˜ +X = max(0, Y +X) +A,
Y˜ + Y = −X,
T : A→ A+Q, T : B → B −Q.
This element, T , is the generator for Z in the decomposition of W (A(1)1 ) ∼= Z n G2 in [18, 47].
Alternatively, we could write this system as a second order difference equation in W = −Y ,
where the resulting system becomes
W + 2W˜ + ˜˜W = max(0, W˜ ) +A,
which coincides with a more standard version of an ultradiscrete version of the first Painleve´
equation [43].
5.4 Pentagons
This case is associated with u-PII. To preserve much of the structure of the two previous cases,
we have parameterize the five rays as follows:
L1 : Y +X −A = 0, L4 : Y +X −B = 0,
L2 : Y = 0, L5 : Y −X − C = 0,
L3 : X = 0,
as depicted in Fig. 15.
A presentation of the group of transformations is
A(1)1 ×A
(1)
1 =
〈
s0, s1, w0, w1 : s
2
i = w
2
i = 1
〉
,
where s0 is a reflection around the line Y = X, and s1 is the same the action of s1 in the
previous section in that above the line Y = 0, we have a reflection, and below the line, we skew
the plane. The generators are
s0 : X → Y, p0 : A→ B, p0 : C → −C,
From Polygons to Ultradiscrete Painleve´ Equations 21
L5
L2
L4L3
L1
Figure 15. The pentagon.
s0 : Y → X, p0 : B → A,
s1 : X → max(0,−Y )−X +B, p1 : A→ B + C, p1 : A2 → A−B,
s1 : Y → Y, p1 : B → B.
As for the other part of the group, w0 swaps L1 and L4 via a transformation that sheers between
the lines L1 and L4, which can be written as
w0 : X → X + max(0, X + Y −A)−max(0, X + Y −B),
w0 : Y → Y + max(0, X + Y −B)−max(0, X + Y −A),
w0 : A→ B, w0 : B → A, w0 : C → C + 2A− 2B,
while w1 is a piecewise linear sheering transformation swapping L2 and L3, which we write as
w1 : X → X + C + max(0, X, Y − C)−max(0, X, Y ),
w1 : X → Y + max(C,X, Y − C)−max(0, X, Y ),
w1 : A0 → A+ C, w1 : B → B + C, s1 : C → −C.
Tracing around the figure reveals that Q is given by
Q = B + C −A.
When C = A−B, we obtain the invariant
H(X,Y ) = max(−X − Y,−X,−Y,X −B, Y −A).
One simple translation is the composition, T = (s0 ◦ s1)2, which can be written as
X˜ +X = max(0,−Y ) +B, (5.1a)
Y˜ + Y = max(0,−X˜) +B + C, (5.1b)
T : A→ A+Q, T : B → B +Q, (5.1c)
where the other obvious translation, (w0 ◦ w1), commutes with T . This system is called u-PII.
Exact solutions of (5.1) were studied in [26].
22 C.M. Ormerod and Y. Yamada
L6 L5
L3
L1
L4
L2
Figure 16. The spiral diagram for the discrete Painleve´ equation with A(1)2 +A
(1)
1 symmetry.
5.5 Hexagons
The tropical representation for W (A(1)2 + A
(1)
1 ) was one of the first to be written down [19].
There are a number of equivalent ways of obtaining a hexagon as a cubic plane curve, we choose
to parameterize our rays so that our presentation coincides with the presentation of Noumi et
al. [19]. In particular, our rays are parameterized as follows:
L1 : Y = 0, L4 : Y −B2 = 0,
L2 : Y −X −B1 = 0, L5 : Y −X +A0 −B1 = 0,
L3 : X = 0, L6 : X +A1 = 0,
which is depicted in Fig. 16.
The group of transformations preserving these spiral diagrams is of the affine Weyl type
W
(
A(1)2 +A
(1)
1
)
=
〈
s0, s1, s2, r0, r1 : s
2
i = r
2
i = (sisi+1)
3〉.
We have a natural A(1)2 group acting on the pairs of lines opposite to each other, in particular,
if we denote the piecewise linear transformation that shears the space between two lines (as in
Fig. 7), Li and Lj , by σi,j , then we let s0 = σ2,5, s1 = σ3,6 and s2 = σ1,4. The action of these
elements may be written as
s0 : X → X + max(X +B1, Y )−max(B1 +X,A0 + Y ),
s0 : Y → A0 + Y + max(X +B1, Y )−max(B1 +X,A0 + Y ),
s1 : X → A1 +X, s1 : Y → Y + max(0, A1 +X)−max(0, X),
s2 : X → X + max(A2, Y )−A2 −max(0, Y ), s2 : Y → Y −A2,
where the action on the parameters is
si : Ai → −Ai, si : Aj → Aj − 2Ai.
The action of the W
(
A(1)1
)
= 〈r0, r1〉 component is as follows:
r0 : X → X + max(X,X + Y −A2, Y −A1 −A2)
−max(X,A0 −B1 + Y,A0 +A1 −B1 +X + Y ),
r0 : Y → Y + max(X,X + Y −A2, Y −A1 −A2)
From Polygons to Ultradiscrete Painleve´ Equations 23
−max(X,A0 −B1 + Y,A0 +A1 −B1 +X + Y ),
r1 : X → X + max(X +B1, Y, B1)−max(X,Y, 0),
r1 : Y → Y + max(X,Y −B1,−B1)−max(X,Y, 0).
The Dynkin diagram automorphisms comprise of a reflection around L2, which swaps L1 with L3
and another that swaps L4 with L6, and hence, are given by
p1 : A0,1,2 → A1,2,0, p1 : B0,1 → B1,0,
p1 : X → Y −A2, p1 : Y → A0 −B1 + Y −X,
p2 : A0,1,2 → −A0,2,1, p2 : B0,1 → −B1,0,
p2 : X → A2 − Y, p2 : Y → −A1 −X.
These generators have been chosen to coincide with the original presentation of Noumi et al. [19].
The transformations p1 and p2 satisfy the relations
p31 = p
2
2 = p
−1
1 ◦ si+1 ◦ p1 ◦ si = p2 ◦ ri+1 ◦ p1 ◦ ri = 1.
We find the value of Q is
Q = A0 +A1 +A2 = B0 +B1,
which, when Q = 0, gives invariant curves arising as the level sets of
H(X,Y ) = max(−A1 −B1 −X,X,A2 − Y,A2 +X − Y, Y −B1, Y −A1 −B1 −X).
We have two distinct evolution equations corresponding to different lattice directions. Firstly,
we have the translation T1 = p1 ◦ s2 ◦ s1, which sends (X,Y ) to (X˜, Y˜ ), related via
X˜ − Y = B0 + max(B1 +X,Y )−max(X +A1 +A2, Y +B0),
Y˜ − Y +X = A0 +A2 −B1 + max(X˜, 0)−max(X˜, A0 +A2),
T1 : A0 → A0 +Q, T1 : A1 → A1 −Q,
which corresponds to a version of u-PIII [19]. Secondly, we have T2 = p2 ◦r0, which sends (X,Y )
to (Xˆ, Yˆ ), where
Xˆ + Y = B1 +A2 + max(0, X, Y )−max(0, Y,X +B1),
Yˆ +X = A0 −B0 + max(0, X, Y )−max(B1, Y,X +B1),
T2 : B0 → B0 +Q, T2 : B1 → B1 −Q.
which corresponds to a version of u-PIV [19].
5.6 Septagons
This case is associated with u-PV [43]. There are seven rays, specified as follows:
L1 : Y = 0, L4 : X = 0,
L2 : Y −B0 = 0, L5 : X −B3 = 0,
L3 : Y +B1 = 0, L6 : X +B4 = 0,
L7 : Y −X +B1 +B2 +B3 +B4 = 0,
which we depict in Fig. 17.
The group of transformations is of affine Weyl type
W
(
A(1)4
)
= 〈s0, . . . , s4〉.
Rather than writing each relation, a presentation may be derived from the groups corresponding
Dynkin diagram, which is shown below:
24 C.M. Ormerod and Y. Yamada
L6
L7
L5L4
L3
L1
L2
Figure 17. The spiral diagram for the case of the ultradiscrete Painleve´ equation with A(1)4 symmetry.
1 2
3
4
0
From the Dynkin diagram, the action of si on is specified by
si : Bj =



−Bi if i = j,
Bj +Bi if i 6= j and node i is adjacent to node j,
Bj otherwise.
(5.2)
From this point, we will choose parameterizations of rays so that that the action of si on the
parameters determined by the Dynkin diagram in this way.
The first action is one that interchanges L1 and L2 by the piecewise linear shearing transfor-
mation
s0 : X → X + max(Y,B0)−B0 −max(0, Y ), s0 : Y → Y −B0.
The second transformation is a simple translation,
s1 : Y → Y +B1,
which has the effect of moving L3 to L1, hence, redefining L1 and L3. The transformation s2
has the form
s2 : X → X + max(B2, B2 +X,Y )−max(0, X, Y ),
s2 : Y → Y + max(0, B2 +X,Y )−B2 −max(0, X, Y ),
while s3 simply is a translation in X that redefines L4 and is given by
s3 : X → X −B3.
The last transformation is similar to s0, but applied to the lines L4 and L5,
s4 : X → B4 +X, s4 : Y → max(−X,B4)− Y,
From Polygons to Ultradiscrete Painleve´ Equations 25
The Dynkin diagram automorphisms are generated by a rotation of the nodes
p1 : Bi → Bi+1, p1 : X → max(0, X, Y )−X − Y, p1 : Y → max(0, X)− Y,
and a reflection
p2 : B0,1,2,3,4 → −B2,1,0,4,3, p2 : Y → max(0, X)− Y,
Tracing around the parameters provides the variable, Q, given by
Q = B0 +B1 +B2 +B3 +B4.
In the autonomous limit, when Q = 0, the spiral diagram degenerates to a foliation by tropical
cubic plane curves, specified by the level sets of
H(X,Y ) = max
(
Y,max(0, B1)−B1 −B4 −X,Y −B4 −X,
max(0, B3)−B1 −B3 −B4 − Y,−B1 −B4 −X − Y,
X −B1 −B3 −B4 − Y
)
.
The translation expressed as the composition
T = s4 ◦ s3 ◦ s2 ◦ s1 ◦ p1,
corresponds to the evolution equation
X˜ +X = B3 + max(0, Y˜ ) + max(0, Y˜ +B1)−max(B0, Q+ Y˜ ),
Y˜ + Y = −B1 −B3 + max(0, X) + max(A3, X)−max(0, X +B4),
T : A0 → A0 −Q, T : A4 → A4 +Q,
which is known as the ultradiscrete version of the fifth Painleve´ equation [43].
5.7 Octagons
The biquadratic invariants obtained in the autonomous limit of q-PVI in Section 3 are not
naturally mapped to cubic plane curves. However, under a simple transformation, we can
present an equivalent system based on octagons arising as cubic plane curves whose rays are
parameterized as follows:
L1 : X = 0, L5 : Y = 0,
L2 : X −B2 = 0, L6 : Y +B5 = 0,
L3 : X −B1 −B2 = 0, L7 : Y −X −B3 = 0,
L4 : X +B0 = 0, L8 : Y −X −B3 −B4 = 0,
which is depicted in Fig. 18
The group of transformations preserving these spiral diagrams is of affine Weyl type
W
(
D(1)5
)
= 〈s0, . . . , s5〉.
A presentation may be derived from the Dynkin diagram below:
1 2 3 4
0 5
26 C.M. Ormerod and Y. Yamada
L4
L5
L4
L2L1
L3
L7
L8
Figure 18. The spiral diagram for the case of the ultradiscrete Painleve´ equation with D(1)5 symmetry.
We analogously specify the generators as we did before, where we denote the generators (in
terms of σi,j which swaps Li with Lj),
s0 = σ1,4, s1 = σ3,4, s2 = σ1,3,
s3 = σ5,7, s4 = σ7,8, s5 = σ5,6.
These generators may be written
s0 : X → X +B0, s0 : Y → Y + max(0, X +B0)−max(0, X),
s2 : X → X −B2, s5 : Y → Y +B5,
s3 : X → X + max(B3 + max(0, X), Y )−max(0, X, Y ),
s3 : Y → X + max(0, X +B3, Y )−max(0, X, Y )−B3,
and the Dynkin diagram automorphisms, p1 and p2, are
p1 : B0,1,2,3,4,5 → −B5,4,3,2,1,0,
p1 : X → max(0, X)− Y, p1 : Y → max(0, X, Y )−X − Y,
p2 : B0,1,2,3,4,5 → −B0,1,2,3,5,4,
p2 : X → −X, p2 : Y → Y −X −B3,
Tracing around the particular values gives us the variable
Q = B0 +B1 + 2B2 + 2B3 +B4 +B5
In particular, when Q = 0, we obtain the invariant
H(X,Y ) = max
(
max(0,−B5)−X,Y −X,max(0,−B1,−B1 −B2)−B5 − Y,
B0 + Y,max(0,−B1,−B1 −B2) +X −B2 −B5 − Y,−b5 −X − Y,
B0 +B3 + max(0, B4) +X, 2X − Y −B1 − 2B2 −B5
)
.
The usual translation that is associated with the dynamics of u-PVI and the symmetry QRT
equation is the action of
T = p2 ◦ p1 ◦ p2 ◦ s1 ◦ s2 ◦ s3 ◦ s5 ◦ s4 ◦ s3 ◦ s2 ◦ s1.
From Polygons to Ultradiscrete Painleve´ Equations 27
L9
L8
L7
L2L1
L3
L4
L5
L6
Figure 19. The spiral diagram for the case of the ultradiscrete Painleve´ equation with E(1)6 symmetry.
To express the evolution of this system in a manner closer to that of (3.2), we invert the
transformation that was used to express the invariant as a tropical cubic curve. This is done by
letting
Z = max(0, X) + max(0, X −B2)−max(0, X +B0)− Y,
which means the evolution in terms of X and Z is expressed as
X + X˜ = B2 −B5 + max(0, Z)−max(0, Z +B0 +B2 +B3)
+ max(B5, Z)−max(0, Z +B0 +B2 +B3 +B4),
Z + Z˜ = B5 −B2 + max(0, X˜)−max(0, X˜ +Q+B0)
+ max(B2, X˜)−max(0, X˜ +Q−B1 −B2),
which is equivalent to (3.2) above.
5.8 Enneagons
It is at this point we go beyond the QRT maps defined by biquadratic cases [29, 46]. We
exploit the translational freedom to parameterize two rays coincide with the y-axis and x-axis
respectively. The remaining ray are parameterized as follows:
L1 : X = 0, L6 : Y +B0 +B6 = 0,
L2 : X −B2 = 0, L7 : Y −X +B3 = 0,
L3 : X −B1 −B2 = 0, L8 : Y −X +B3 +B4 = 0,
L4 : Y = 0, L9 : Y −X +B3 +B4 +B5 = 0,
L5 : Y +B6 = 0.
The relevant spiral diagram is of irregular enneagons, depicted in Fig. 19.
The group of transformations are of affine Weyl type
W
(
E(1)6
)
= 〈s0, . . . , s6〉.
The presentation, and action on the parameters, is specified by the Dynkin diagram below:
28 C.M. Ormerod and Y. Yamada
1 2 3 4 5
6
0
We may now parameterize the affine Weyl group actions by first specifying the generators
that have a little effect on X and Y , by letting s0 = σ5,6, s1 = σ2,3, s2 = σ1,2, s4 = σ7,8,
s5 = σ8,9 and s6 = σ4,5. In these cases, the effect on X and Y are trivial, except for s2 and s6,
which have the effect
s2 : X → X −B2, s6 : Y → Y +B6.
The action of s3 is given by
s3 : X → X + max(B3, B3 +X,Y )−max(0, X, Y ),
s3 : Y → Y −B3 + max(0, B3 +X,Y )−max(0, X, Y ).
The Dynkin diagram automorphisms are given by
p1 : B0,1,2,3,4,5,6 → −B5,1,2,3,6,0,4,
p1 : X → −X, p2 : Y → Y −X −B3,
p2 : B0,1,2,3,4,5,6 → −B1,0,6,3,4,5,2,
p2 : X → Y, p2 : Y → X.
Tracing around the enneagon, we obtain the variable
Q = B0 +B1 + 2B2 + 3B3 + 2B4 +B5 +B6.
In the autonomous limit, when Q = 0, this spiral diagram degenerates to give a fibration by
cubic plane curves, which may be expressed as the tropical curves that arise as the level sets of
H(X,Y ) = max
(
2X − Y −B1 −B2,max(0,−B2,−B1 −B2) +X − Y −B2,
max(0,−B5,−B4 −B5) +X −B1 − 2B2 −B3,max(0,−B1,−B2 −B2)− Y,
Y + max(0,−B4,−B4 −B5)−B1 − 2B2 − 2B3 −B4, 2Y +B0 + 2B6 −X,
Y +B6 + max(0, B0, B0 +B6)−X,max(0, B6, B0 +B6)−X,−X − Y
)
.
The translation that is associated with the dynamics of the discrete Painleve´ equation in this
case is given by
T = p1 ◦ p2 ◦ s1 ◦ s2 ◦ s3 ◦ s4 ◦ s6 ◦ s0 ◦ s3 ◦ s2 ◦ s1 ◦ s6 ◦ s3 ◦ s2 ◦ s4 ◦ s3 ◦ s6 ◦ s0.
Though the evolution equation is very complicated, the action can be evaluated quite easily by
the geometric method in [16]. In the case of a tropical cubic genus one curve, the action of T can
be describes as follows: we choose two two rays, say Li and Lj , and let T move Li to the point
in which the other rays and T (Li) define a pencil of tropical cubic genus one curves that foliate
the plance, i.e., rather than spirals, we have closed curves. Any point, P ∈ TP2, is now on some
closed genus one cubic curve, C, in the pencil. We send P to T (P ), so that T (P ) satisfies
T (P ) + T (Li) = P + Lj , (5.3)
where we interpret T (Li) and Li in terms of the unique stable intersection of T (Li) and Lj
with C respectively and the addition is in accordance with the group law on C (see [5]). Finally
we send Lj to a point in which T (Lj) satisfies
Li + Lj = T (Li) + T (Lj),
on C. We have illustrated this in Fig. 20.
From Polygons to Ultradiscrete Painleve´ Equations 29
Li
T (Li) Lj
P
Li
T (Li) Lj
T (P )
Figure 20. This is a pictorial represection of (5.3) where the dashed lines intersect the polygon at the
four fixed points (in red) and P and T (Li) on the left and T (P ) and Lj on the right.
5.9 Decagons
The rational surface of type A(1)1 was obtained by blowing up three points on a line and six on
a quadratic curve [47]. The resulting surface is rationally equivalent to the surface obtained by
blowing up four points at lines at infinity and two points at on the third line at infinity. Hence,
in the discrete setting, the underlying surface and the symmetries W
(
E(1)7
)
obtained here are
equivalent up to a rational transformation to those of [47].
The most general tropical cubic plane curve is a enneagon, hence, to describe the decagon
spirals, we need to consider spiral degenerations of quartic plane curves with two rays of order
two. We choose to parameterize these rays as follows:
L1 : X +B4 = 0, L5 : Y = 0,
L2 : X +B4 +B5 = 0, L6 : Y −B3 = 0,
L3 : X +B4 +B5 +B6 = 0, L7 : Y −B3 −B2 = 0,
L4 : X +B4 +B5 +B6 +B7 = 0, L8 : Y −B3 −B2 −B1 = 0,
L9 : Y −X −B0 = 0, L10 : Y −X = 0.
It should be noted that L9 and L10 are of order 2. With these considerations, the resulting
system of spiraling polygons is depicted in Fig. 21.
The resulting group of transformations is of affine Weyl type
W
(
E(1)7
)
= 〈s0, . . . , s7〉.
This groups Dynkin diagram is below:
1 2 3 4 5 6 7
0
The reflections are given by s0 = σ9,10, s1 = σ7,8, s2 = σ6,7, s3 = σ5,6, s5 = σ1,2, s6 = σ2,3
and s7 = σ3,4, hence, the nontrivial actions are given by
s0 : X → X +B0, s3 : Y → Y +B3, s3 : X → X +B3,
30 C.M. Ormerod and Y. Yamada
Figure 21. A spiral diagram for the E(1)7 case.
s4 : X → X −B4 + max(B4, X, Y )−max(B4, X,B4 + Y ),
s4 : Y → Y + max(0, X, Y )−max(B4, X,B4 + Y ).
We also have a single Dynkin diagram automorphism. This Dynkin diagram automorphism has
the effect of sending X to −X, and Y to −Y , which swaps all the eight first order rays, however,
it also has the effect of reflecting the two rays, L9 and L10, in opposite direction, hence, we
compose this a transformation of the form of ιA along L9 and L10, giving
p1 : X → Y +B4 −max(Y,X)−max(Y,B0 +X),
p1 : Y → X +B0 +B4 −max(Y,X)−max(Y,B0 +X),
p1 : B0,1,2,3,4,5,6,7 → B0,7,6,5,4,3,2,1.
Tracing around the spirals reveals that
Q = 2B0 +B1 + 2B2 + 3B3 + 4B4 + 3B5 + 2B6 +B7,
which in the autonomous limit, when Q = 0, gives a closing of spirals to give the following
invariant
H(X,Y ) = max
(
0, Y + µ1, 2Y + µ2, X + µ5, 2X + µ6,
max(Y + µ3, X −B0 + µ7) + max(Y,X +B0) + max(X,Y ),
2 max(Y,X +B0) + 2 max(X,Y ) + µ4
)
−X − Y,
where the values of µi are defined by
max(0, X) + max(0, B3 +X) + max(0, B3 +B2 +X)
+ max(0, B3 +B2 +B1 +X) = max(0, µ1 +X,µ2 + 2X,µ3 + 3X,µ4 + 4X),
max(0, X −B4) + max(0, X −B4 −B5) + max(0, X −B4 −B5 −B6)
+ max(0, X −B4 −B5 −B6 −B7) = max(0, µ5 +X,µ6 + 2X,µ7 + 3X,µ8 + 4X).
A translation associated with the ultradiscrete Painleve´ equation is given by
T = p1 ◦ s1 ◦ s2 ◦ s3 ◦ s4 ◦ s0 ◦ s5 ◦ s4 ◦ s3 ◦ s2 ◦ s1 ◦ s6 ◦ s5 ◦ s4 ◦ s0
◦ s3 ◦ s2 ◦ s4 ◦ s3 ◦ s5 ◦ s4 ◦ s0 ◦ s6 ◦ s5 ◦ s4 ◦ s3 ◦ s2 ◦ s1,
From Polygons to Ultradiscrete Painleve´ Equations 31
L1 L2
L4
L5
L6
L7
L8
L9
L10L11
L12
Figure 22. A spiral diagram for the E(1)8 case.
whose action on T2 is too complicated to be written here. However, the action of T may be
described by the theory of [16], where the evolution takes the form
T (P ) + T (Li) = P + Lj , (5.4)
where the addition here is defined in terms of the group law on a tropical quartic genus one
curve.
5.10 Undecagon
The blow-up points in the original classification of Sakai [47] on P2 lie on a single nodal cu-
bic. This configuration is birationally equivalent (by a series of blow-ups and blow-downs) to
a configuration in P2 in which there are three order two singularities on the line at y = 0, two or-
der three singularities on the line x = 0 and six order one singularities on the line z = 0. Hence,
in the discrete setting, the underlying surface and the symmetries of affine Weyl type E(1)8
obtained here are equivalent up to a rational transformation to those of [47].
The tropical analogue requires we have a configuration of two, three and six rays, which may
be parameterized as follows:
L1 : X = 0, L7 : Y −X −B3 −B4 = 0,
L2 : X −B0 = 0, L8 : Y −X −B3 −B4 −B5 = 0,
L3 : Y = 0, L9 : Y −X −B3 −B4 −B5 −B6 = 0,
L4 : Y +B2 = 0, L10 : Y −X −B3 −B4 −B5 −B6 −B7 = 0,
L5 : Y +B1 +B2 = 0, L11 : Y −X −B3 −B4 −B5 −B6 −B7 −B8 = 0,
L6 : Y −X −B3 = 0,
where L1 and L2 are of order 3, L4, L5 and L6 are of order 2 and the remaining rays are order 1.
Such a configuration is depicted in Fig. 22.
The top case of the multiplicative type Painleve´ equations of [47] is one that has a symmetry
group of type
W
(
E(1)8
)
= 〈s0, . . . , s8〉.
32 C.M. Ormerod and Y. Yamada
A presentation may be derived from the groups corresponding Dynkin diagram, which is shown
below:
1 2 3 4 5 6 7 8
0
There are no Dynkin diagram automorphisms. We may specify the elemtents in terms of σi,j
as s0 = σ0,3, s1 = σ4,5, s2 = σ1,2, s4 = σ6,7, s5 = σ7,8 ,s6 = σ8,9, s7 = σ9,10 and s8 = σ10,11. The
nontrivial actions are given by
s0 : X → X −B0, s2 : Y → Y +B2,
s3 : X → X + max(B3, B3 +X,Y )−max(0, X, Y ),
s3 : Y → Y −B3 + max(0, B3 +X,Y )−max(0, X, Y ).
By tracing around the figure, we find that
Q = 3B0 + 2B1 + 4B2 + 6B3 + 5B4 + 4B5 + 3B6 + 2B7 +B8.
In the autonomous limit, this becomes a foliation of tropical sextic curves, specified by the level
sets of
H(X,Y ) = max{iX + jY + ci,j | 0 ≤ i, 0 ≤ j, i+ j ≤ 6} − 2X − 3Y,
where
c0,0 = 2λ3, c0,2 = 2λ2, c0,4 = 2λ1,
c1,1 = max(2κ2 + µ5, κ1 − κ2 + λ2 + λ3),
c1,2 = max(κ1 − κ2 + 2λ2, 2κ2 + µ5 + λ2 − λ3, µ1 + λ3),
c1,3 = max(µ1 + λ2, κ1 − κ2 + λ1 + λ2, 2κ2 + µ5 + λ1 − λ3),
c1,4 = max(µ1 + λ1, κ1 − κ2 + λ2, 2κ2 + µ5 − λ3), c1,5 = µ1,
c2,1 = max(κ1 + κ2 + µ5, 2κ1 − 2κ2 + λ2 + λ3),
c2,2 = max(κ2 + µ4, 2λ2 − κ2, κ1 + κ2 + µ5 + λ2 − λ3, κ1 − κ2 + µ1 + λ3,
2κ1 − 2κ2 + λ1 + λ3),
c2,4 = µ2, c3,0 = 3κ1 − 3κ2 + 2λ3, c3,1 = max(2κ1 + µ5, κ1 − 2κ2 + λ2 + λ3),
c3,2 = max(κ1 + µ4, κ2 + µ5 + λ2 − λ3,−κ2 + µ1 + λ3, κ1 − 2κ2 + λ1 + λ3),
c3,3 = µ3, c4,1 = max(κ1 + µ5,−2κ2 + λ2 + λ3),
c4,2 = µ4, c5,1 = µ5, c6,0 = µ6, c0,1 = c0,3 = c2,3 = −∞,
max(0, X + κ1, 2X + κ2) = max(0, X) + max(0, X +B0),
max(0, X + λ1, 2X + λ2, 3X + λ3) = max(0, X) + max(0, X −B2)
+ max(0, X −B1 −B2),
max(0, X + µ1, . . . , 6X + µ6) = max(0, X +B3)
+ max(0, X +B3 +B4) + · · ·+ max(0, X +B3 +B4 + · · ·+B8).
The translation is the composition
T = s3 ◦ s2 ◦ s4 ◦ s3 ◦ s1 ◦ s2 ◦ s5 ◦ s4 ◦ s3 ◦ s6 ◦ s5 ◦ s4 ◦ s0 ◦ s3 ◦ s2 ◦ s1
◦ s7 ◦ s6 ◦ s5 ◦ s4 ◦ s3 ◦ s2 ◦ s8 ◦ s7 ◦ s6 ◦ s5 ◦ s4 ◦ s3 ◦ s0 ◦ s3 ◦ s4
◦ s5 ◦ s6 ◦ s7 ◦ s8 ◦ s2 ◦ s3 ◦ s4 ◦ s5 ◦ s6 ◦ s7 ◦ s1 ◦ s2 ◦ s3 ◦ s0 ◦ s4
◦ s5 ◦ s6 ◦ s3 ◦ s4 ◦ s5 ◦ s2 ◦ s1 ◦ s3 ◦ s4 ◦ s2 ◦ s3 ◦ s0.
From Polygons to Ultradiscrete Painleve´ Equations 33
s0 wl wpu1uq
v1
vr
Figure 23. A Dynkin diagram of type Tp,q,r.
Once again, the evolution is too complicated in its tropical form to give here. However, the
geometric interpretation is that the evolution is defined as
T (P ) + T (Li) = P + Lj , (5.5)
where the addition is with respect to the group law on a tropical sextic genus one curve.
6 Discussion of dodecagons, triskaidecagons and higher
We wish to breifly discuss some of the difficulties extending the above arguments to more than
eleven sides. We have two constructions that we believed were related; tropical maps of the
plane arising from polygons with greater than 11 sides and tropical birational representations
of the Weyl group W (Tp,q,r) constructed in [54], whose Dynkin diagram is given in Fig. 23.
Each one of the polygons we have considered so far arise from tropical genus one curves. If
we go to a higher number of sides, a simple combinatorial argument based on (3.7) shows us
that the higher sided polygons must come from higher genus cases. The simplest example is the
autonomous system defined on a pencil of tropic quartics such that there are four distinct rays
of order one in each direction. Suppose we parameterize these by
L1 : X = 0, L7 : Y +B7 +B8 = 0,
L2 : X −B0 = 0, L8 : Y +B7 +B8 +B9 = 0,
L3 : X −B0 −B1 = 0, L9 : Y −X +B3 = 0,
L4 : X −B0 −B1 −B2 = 0, L10 : Y −X +B3 +B4 = 0,
L5 : Y = 0, L11 : Y −X +B3 +B4 +B5 = 0,
L6 : Y +B7 = 0, L12 : Y −X +B3 +B4 +B5 +B6 = 0,
as labelled in Fig. 24.
The diagram in Fig. 24 has been obtained by following a path in which the rays have been
fixed, and follow the level curves of a biquartic invariant where three of the parameters, the
coefficients of X +Y , 2X +Y and X + 2Y , have been set to −∞. Tracing around the diagram,
we find that Q is given by
Q = 3B0 + 2B1 +B2 + 4B3 + 3B4 + 2B5 +B6 + 3B7 + 2B8 +B9.
Define a translation, T , by letting T (Li) move to a point that defines a pencil of closed polygons.
For any point P , we have a unique closed curve, C, intersecting with P . The evolution defined
by (5.3), (5.4) and (5.5) were in terms of a group law on genus one curves, however, these
resulting closed curves in this more general setting are no longer of genus one, hence, describing
the group structure on such curves is not so straightforward.
To obtain a higher number of sides (with the constraint that all rays of the same form are of
the same order), we may realize 13-sided polygons as specializations of tropical curves of degree
34 C.M. Ormerod and Y. Yamada
L1 L2
L3
L4
L9
L10L11L12
L6
L5
L7
L8
Figure 24. A model for the dodecagon.
twelve and 14-sided polygons as specializations of tropical curves of degree six. A rudimentary
search reveals n-agons for all n up to 30 sides.
The problem is that in the autonomous limit, the naive extension to W (T4,4,4) using canonical
permutations and an analogue s3 in the E
(1)
8 case does not preserve all the required quartic
plane curves in the pencil constructed. The action of this generator generally gives a curve
of degree five, hence, it is not s3 invariant. It seems likely that this fails to preserve all the
required degenerate curves when the closed piecewise linear curve become small. It seems that
the dynamics we describe may have an interpretation in terms of the addition on some tropical
hyper-elliptic curve, for example [11] where certain tropical dynamics was studied by using the
tropical addition formulae on the spectral curve of hyper-elliptic type.
7 Conclusion
What has been presented is a way of naturally obtaining a group of transformations that preserve
the structure of a spiral diagram. It is possible to extend this to cases that do not arise as
ultradiscrete Painleve´ equations, however the invariants seem more elusive. The autonomous
limits do not necessarily result in foliations of genus one curves. Where the role of the addition
law on cubic plane curves in the Painleve´ equations and QRT maps is central [16], perhaps
similar integrable systems could be based on the addition laws for hyperelliptic curves, which
are in general, much more complicated [4, 11].
Another possible direction is to explore the tropical Cremona transformations more thor-
oughly. Interesting tropical versions of del Pezzo surfaces have emerged with W (E6) and W (E7)
symmetry during the write-up of this paper [44]. A homological approach that follows [7, 24,
27, 28] more closely would also be of interest.
Acknowledgements
Christopher M. Ormerod would like to acknowledge Eric Rains for his helpful discussions. Y. Ya-
mada is supported by JSPS KAKENHI Grant Number 26287018.
From Polygons to Ultradiscrete Painleve´ Equations 35
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