The projective translation equation

The projective translation equation and unramified -dimensional flows with rational vector fields


Let . Previously we have found all rational solutions of the -dimensional projective translation equation, or PrTE, ; here is a pair of two (real or complex) functions. Solutions of this functional equation are called projective flows. A vector field of a rational flow is a pair of -homogenic rational functions. On the other hand, only special pairs of -homogenic rational functions give rise to rational flows. In this paper we are interested in all non-singular (satisfying the boundary condition) and unramified (without branching points, i.e. single-valued functions in ) projective flows whose vector field is still rational. If an orbit of the flow is given by homogeneous rational function of degree , then is called the level of the flow. We prove that, up to conjugation with -homogenic birational plane transformation, these are of types: 1) the identity flow; 2) one flow for each non-negative integer - these flows are rational of level ; 3) the level exponential flow, which is also conjugate to the level tangent flow; 4) the level flow expressable in terms of Dixonian (equianharmonic) elliptic functions; 5) the level flow expressable in terms of lemniscatic elliptic functions; 6) the level flow expressable in terms of Dixonian elliptic functions again. This reveals another aspect of the PrTE: in the latter four cases this equation is equivalent and provides a uniform framework to addition formulas for exponential, tangent, or special elliptic functions (also addition formulas for polynomials and the logarithm, though the latter appears only in branched flows). Moreover, the PrTE turns out to have a connection with Pólya-Eggenberger urn models. Another purpose of this study is expository, and we provide the list of open problems and directions in the theory of PrTE; for example, we define the notion of quasi-rational projective flows which includes curves of arbitrary genus. This study, though seemingly analytic, is in fact algebraic and our method reduces to algebraic transformations in various quotient rings of rational function fields.

Key words and phrases:
Projective translation equation, flows, rational vector fields, iterative functional equation, elliptic curves, elliptic functions, Dixonian elliptic functions, linear PDE’s, finite group representations, hypergeometric functions
2010 Mathematics Subject Classification:
Primary 39B12, 33E05, 35F05; Secondary 14H52, 14H05, 14E05
The author gratefully acknowledges support by the Lithuanian Science Council whose postdoctoral fellowship is being funded by European Union Structural Funds project “Postdoctoral Fellowship Implementation in Lithuania”.

1. Introduction

For convenience reasons, we write instead of . We also write . This paper is a continuation of [Al10, Al12]. It is completely independent from the first, and also mostly independent from the second paper, apart from the ([Al12], Section 4, steps I and III), which are crucial for the current work. The main needed steps will be summarized here in the Subsection 1.3. The current paper is an introduction to the prospective paper [Alpr2] where general plane vector fields are treated in the framework of quasi-rational flows; see the Subsection 5.4.

1.1. Background

The general “affine” translation equation is a functional equation of the type

where . Note that this is not the most general form of the translation equation; to get the feeling of the variety of structure and methods that underlies this equation, the reader may consult [Ac66, Mo73, Mo95, FR08, FR10]. Our research concentrates on the special case of this equation, where is of the form ; this choice is not accidental, since it exhibits several fascinating features not encountered in the general affine solutions (“affine flows”). This special case was first introduced in [Al10] where we considered the -dimensional equation from the topologic point of view in the simplest case of the sphere .

In the paper [Al12] we solved the following problem: find all rational solutions of the -dimensional projective translation equation (PrTE for short)


Here is a pair of rational functions in two real or complex variables. It appears that, up to conjugation with a -homogenic birational plane transformation (-BIR in short, for the structure of these see the Appendix in [Al12]), all rational solutions of this equation are as follows: the zero flow , two singular flows and , an identity flow , and one non-singular flow for each non-negative integer , called the level of the flow; this solution is given by below. The non-singular solution of this equation is called the projective flow. Non-singularity means that a flow satisfies the boundary condition


Thus, each rational projective flow is automatically a birational plane transformation whose inverse is given by .

The main object associated with a solution is its vector field given by


Vector field is necessarily a pair of -homogenic functions. For smooth functions, the functional equation (1) implies the PDE


and the same PDE for , with boundary conditions

In principle, the PDE (4) with the above boundary conditions is equivalent to (1); but the proof of this equivalence is possible only in each case separately, since there arises the complication to determine the definition domain of the flow - for example, explore below. Thus, not all functions in two complex variables can be iterated without restrictions - this restriction is explicitly present in (1), but only implicitly present in (4), and only a posteriori, after we have found the solution, whether in analytic form in terms of known special functions, or via an independent analysis.

Each point under a flow possesses the orbit, which is either a curve (when we deal with flows over ) or a surface (complex curve over ), or a single point; the orbit is defined by

For rational flows of level , there exists an th degree homogenic rational function such that the orbits are the curves , . We will also refer to non-rational flows as also being of level if the orbits are given by th degree homogeneous rational function. It appears that the vector field of a rational flow is a pair of -homogenic rational functions. The main theorem in [Al12] gives the first structural result for the rational flows; many more properties of the flows which take into account not only the level but the birational conjugation itself will be treated in [Alpr1]. The main idea of the proof (which is rather lengthy) is that using conjugation with -BIR, the vector field can be reduced step by step, leading to rational -homogenic functions with numerators of smaller degree, and eventually, provided we do not encounter an obstruction, to quadratic forms. The obstruction arises when we hit the vector field whose both coordinates are proportional. So, we are left to find all rational flows whose vector field is given by a pair of two quadratic forms in two variables, or whose one of the coordinates vanish (this is equivalent, after a linear conjugation, to the obstruction). The first case is a rewarding part of the proof. For example, we find that some of these pairs of quadratic forms lead to rational solutions, while others should be discarded since they arise from non-rational flows. In this paper we take a closer look at non-rational solutions of (1) whose vector field is a pair of quadratic forms. In general, every vector field whose orbit is given by a homogeneous rational function of degree gives rise, generally, not to a rational flow but to a quasi-rational flow; see the Subsection 5.4 and [Alpr2].

1.2. Addition formulas. Motivation

Consider the following two-variable two-dimensional functions:

(Here in each case stands for the corresponding function, , , and so on). We can check directly that these functions satisfy the PrTE (1). The vector fields and the equations for the orbits of these flows are given by

: , ;
: , ;
: , ;
: , .

(In the last case the orbits are non-algebraic curves). In fact, replace the known function in the expression of with the unknown (this is shown above). Now require that the so obtained function satisfies (1). Eventually, this turns out to be equivalent to nothing else but the standard addition formulas:

The first two can be restated in a symmetric way; that is, if , then

These are superior to the former, since if we know additionally that and , then these identities also encode the symmetry properties and . In [Al12] we found out that the -dimensional PrTE is directly related to birational transformations of , as opposed to general “affine” translation equation which is tied to birational affine transformations of , albeit this dependency is of different nature, the first case being much more involved. Now we see another fascinating feature of (1); namely, the PrTE provides a uniform framework for addition formulas for certain functions: abelian - , ; algebraic - , if ; integrals over algebraic - . This is the simplest, -dimensional case of the theory. As we will see in Theorem 1, the above list will not include trigonometric functions and , but will include rather special elliptic functions related to regular hexagonal and square lattices. The higher dimensional case even in the setting of [Al12] (that is, classification of higher dimensional rational projective flows) is open and very promising; see the Subsection 5.1.

Now we formulate the main problem of this paper.

Problem 1.

Find all flows, that is, bivariate functions satisfying (1), which are defined for , whose vector field is rational, and which are single-valued functions, i.e. without branching points.

Thus, the flow has an infinite branching point at . The flow has an infinite branching if and a finite one if . So, our chief interest is only the case , and this was dealt with in [Al12]. On the other hand, the flows and are single-valued functions for all . With the help of linear conjugation we can force the exponential and tangent flows to be symmetric with respect to the linear involution . Thus, we summarize this as

Proposition 1.


These functions are symmetric projective flows. Both and satisfy the PDE (4), where in the exponential case, and in the tangential case, respectively. The projective flow property is equivalent to addition formulas for corresponding functions.

Note however that these two cases are conjugate and will fall under the same item in the classification (see Theorem 1). In fact, let us define the -BIR by


and the latter projective flow is linearly conjugate (over ) to .

1.3. Quadratic forms as vector fields of projective flows

We will see later that along with , and , there are three more pairs of functions which complete the picture. As a crucial part of our investigations, let us make a brief summary of the Step II of the proof of the main theorem in [Al12]. Let the vector field of the projective flow be given by , where both coordinates are -homogenic rational functions, and the common denominator has a degree . We found that, unless and are proportional, there exists a -BIR such that the vector field of the flow is a pair of rational functions with lowered degree in the common denominator. Thus, we are left to consider cases where either and are proportional, or they both are quadratic forms. If the first statement holds, then we see that (after a linear conjugation) one can confine to the case . This implies . If were a rational function, then this would necessarily mean that is a Jonquières transformation; this would imply that is in fact a quadratic form, and calculations in ([Al12], Step II) provide the complete solution. The flows and arise exactly from this analysis in cases and , respectively. In the setting of the Problem 1, the pair with the vector field ought no longer be rational, the implication that is a Jonquières transformation is irrelevant, and we need to provide an independent analysis of the solution of (4) in case is any -homogenic rational function, and . This is accomplished in the Subsection 4.4.

Suppose now that both and are quadratic forms, see ([Al12], Step III). We found that if is a cube of a linear polynomial, then this always leads to the flow with a ramification of the type . If is not a cube, then the vector field, with the help of linear conjugation, can be transformed into

The cases or lead to rational solutions or flows with ramification of the type , . So, let . If or , this again leads to a ramification of the type . So, after a linear conjugation, we may assume that

The flow with this vector field has ramifications of both types and , so, if there is none, we get the arithmetic condition


If , then again, , otherwise the flow is ramified, and gives exactly the rational flow of level . Assume . Then the vector field with the help of linear conjugation can be transformed into the vector field


Since this is also unramified, we get another arithmetic condition


The case leads to a rational or algebraically ramified flow. It is fascinating that if , and , then all pairs which satisfy as simple arithmetic conditions as (5) and (7) are encoding elliptic unramified flows! More precisely, there are exactly such pairs (standard exercise):

The symbol means that the two pairs a linearly conjugate via (6), and means that the flow is self conjugate. The pairs and are also linearly conjugate with the help of the involution . So, there are three equivalence classes of flows (shown as rows above), consisting of , and pairs respectively; in each class any two flows are linearly conjugate, and we will show that all three arise form elliptic flows. We will choose such representatives: , , and . Most of this paper deals with the first case, the fascinating vector field


which is the class on its own and thus it has exactly the -fold symmetry: for every ,


for the definitions, see the property (SYMM), Subsection 3.2. In the setting of [Al12], all the tricks which ruled out other vector fields as arising from non-rational flows (as a rule, these tricks constituted in showing that corresponding flows have branching points, and rational flows, obviously, cannot have these) were not applicable in all these exceptional cases of , and it was still not clear why the solution of (4) in case (8), for example, which is exactly the function (see the Subsection 3.2), cannot be a rational function. And it appears that it is not; since, if we put , then, as the property (ELL), Subsection 3.2, implies, one has

and thus parametrizes the elliptic curve and cannot be a rational function. In this case is not rational itself. Two other vector fields with and , whose orbits are also elliptic curves, are examined in the Subsection 4.5.

2. Auxiliary functions

Now we make an interlude and introduce functions which will be crucial in our study of the vector field (8). The material is just a collection of various facts from the literature; our contribution to this topic is an introduction of special elliptic functions and for which we prove Proposition 3.

2.1. The special hypergeometric series

Let us introduce our first auxiliary function [BAT]


It satisfies , and the linear ODE


The derivative of this gives the second order ODE

which coincided with Euler’s hypergeometric differential equation for . Since , the function is invariant under Pfaff’s transformation and thus it satisfies the functional equation [BAT]

this can be verified easily using the integral representation (10). Of course, this functional equation is satisfied by all hypergeometric functions of the form . For example, when , this hypergeometric function reduces to . The change of variables in (10) gives


The appearance of the symmetric group in our investigations - see (SYMM), the Subsection 3.2 - can be explained from several points of view; here is one of them.

The special case of Kummer’s theory for hypergeometric series [BAT] is the following fact, which is easily checked in our case: the differential equation (11) has the following three solutions:

for , ;
for , ;
for , .

This can be treated as the complete description of the differential equation (11) on the line . The three singular points divide this circle into three parts. For each interval , , , the general solution of (11) in the interval with a floating boundary condition is given by

The symmetric group acts on the set by permutations. This corresponds to the action of Möbius transformations on , and as follows.

The first entry, for instance, means the following: the map under consideration interchanges and but leaves the function intact. Other two elements of (the last one is the identity) are obtained from the above. So, for example, the cycle corresponds to the transformation

These correspondences should find their analogues in the setting of the Subsection 5.1, at least in case of symmetric groups , .

2.2. The Dixonian elliptic functions

For the general theory of elliptic functions we may refer to the classical book (in Russian) [Ah48]. Let . The functions , were introduced by Dixon [Di90] as a pair of functions which parematrize the Fermat cubic . These are in fact special elliptic functions satisfying the following

Proposition 2.

[Di90, FCF05] The Dixonian elliptic functions have these properties

the last holds for . Here the contant (the period, according to the notion of Zagier-Kontsevich) is given by

The last two constants are of great importance in the current paper. The can be numerically calculated by the fast converging series

The full lattice of periods for both and is given by . Let be the fundamental parallelogram. Consider special points

In terms of the theory of elliptic functions, both and are order elliptic functions, and so each of them attains every value in in exactly thrice, counting multiplicities. The simple poles of both and are , and . The function has simple zeros at , and , while the values at , and are, respectively, . Likewise, has simple zeros at , and , and the values at , and are, respectively, (triple value), , . These properties follow from Proposition 2. The authors in [FCF05], for the convenience reasons, introduce the hyperbolic versions of these functions, given by , . These functions parametrize the “Fermat hyperbola” . For our purposes, we will need yet another version of these functions, and here we introduce

These are order elliptic function with the same period lattice . Moreover, we have

Proposition 3.

The functions and have these properties:

These are verified using Proposition 2.

3. The results

3.1. Classification

The first main result of this paper is the complete solution of the Problem 1.

Theorem 1.

Let be a smooth projective flow such that (2) holds, and its vector field, defined by (3), is a pair of -homogenic rational functions. Suppose that both and are defined on , except for a countable set of isolated curves each, where they might have poles, and that in their definition domains are single-valued analytic functions. Then there exists a -BIR , such that is one of the following canonic projective flows:

  • ;

  • for , the level flow, whose orbits are given by ; only in the latter two cases the flow is rational;

  • , the level flow whose orbits are given by ; it is conjugate to the flow , also the level flow and also with orbits given by (for these two, see Proposition 1);

  • , the level flow whose vector field is , orbits are given by , and this flow is algebraically expressable in terms of Dixonian elliptic functions (see Theorem 2);

  • , the level flow whose vector field is , orbits are given by , and this flow is expressable in terms of lemniscatic elliptic functions (with quadratic period lattice);

  • , the level flow whose vector field is , orbits are given by , and this flow is expressable in terms of Dixonian elliptic functions again (for the last two, see the Subsection 4.5).

Our main concern of this paper is the case 4); the last two cases will be covered in the Subsection 4.5. The case 4) is particularly interesting since it has an additional -fold symmetry, and now we will concentrate on it.

3.2. The -superflow

(For the explanation of the title, see the Subsection 5.1). So, we investigate the fascinating function which possesses these main properties.

  • (PDE). The first coordinate satisfies the partial differential equation


    with the boundary condition

  • (FLOW). The function satisfies the iterative functional equation

  • (SYMM). If , let . Consider the element subgroup of , call it , whose elements are


    then , . The function possesses the -fold symmetry: for every , we have

    Of course, involutions and generate the whole group , so only two of these invariance properties are independent. The invariance under tells us that , that is, the second coordinate is just the flip of the first, and the invariance under implies the identities

  • (ELL). In case are fixed, , the pair of functions

    parametrize the elliptic curve ; thus,

    It turns out that three exceptional lines , and correspond to three ramification points of the algebraic function ; namely, , and .

Figure 1. The vector field of the flow is given by . Here we show the normalized vector field, meaning that all arrows are adjusted to have the same length. A selected orbit shown is the elliptic curve A point travels this orbit with a convention that, for example, if it goes up via the left side of the top-right branch, it reappears on the left side of the bottom-right branch, and so on. So, we always mind the asymptote.

3.3. Basic properties of

The PDE (13) with the boundary condition (14) has, as already mentioned, the unique solution . In this subsection we will derive few computational results. As was proved in [Al12], we formally have


and the homogenic functions can be recurrently calculated by


here, as before, , . This recursion is essentially equivalent to the PDE (13). The above also holds for if we make a natural convention that . Thus, we can calculate these polynomials, as presented in the Table 1.