# Periodic solutions from Lie symmetries for the generalized Chen-Lee-Liu
equation

Andronikos Paliathanasis
Institute of Systems Science, Durban University of Technology
PO Box 1334, Durban 4000, Republic of South Africa
Instituto de Ciencias Fisicas y Matematicas,
Universidad Austral de Chile, Valdivia 5090000, Chile
Email<EMAIL_ADDRESS>
###### Abstract

The nonlinear generalized Chen-Lee-Liu 1+1 evolution equation which describes
the propagation of an optical pulse inside a monomode fiber is studied by
using the method of Lie symmetries and the singularity analysis. Specifically,
we determine the Lie point symmetries of the Chen-Lee-Liu equation and we
reduce the equation by using the Lie invariants in order to determine
similarity solutions. The solutions that we found have periodic behaviour and
describe optical solitons. Furthermore, the singularity analysis is applied in
order to write algebraic solutions of the Chen-Lee-Liu with the use of Laurent
expansions. The latter analysis support the result for the existence of
periodic behaviour of the solutions.

Keywords: Lie symmetries; similarity solutions; singularity analysis; optics;
Chen-Lee-Liu equation

##  1 Introduction

Lie symmetry analysis is a powerful method for the study of nonlinear
differential equations. The main feature of Sophus Lie approach is that the
existence of a symmetry vector for a given differential equation indicates the
existence of an invariant surface which can be applied for the construction of
a similarity transformation in order to simplify the differential equation
under the so-called reduction process [1, 2, 3, 4]. In addition, Lie
symmetries can be used to determine algebraic equivalent systems [5] and give
linearization criteria for nonlinear differential equations [6, 7, 8].
Furthermore, Lie symmetries can be applied to construct conservation laws [9,
10, 11]; to determine new solutions from old solutions [12] and many other
[4]. There are various important results in the literature on the application
of Lie’s theory in mathematical physics and applied mathematics [13, 14, 15,
16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26].

On the other hand, singularity analysis is an alternative way to study the
integrability of nonlinear differential equations The main requirement for the
singularity analysis is the existence of movable singularities in the
differential equation. Singularity analysis is associated with the French
school led by Painlevé [27, 28, 29] and their approach was actually inspired
by the successful application to the determination of the third integrable
case of Euler’s equations for a spinning top by Kowalevskaya [30]. For modern
approaches of the singularity analysis we refer the reader in [31, 32, 33, 34]
and references therein, while some applications on partial differential
equations can be found in [35, 36, 37].

In this work we are interesting on the determination of exact and analytic
solutions for the generalized Chen-Lee-Liu equation (gCLL) is [38]

$iq_{t}+\frac{1}{2}q_{xx}-\left|q\right|^{2}q+i\delta\left|q\right|^{2}q_{x}=0,$
(1)

with the application of the Lie’s theory and of the singularity analysis.
Equation (1) under the Madelung transformation
$q\left(t,x\right)=\rho\left(t,x\right)\exp\left(i\int
u\left(t,x^{\prime}\right)dx^{\prime}\right)$ the latter equation can be
written as the following system of evolution equations

$\displaystyle\rho_{t}+\left(\rho u+\frac{1}{2}\delta\rho^{2}\right)_{x}$
$\displaystyle=$ $\displaystyle 0,$ (2) $\displaystyle
u_{t}+uu_{x}+\rho_{x}+\delta\left(\rho
u\right)_{x}+\left(\frac{\rho_{x}^{2}}{8\rho^{2}}-\frac{\rho_{xx}}{4\rho}\right)_{x}$
$\displaystyle=$ $\displaystyle 0,$ (3)

in which $\rho\left(t,x\right)$ is the intensity variable and chirp variables,
while the positive parameter $\delta\ $is associated with the self-steepening
phenomena [39]. The nonlinear evolution equation, has been proposed for the
description of an optical pulse inside a monomode fiber. For other
applications of the equation (1) we refer the reader in [40]. The solitons
provided by the gCLL equation has been found to be essential for the
description of phenomena in optical fiber theory [38, 41, 42, 43], while some
experimental evidences are presented in [44],

In the following Sections we determine the Lie point symmetries for the real
function system (2), (3). For the admitted Lie point symmetries we constructed
the algebraic structure and we find the one-dimensional optimal system.
Moreover, we find all possible similarity transformations which are used to
reduce the differential equation and write the equivalent system. We find four
independent similarity solutions. In particular, we determine static
solutions, stationary solutions, travel-wave solutions and scaling solutions.
We observe that the solution are periodic in the similarity variables which
indicate the existence of kink solutions in the original variables.
Furthermore, we apply the singularity analysis such that to write the analytic
solution of the resulting system with the use of Laurent expansions. While the
original system (2), (3) is investigated if possess the Painlevé property. At
this point it is important to mention some previous studies of Lie’s theory on
optical physics [45, 46, 47], while only recently the algebraic properties of
the Chen-Lee-Liu were studied in [48]. The plan of the paper is as follows.

In Section 2, we determine the Lie symmetries and the similarity
transformations for the gCLL equations, while we construct the similarity
solutions. In Section 3 we show that the gCLL equation possess the Painlevé
property and we write the generic solution by using a Right Painlevé Series.
Finally, in Section 4 we discuss our results and we draw our conclusions. In
Appendices A, B and C we present the main mathematical theory of the tools
that are applied in this work.

##  2 Lie symmetries and similarity transformations

For the system of the 1+1 evolution partial differential equations (2), (3) we
apply the Lie theory in order to determine the generator [4, 1, 2]

$X=\xi^{t}\left(t,x,\rho,u\right)\partial_{t}+\xi^{x}\left(t,x,\rho,u\right)\partial_{x}+\eta^{\rho}\left(t,x,\rho,u\right)\partial_{\rho}+\eta^{u}\left(t,x,\rho,u\right)\partial_{u},$
(4)

of the infinitesimal one parameter point transformation

$t^{\prime}=t+\varepsilon\xi^{t}\left(t,x,\rho,u\right)~{},~{}x^{\prime}=x+\varepsilon\xi^{x}\left(t,x,\rho,u\right)~{},$
(5)
$\rho^{\prime}=\rho+\varepsilon\eta^{\rho}\left(t,x,\rho,u\right)~{},~{}u^{\prime}=u+\varepsilon\eta^{u}\left(t,x,\rho,u\right)~{},$
(6)

which keeps invariant the system (2), (3).

The possible generators are derived to be the following three

$X_{1}=\partial_{t}~{},~{}X_{2}=\partial_{x}~{}\text{and
}X_{3}=2t\partial_{t}+\left(\delta
x-t\right)\partial_{x}-\rho\partial_{\rho}-\left(1+u\delta\right)\partial_{u}.$
(7)

The latter Lie point symmetries form the $A_{3,2}$ Lie algebra in the Morozov-
Mubarakzyanov Classification Scheme [49, 50, 51, 52]. The commutators and the
adjoint representation of the admitted Lie point symmetries are presented in
Tables 1 and 2.

In order to proceed with the derivation of all the possible independent
similarity transformations which reduce the system (2), (3) we should
determine the one-dimensional optimal system. Straightforward, from Table 2 we
find the one-dimensional optimal system [3]

$\left\\{X_{1}\right\\}~{},~{}\left\\{X_{2}\right\\}~{},~{}\left\\{X_{1}+\alpha
X_{2}\right\\}\text{ and }\left\\{X_{3}\right\\}\text{.}$ (8)

Thus we continue our analysis by applying the Lie point symmetries in order to
reduce the system of partial differential equations (2), (3) into a system of
ordinary differential equations.

Table 1: Commutators of the admitted Lie symmetries $\left[\mathbf{X}_{I},\mathbf{X}_{J}\right]$ | $\mathbf{X}_{1}$ | $\mathbf{X}_{2}$ | $\mathbf{X}_{3}$
---|---|---|---
$\mathbf{X}_{1}$ | $0$ | $0$ | $2X_{1}-X_{2}$
$\mathbf{X}_{2}$ | $0$ | $0$ | $X_{2}$
$\mathbf{X}_{3}$ | $\,-2X_{1}+X_{2}$ | $-2X_{2}$ | $0$
Table 2: Adjoint representation of the admitted Lie algebra $Ad\left(e^{\left(\varepsilon\mathbf{X}_{i}\right)}\right)\mathbf{X}_{j}$ | $\mathbf{X}_{1}$ | $\mathbf{X}_{2}$ | $\mathbf{X}_{3}$
---|---|---|---
$\mathbf{X}_{1}$ | $X_{1}$ | $X_{2}$ | $2\varepsilon\delta X_{1}-\varepsilon X_{2}+X_{3}$
$\mathbf{X}_{2}$ | $X_{1}$ | $X_{2}$ | $\varepsilon\delta X_{2}+X_{3}$
$\mathbf{X}_{3}$ | $e^{-2\delta\varepsilon}X_{1}-\frac{e^{-\delta\varepsilon}\left(e^{-\delta\varepsilon}-1\right)}{\delta}X_{2}$ | $e^{-\delta\varepsilon}X_{2}$ | $X_{3}$

###  2.1 Similarity transformations

We proceed with the application of the similarity transformations.

####  2.1.1 Lie symmetry $X_{1}$

The application of the Lie point symmetry $X_{1}$ leads to the static solution
$u=u\left(t\right),~{}\rho=\rho\left(t\right)$ where the reduced equations are
determined to be

$\left(\rho u+\frac{1}{2}\delta\rho^{2}\right)_{,x}=0~{},$ (9)
$uu_{x}+\rho_{x}+\delta\left(\rho
u\right)_{x}+\left(\frac{\rho_{x}^{2}}{8\rho^{2}}-\frac{\rho_{xx}}{4\rho}\right)_{x}=0~{},$
(10)

or equivalently

$\rho u+\frac{1}{2}\delta\rho^{2}=c_{1},$ (11)
$\frac{1}{2}u^{2}+\rho+\delta\left(\rho
u\right)+\left(\frac{\rho_{x}^{2}}{8\rho^{2}}-\frac{\rho_{xx}}{4\rho}\right)=c_{2}~{},$
(12)

where now $c_{1},~{}c_{2}$ are two integration constants.

Hence, from (10) it follows

$u=-\frac{1}{2}\delta\rho^{2}+\frac{c_{1}}{\rho},$ (13)

that is, equation (12) reads

$\varrho_{xx}+\frac{3}{4}\delta^{2}\varrho^{5}-2\varrho^{3}+C_{2}\varrho-\frac{c_{1}^{2}}{\varrho^{3}}=0~{},$
(14)

where $\rho=\varrho^{2}\left(x\right)$ and $C_{2}=\left(\frac{c_{2}}{2}-\delta
c_{1}\right)$.

Equation (14) admits one Lie point symmetry, the vector field $X_{2}$, which
is a reduced symmetry vector. Moreover, equation (14) can be integrated by
quadratures, that is

$\frac{1}{2}\left(\varrho_{x}\right)^{2}-\frac{\delta}{8}\varrho^{6}-\frac{1}{4}\varrho^{4}+\frac{1}{2}\left(\frac{c_{2}}{2}-\delta
c_{1}\right)\varrho^{2}+\frac{c_{1}^{2}}{2\varrho^{2}}=c_{3}~{},$ (15)

where $c_{3}$ is a third integration constant.

In Fig. 1 we present the phase portrait of equation (14) from where it is
clear that there are attractors in the dynamical system which provide a
periodic behaviour.

Figure 1: Phase portrait for equation (14) for various values of the free
parameters $\left\\{\delta,c_{1},C_{2}\right\\}$.

In order to write the analytic solution of equation (14) and understand the
periodic behaviour of the solution, we apply the singularity analysis.
Specifically we apply the ARS algorithm. We search for the singular behaviour
$\varrho\left(x\right)=\varrho_{0}x^{p}$ from where we follows that the
leading-order behaviour has $p=-\frac{1}{2}$ and
$\left(\varrho_{0}\right)^{2}=\pm\frac{i}{\delta},$ it is clear from here that
a periodic behaviour will follow.

The second step of the ARS algorithms is the determination of the resonances,
we do that by replacing
$\varrho\left(x\right)=\varrho_{0}x^{-\frac{1}{2}}+mx^{-\frac{1}{2}+S}$ in
(14) and linearize around $m=0$, it follows the algebraic equation
$\left(S+1\right)\left(S-3\right)=0$, which gives the resonances $S=-1$ and
$S=3$.

Finally we write the Laurent expansion

$\varrho\left(x\right)=\varrho_{0}x^{-\frac{1}{2}}+\varrho_{1}x^{\frac{1}{2}}+\varrho_{2}x^{\frac{3}{2}}+...~{},$
(16)

from where we test that it is a solution of equation (12) with integration
constant the $\varrho_{3}$ and

$\left(\varrho_{0}\right)^{2}=\pm\frac{i}{\delta}~{},~{}\varrho_{1}=-\frac{\left(-1\right)^{\frac{3}{4}}}{2\delta^{\frac{3}{2}}}~{},~{}\varrho_{2}=-\frac{\left(-1\right)^{1/4}\left(9-8\bar{C}_{2}\delta\right)}{24\delta^{\frac{5}{2}}},~{}...~{}.$

We remark that the second integration constant is the position of the
singularity $x_{0}$.

####  2.1.2 Lie symmetry $X_{2}$

Reduction with respect to the vector field $X_{2}$ provides the stationary
solutions $u=u\left(t\right),~{}\rho=\rho\left(t\right)$ where $u_{t}=0$ and
$\rho_{t}=0$, that is $u\left(t,x\right)=u_{0},~{}\rho\left(t,x\right)=u_{0}$.

####  2.1.3 Lie symmetry $X_{1}+\alpha X_{2}$

The application of the Lie symmetry vector $X_{1}+\alpha X_{2}$ provides a
travel-wave solution. Indeed we find
$u=u\left(z\right),~{}\rho=\varrho^{2}\left(z\right)$ with $z=x-\alpha t$,
where now the reduced system is

$\left(u-\alpha\right)\varrho^{2}+\frac{1}{2}\delta\varrho^{4}=c_{1}~{},$ (17)
$\varrho_{zz}+2\left(\alpha-\delta\varrho^{2}\right)\rho
u-u^{2}\varrho-\varrho^{3}+c_{2}\varrho=0~{},$ (18)

where $c_{1},~{}c_{2}$ are two integration constants.

With the use of (17), equation (18) reads as

$\varrho_{zz}+\frac{3}{4}\delta^{2}\varrho^{5}-2\left(1+\alpha\delta\right)\varrho^{3}+\bar{C}_{2}\varrho-\frac{c_{1}^{2}}{\varrho^{3}}=0~{},$
(19)

with $\bar{C}_{2}=\left(c_{2}-\alpha-\delta c_{1}\right)$.

Equation (19) is of the form of the static solution (14) and can be integrated
by quadratures, the only main difference is the coefficient of the
$\varrho^{3}$, where now it is $\left(1+\alpha\delta\right)$. The latter
quantity can be positive or negative, either if the travel wave travel of the
left to the right or from the right to the left.

In Fig. 2 we present the phase portrait for the equation (19) and
$\left(1+\alpha\delta\right)<0,$ specifically for
$\left(1+\alpha\delta\right)=-1$, and for the rest of the constants to have
the same values as in Fig. 1. Again we observe a similar behaviour and the
existence of periodic solutions which correspond to travel-wave solutions.

Figure 2: Phase portrait for equation (19) for various values of the free
parameters $\left\\{\delta,c_{1},\bar{C}_{2}\right\\}$ and
$\left(1+\alpha\delta\right)=-1$.

As in the case of the static solution, the application of the singularity
analysis provides the analytic solution expressed by the Laurent expansion

$\varrho\left(x\right)=\varrho_{0}x^{-\frac{1}{2}}+\varrho_{1}x^{\frac{1}{2}}+\varrho_{2}x^{\frac{3}{2}}+...~{},$
(20)

where now

$\left(\varrho_{0}\right)^{2}=\pm\frac{i}{\delta}~{},~{}\varrho_{1}=-\frac{\left(-1\right)^{\frac{3}{4}}}{2\delta^{\frac{3}{2}}}\left(1+\alpha\delta\right)~{},~{}\varrho_{2}=-\frac{\left(-1\right)^{1/4}\left(9\left(1+\alpha\delta\right)^{2}-8\bar{C}_{2}\delta\right)}{24\delta^{\frac{5}{2}}},~{}...~{}.$
(21)

and $\varrho_{3}$ is an arbitrary constant.

In Fig. 3 we present the numerical solution of (19) where we observe the
existence of travel-waves, i.e. a kink solutions, for the gCLL equation.

Figure 3: Numerical simulation of equation (25) for various values of the free
parameters $\left\\{c_{1},C_{2},\left(1+\alpha\delta\right)\right\\}$ and
initial conditions $\varrho\left(0\right)=1$ and
$\varrho_{z}\left(0\right)=0.$ The plots of the first row are for (left to
right)
$\left\\{1,1,0.5\right\\},~{}\left\\{1,1,1\right\\},~{}\left\\{1,1,3\right\\}$,
the plots of the second row are for
$\left\\{2,0,0.5\right\\},~{}\left\\{2,0,1\right\\}$, $\left\\{2,0,3\right\\}$
and the plots of the third row are for
$\left\\{0.1,0.1,0.5\right\\},~{}\left\\{0.1,0.1,1\right\\}$ and
$\left\\{0.1,0.1,3\right\\}$.

####  2.1.4 Lie symmetry $X_{3}$

We proceed with the application of the similarity transformation provided by
the scaling symmetry $X_{3}$. The similarity transformation is

$u\left(t,x\right)=-\frac{1}{\delta}+\frac{1}{\sqrt{t}}U\left(\sigma\right)~{},~{}\rho\left(t,x\right)=\frac{1}{\sqrt{t}}R^{2}\left(\sigma\right)~{}\text{with}~{}\sigma=\left(\frac{x}{\sqrt{t}}+\frac{\sqrt{t}}{\delta}\right)\text{.}$
(22)

Therefore, the reduced system after the application of the latter similarity
transformation is derived to be

$2RU-\sigma R+\delta R^{2}=c_{1},$ (23)
$R_{\sigma\sigma}-U^{2}R+\left(\sigma-2\delta R^{2}\right)UR+c_{2}R=0,$ (24)

where $c_{1},~{}c_{2}$ are two integration constants.

Therefore, we end with only one differential equation

$R_{\sigma\sigma}+\frac{3}{4}\delta^{2}R^{5}-2\left(1+\alpha\delta\right)R^{3}+\left(C_{2}+\frac{\sigma^{2}}{4}\right)R-\frac{c_{1}^{2}}{R^{3}}=0$
(25)

where now $C_{2}=\left(\frac{c_{2}}{2}-\delta c_{1}\right)$. The main
difference with the previous reduction is that the coefficient of the linear
term is not a constant.

Easily, we observe that the resulting solution is again periodic solution but
in this case around a central which moves. In Fig. 4 we present numerical
simulation for equation (25) for the initial condition
$R\left(0\right)=1~{},~{}R_{\sigma}\left(0\right)=0\,\ $and for various values
of the free parameters
$\left\\{c_{1},C_{2},\left(1+\alpha\delta\right)\right\\}$, from the figures
it is clear the periodic behaviour of the scaling solution, that is nothing
else than a kink solution.

For small values of $R$, the dominant terms are the

$R_{\sigma\sigma}+\left(C_{2}+\frac{\sigma^{2}}{4}\right)R-\frac{c_{1}^{2}}{R^{3}}=0$
(26)

which is nothing else than the Ermakov-Pinney equation [58, 59, 60].

Figure 4: Numerical simulation of equation (25) for various values of the free
parameters $\left\\{c_{1},C_{2},\left(1+\alpha\delta\right)\right\\}$ and
initial conditions $R\left(0\right)=1$ and $R_{\sigma}\left(0\right)=0.$ The
plots of the first row are for (left to right)
$\left\\{1,1,0.5\right\\},~{}\left\\{1,1,1\right\\},~{}\left\\{1,1,3\right\\}$,
the plots of the second row are for
$\left\\{2,0,0.5\right\\},~{}\left\\{2,0,1\right\\}$, $\left\\{2,0,3\right\\}$
and the plots of the third row are for
$\left\\{0.1,0.1,0.5\right\\},~{}\left\\{0.1,0.1,1\right\\}$ and
$\left\\{0.1,0.1,3\right\\}$.

##  3 Singularity analysis

Let us now apply the ARS algorithm to study if the system (2), (3) possess the
Painlevé property.

In order to determine the singular behaviour we replace in (2), (3) the
following expression

$\rho\left(t,x\right)=\rho_{0}\left(t,x\right)\phi\left(t,x\right)^{p}~{},~{}u\left(t,x\right)=u_{0}\left(t,x\right)\phi\left(t,x\right)^{q}$

where $\phi\left(t,x\right)$ is a singular function. Hence, it follows that
the leading order terms provide $\left\\{p,q\right\\}=\left(-1,-1\right)$,
where

$\rho_{0}\left(t,x\right)=-\frac{1}{\delta^{2}}\phi_{x}^{2}~{},~{}u_{0}\left(t,x\right)=-\frac{1}{4}\phi_{x}^{2}\text{.}$

For the second step of the ARS algorithm, we determine the resonances which
are $\left\\{s_{1},s_{2},s_{3},s_{4}\right\\}=\left\\{-1,2,2,3\right\\}$ from
where we conclude that the solution is given by Right Painlevé Series. Indeed
the solutions is expressed as

$\rho\left(t,x\right)=\rho_{0}\left(t,x\right)\phi\left(t,x\right)^{-1}+\rho_{1}\left(t,x\right)+\rho_{2}\left(t,x\right)\phi\left(t,x\right)+\rho_{3}\left(t,x\right)\phi\left(t,x\right)^{3}+...~{},$
(27)
$u\left(t,x\right)=u_{0}\left(t,x\right)\phi\left(t,x\right)^{-1}+u_{1}\left(t,x\right)+u_{2}\left(t,x\right)\phi\left(t,x\right)+u_{3}\left(t,x\right)\phi\left(t,x\right)^{3}+...~{}.$
(28)

where the consistency test gives that functions
$\rho_{2}\left(t,x\right),~{}\rho_{3}\left(t,x\right)$ and
$u_{2}\left(t,x\right)$ are arbitrary.

We conclude that the generalized Chen-Lee-Liu equation possess the Painlevé
property and its solution is expressed by Right Laurent expansions. The latter
expressions can describe Kink solutions for specific initial conditions.

##  4 Conclusion

In this work we determined exact and analytic periodic solutions for the gCLL
equation with the use of Lie symmetries. The gCLL equation admits a three-
dimensional Lie algebra, which leads to four different similarity
transformations. In particular we found static, stationary, travel-wave and
scaling similarity solutions.

Except from the stationary solution, which is not of interest, the other
solutions are periodic solutions. It is interesting that in all cases for
small values of the intensity variable $\rho\left(t,x\right)$ the gCLL is
reduced to the Ermakov-Pinney equation, which is a well-known integrable
system.

Moreover, we investigated if the gCLL possess the Painlevé property, for that
we applied the ARS algorithm where we were able to write the analytic solution
of the gCLL with the use of Right Painlevé Series.

The periodic solutions, provided by the similarity transformations, are direct
related with the existence of dark and bright solitons for the nonlinear
differential equation [61, 62, 63]. Optical solitons are exact solutions of
mathematical models with direct applications in the information transfer in
optical fibers [64]. As far the results or our analysis, are concerned, the
travel-wave solution which was found before for the gCLL describes a 1-soliton
solution known as kink solution. Furthermore, the scaling solution is also a
kink solution, where now the amplitude of the oscillation is not a constant.

The determination of these new kink solutions is essential for the physical
viability of the model. A study of the properties of the kink solutions and
their real world applications extends the scopus of this study and will be
performed in a future study.

##  Appendix A Lie symmetries

We briefly discuss the main definition and algorithm for the determination of
Lie point symmetries. Consider the one-dimensional parameter point
transformation

$\displaystyle\bar{x}^{k}$
$\displaystyle=x^{k}+\epsilon\xi^{i}\left(x^{k},u\right),$ (29)
$\displaystyle\bar{\eta}$
$\displaystyle=\eta+\epsilon\eta\left(x^{k},u\right).$ (30)

with generator
$X=\xi^{i}\left(x^{k},u\right)\partial_{i}+\eta\left(x^{k},u\right)\partial_{u}$,
then the differential equation
$\mathbf{H}\left(x^{k},u,u_{i},u_{ij},...,u_{i_{1}i_{2}...i_{n}}\right)$ is
invariant under the action of the one parameter point transformation if and
only if

$\lim_{\varepsilon\rightarrow
0}\frac{\mathbf{\bar{H}}\left(\bar{x}^{k},\bar{u},\bar{u}_{i},\bar{u}_{ij},...,\bar{u}_{i_{1}i_{2}...i_{n}}\right)-\mathbf{H}\left(x^{k},u,u_{i},u_{ij},...,u_{i_{1}i_{2}...i_{n}}\right)}{\varepsilon}=0,$

or equivalently, if there exists a function $\lambda$ such that the following
condition to be true [4, 1, 2]

$X^{\left[n\right]}\mathbf{H}-\lambda\mathbf{H=0}$ (31)

where $X^{\left[n\right]}$ is called the n-th prolongation/extension of
$X~{}$in the jet-space defined as

$X^{\left[n\right]}=X+\left(D_{i}\eta-
u_{,k}D_{i}\xi^{k}\right)\partial_{u_{i}}+\left(D_{i}\eta_{j}^{\left[i\right]}-u_{jk}D_{i}\xi^{k}\right)\partial_{u_{ij}}+...+\left(D_{i}\eta_{i_{1}i_{2}...i_{n-1}}^{\left[i\right]}-u_{i_{1}i_{2}...k}D_{i_{n}}\xi^{k}\right)\partial_{u_{i_{1}i_{2}...i_{n}}}.$
(32)

If $X$ is a symmetry vector for the differential equation $\mathbf{H}$, then
we can always find a coordinate transformation such that the symmetry vector
to be written in the canonical coordinates, i.e. $X=\partial_{x^{n}}$, where
the differential equation is

$\mathbf{H=H}\left(x^{\mu},u,u_{i},u_{ij},...,u_{i_{1}i_{2}...i_{n}}\right)~{},~{}\mu\neq
n,$

clear from the last expression it follows $\partial_{x^{\mu}}H=0$. The
coordinate transformation which leads to the canonical coordinates it is
called similarity transformation and it is mainly applied for the reduction of
the differential equation,

##  Appendix B One-dimensional optimal system

Let a given differential equation admit as Lie symmetries the elements
$\left\\{X_{1},~{}X_{2},~{}...~{}X_{n}\right\\}$ of the $n$-dimensional Lie
algebra $G_{n}$ with structure constants $C_{jk}^{i}$. The two symmetry
vectors $Z,~{}W$ defined as

$Z=\sum_{i=1}^{n}a_{i}X_{i}~{},~{}W=\sum_{i=1}^{n}b_{i}X_{i}~{},~{}\text{\
}a_{i},~{}b_{i}\text{ are constants.}$ (33)

we shall say that are equivalent if and only if [3]

$\mathbf{W}=\sum_{j=i}^{n}Ad\left(\exp\left(\epsilon_{i}X_{i}\right)\right)\mathbf{Z}$
(34)

or

$W=cZ~{},~{}c=const\text{ that is }b_{i}=ca_{i}\text{.}$ (35)

Operator $Ad\left(\exp\left(\epsilon X_{i}\right)\right)X_{j}$ defined as

$Ad\left(\exp\left(\epsilon
X_{i}\right)\right)X_{j}=X_{j}-\epsilon\left[X_{i},X_{j}\right]+\frac{1}{2}\epsilon^{2}\left[X_{i},\left[X_{i},X_{j}\right]\right]+...,$
(36)

is called the adjoint representation.

The determination of all the one-dimensional subalgebras of $G_{n}$ which are
not related through the adjoint representation it is necessary in order to
perform a complete classification of all the possible similarity
transformations, i.e. similarity solutions, for a given differential equation.
This classification is known as the one-dimensional optimal system.

##  Appendix C Singularity analysis

The development of the Painlevé Test for the determination of integrability of
a given equation or system of equations and its systematization been
succinctly summarized by Ablowitz, Ramani and Segur in the so-called ARS
algorithm [53, 54, 55]. The ARS algorithm is constructed by three basic steps,
they are: (a) determine the leading-order term which describes the behaviour
of the solution near the singularity, (b) find the position of the resonances
which shows the existence and the position of the integration constants and
(c) write a Laurent expansion with leading-order term determined in the first
step in order to perform the consistency test and the solution, for a review
on the ARS algorithm and various applications we refer the reader in [56],
while in [57] a discussion between the Lie’s approach and the singularity
analysis is given.

## References

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# A MacDonald Formula for Zeta Functions of Varieties Over Finite Fields

Jonathan A. Huang  Department of Mathematics and Computer Science, Wesleyan
University, Middletown, CT 06459-0128<EMAIL_ADDRESS>
(Date: January 11, 2018)

###### Abstract.

We provide a formula for the generating series of the zeta function $Z(X,t)$
of symmetric powers $\operatorname{Sym}^{n}X$ of varieties over finite fields.
This realizes $Z(X,t)$ as an exponentiable motivic measure whose associated
Kapranov motivic zeta function takes values in $W(R)$ the big Witt ring of
$R=W(\mathbb{Z})$. We apply our formula to compute
$Z(\operatorname{Sym}^{n}X,t)$ in a number of explicit cases. Moreover, we
show that all $\lambda$-ring motivic measures have zeta functions which are
exponentiable. In this setting, the formula for $Z(X,t)$ takes the form of a
MacDonald formula for the zeta function.

###### Key words and phrases:

Grothendieck ring of varieties, motivic measures, Kapranov motivic zeta
function, Witt vectors, $\lambda$-ring.

###### 2010 Mathematics Subject Classification:

11S40,13F35,14C15

##  1\. Introduction

A remarkable formula of MacDonald [8] provides a closed expression for the
generating series of the Poincaré polynomial of the symmetric powers
$\operatorname{Sym}^{n}X$ of a space $X$. Let $X$ be a compact complex
manifold of dimension $m$, and recall the Poincaré polynomial is defined as
$P(X,z)=\sum_{i}(-1)^{i}b_{i}(X)z^{i}$, where
$b_{i}=b_{i}(X)=\dim_{\mathbb{Q}}H^{i}(X,\mathbb{Q})$ are the Betti numbers.
The MacDonald formula for $P(X,z)$ is

$\sum_{n\geqslant
0}P(\operatorname{Sym}^{n}X,z)t^{n}=\frac{(1-z^{1}t)^{b_{1}}(1-z^{3}t)^{b_{3}}\cdots(1-z^{2m-1}t)^{b_{2m-1}}}{(1-t)^{b_{0}}(1-z^{2}t)^{b_{2}}\cdots(1-z^{2m}t)^{b_{2m}}}.$
(1)

Thus the Poincaré polynomial $P(\operatorname{Sym}^{n}X,z)$ may be expressed
directly in terms of invariants associated to $X$. A similar formula for the
Euler characteristic is recovered by setting $z=1$ as $P(X,1)=\chi(X)$.

In this short note we prove an analogous formula for the Hasse-Weil zeta
function $Z(X,t)$ of varieties over finite fields. To provide such a formula,
we consider the zeta function $Z(X,t)$ as an element in the Witt ring
$W(\mathbb{Z})$. As noted by Ramachandran [12], $Z(X,t)$ takes form of an
Euler-Poincaré characteristic as an alternating sum of _Teichmüller elements_
$[\alpha]$:

$Z(X,t)=\sum_{i,j}(-1)^{i+1}[\alpha_{ij}]\in W(\overline{\mathbb{Q}_{\ell}})$

where $\alpha_{ij}$ is the $j$th (inverse) eigenvalue of the Frobenius
operator acting on the $i$th étale cohomology. By definition, the zeta
function $Z(X,t)$ takes values in the ring $W(\mathbb{Z})\subset
W(\overline{\mathbb{Q}_{\ell}})$.

Our main result is a MacDonald formula for $Z(X,t)$:

######  Theorem 1.1.

Given a quasi-projective variety $X\in\operatorname{Var}_{\mathbb{F}_{q}}$, we
have

$\sum_{n\geqslant
0}Z(\operatorname{Sym}^{n}X,t)u^{n}=\sum_{i,j}(-1)^{i+1}[[\;[\alpha_{ij}]\;]]\in
W(W(\overline{\mathbb{Q}_{\ell}}))$

where the $[[\;[\alpha_{ij}]\;]]$ are double Teichmüller elements.

We interpret this result in the setting of motivic zeta functions,
specifically in the context of $\lambda$-ring-valued motivic measures. Let
$K_{0}(\operatorname{Var}_{k})$ be the Grothendieck ring of varieties and
consider an $A$-valued motivic measure
$\mu:K_{0}(\operatorname{Var}_{k})\rightarrow A$. In [5], Kapranov associates
to $\mu$ a motivic zeta function

$\zeta_{\mu}(X,t)=\sum_{n\geqslant 0}\mu(\operatorname{Sym}^{n}X)t^{n},$

an invertible power series with coefficients in $A$. Following Ramachandran
[12], we say that $\mu$ _exponentiates_ or _can be exponentiated_ if
$\zeta_{\mu}$ takes values in the big Witt ring $W(A)$. That is, if the zeta
function $\zeta_{\mu}$ reflects the product structure in
$K_{0}(\operatorname{Var}_{k})$ via the Witt product in $W(A)$. A particularly
interesting case is that of $A$-valued motivic measures where $A$ is a
$\lambda$-ring. It is shown by Ramachandran and Tabuada [13] that if such a
measure is a pre-$\lambda$-ring map on $K_{0}(\operatorname{Var}_{k})$, then
it exponentiates (see Proposition 5.2). We refer to such measures $\mu$ as
$\lambda$-measures.

The zeta function of such a measure is a ring homomorphism into $W(A)$ and
thus induces a zeta function measure $\mu_{Z}([X])=\zeta_{\mu}(X,t)$. We prove
that if $\mu$ is a $\lambda$-measure, then $\mu_{Z}$ is also a
$\lambda$-measure (Theorem 5.6); this involves a study of the $\lambda$-ring
structure of the big Witt ring $W(A)$. The classical MacDonald formulae show
that the Euler characteristic and Poincaré polynomial define
$\lambda$-measures. The formula in Theorem 1.1 shows that the zeta function
$Z(X,t)$ for varieties over finite fields defines a $\lambda$-measure.

Analogous MacDonald formulae can be proved in other contexts. Let
$\mathcal{C}$ be a $k$-linear tensor category, in the sense of a $k$-linear
additive pseudo-abelian symmetric monoidal category where $\otimes$ is
$k$-linear. As shown by Heinloth [4], the Grothendieck ring
$K_{0}(\mathcal{C})$ is a $\lambda$-ring. Del Baño Rollin [2] and Maxim-
Schürmann [10] prove examples of closed formulae for various generating series
of measures taking values in various $K(\mathcal{C})$; these follow directly
from the opposite $\lambda$-ring structure map $\sigma_{t}$. The motivic
measures described in [2] and [10] are also $\lambda$-measures.

After reviewing some preliminaries on $\lambda$-rings and Witt rings in
Section 2, we prove the formula for zeta functions in Section 3 and provide
some explicit computations of zeta functions of various
$\operatorname{Sym}^{n}X$ in Section 4. Finally in Section 5, we provide some
context for these results in the theory of $\lambda$-ring-valued motivic
measures and explore related results for motivic zeta function measures.

### Acknowledgements

The author would like to thank Niranjan Ramachandran for suggesting this
project and for eagerly and openly sharing his ideas and insights. The author
is also indebted to the Department of Mathematics at the University of
Maryland, College Park—in particular to helpful discussions with and
encouragement from Larry Washington and Jonathan Rosenberg.

##  2\. Lambda rings and Witt rings

We begin by describing the $\lambda$-ring structure on the big ring of Witt
vectors. Our presentation here of the big Witt ring follows Bloch [1] and
Ramachandran [12]. For a more on $\lambda$-rings, the reader may consult
Knutson [6], Yao [14], and Hazewinkel [3].

Let $A$ be a commutative ring with identity. Denote by $\Lambda(A)$ the group
of invertible power series $(1+tA[[t]],\times)$ under usual multiplication of
power series. For a ring homomorphism $f:A\rightarrow A^{\prime}$, let
$\Lambda_{f}:\Lambda(A)\rightarrow\Lambda(A^{\prime})$ be the obvious induced
map. Recall the ghost map $\operatorname{gh}:\Lambda(A)\rightarrow
A^{\mathbb{N}}$ is defined as, for $P\in\Lambda(A)$,

$\operatorname{gh}(P)=(b_{1},b_{2},\ldots)\quad\textrm{where}\quad\frac{t}{P}\frac{dP}{dt}=\sum_{n=1}^{\infty}b_{n}t^{n}.$

This represents the power series $P(t)$ by a series of coordinates
$(b_{1},b_{2},\ldots)$ called ghost coordinates
$\operatorname{gh}_{n}(P)=b_{n}$. The ghost map is a functorial group
homomorphism where $A^{\mathbb{N}}$ has a pointwise addition. Moreover, the
relation between the power series coefficients and ghost coordinates can be
made explicit:

######  Lemma 2.1.

For $P(t)=\sum_{n}a_{n}t^{n}\in\Lambda(A)$ and
$b_{n}=\operatorname{gh}_{n}(P(t))$, we have

$na_{n}=b_{n}+a_{1}b_{n-1}+\cdots a_{n-1}b_{1}.$

This relation uniquely determines $P(t)$ in terms of its ghost coordinates
$b_{n}$ in the case where $A$ has no $\mathbb{Z}$-torsion.

###### Proof.

By definition, $\dfrac{t}{P}\dfrac{dP}{dt}=\sum_{n=1}^{\infty}b_{n}t^{n}$ so
that

$\displaystyle P(t)\left(\sum_{n=1}^{\infty}b_{n}t^{n}\right)$
$\displaystyle=$ $\displaystyle t\frac{dP}{dt}$
$\displaystyle(1+a_{1}t+a_{2}t^{2}+\cdots)(b_{1}t+b_{2}t^{2}+\cdots)$
$\displaystyle=$ $\displaystyle t(a_{1}+2a_{2}t+3a_{3}t^{2}\cdots)$
$\displaystyle=$ $\displaystyle a_{1}t+2a_{2}t^{2}+3a_{3}t^{3}+\cdots$

and we simply identify coefficients. ∎

The big Witt ring $W(A)$ (also called the universal ring of Witt vectors) is
the ring whose underlying additive group is $\Lambda(A)$ and whose
multiplication $*_{W}$ is such that

$(1-at)^{-1}*_{W}(1-bt)^{-1}=(1-abt)^{-1}\quad\textrm{for }a,b\in A$

and the association $A\mapsto W(A)$ is functorial. This suffices to define a
commutative ring structure on $W(A)$. The elements $[a]=(1-at)^{-1}$ above are
called Teichmüller elements.

The ghost map $\operatorname{gh}:W(A)\rightarrow A^{\mathbb{N}}$ is a ring
homomorphism where $A^{\mathbb{N}}$ has a pointwise product. That is, for
$P,Q\in W(Z)$,
$gh_{n}(P*_{W}Q)=\operatorname{gh}_{n}(P)\operatorname{gh}_{n}(Q)$ and thus in
the case where $A$ has no $\mathbb{Z}$-torsion, multiplication is determined
by pointwise product of ghost coordinates. Note that for Teichmüller elements,
$\operatorname{gh}_{n}([a])=a^{n}$.

For each positive integer $n$, the Frobenius map
$\operatorname{Fr}_{n}:W(A)\rightarrow W(A)$ is the ring endomorphism defined
as

$\operatorname{Fr}_{n}(P(t))=\prod_{\zeta^{n}=1}P(\zeta t^{1/n}).$

On Teichmüller elements $\operatorname{Fr}_{n}([a])=[a^{n}]$ and on ghost
coordinates, $\operatorname{gh}_{m}(Fr_{n}(P))=\operatorname{gh}_{mn}(P)$.
Clearly, we have
$\operatorname{Fr}_{n}\circ\operatorname{Fr}_{m}=\operatorname{Fr}_{nm}$.

Recall that a pre-$\lambda$-ring $(A,\lambda_{t})$ is a commutative ring with
identity $A$ equipped with a group homomorphism called the structure map
$\lambda_{t}:A\rightarrow\Lambda(A)$, where the coefficients are denoted as
$\lambda_{t}(a)=\sum_{n\geqslant 0}\lambda^{n}(a)t^{n}$. Note in particular
that $\lambda^{0}(a)=1$ and
$\lambda^{n}(a+b)=\sum_{i+j=n}\lambda^{i}(a)\lambda^{j}(b)$. Here,
$\Lambda(A)$ is a pre-$\lambda$-ring with multiplication and a canonical
structure map defined by certain universal polynomials (see Knutson [6, p.
13]). A map of pre-$\lambda$-rings $f:A\rightarrow A^{\prime}$ is a ring
homomorphism respecting the $\lambda$-ring structure maps. If
$\lambda_{t}:A\rightarrow\Lambda(A)$ is a ring homomorphism, then $A$ is
called a $\lambda$-ring; a $\lambda$-ring map is a map of pre-$\lambda$-rings.
$\Lambda(A)$ is a $\lambda$-ring.

The Witt ring $W(A)$ also has a canonical $\lambda$-ring structure111Some
authors refer to a $\lambda$-ring as a special $\lambda$-ring and a
pre-$\lambda$-ring as a $\lambda$-ring. denoted $\bm{\lambda_{u}}$ (see
Knutson [6, p. 18]). On Teichmüller elements, it is given by

$\bm{\lambda_{u}}([a])=[1]+[a]u\in\Lambda(W(A))$

where $\Lambda(W(A))=1+uW(A)[[u]]$. This $\lambda$-ring structure behaves well
in the case where $A$ itself is a $\lambda$-ring. In particular, we have

######  Proposition 2.2.

If $(A,\lambda_{t})$ is a $\lambda$-ring, then the opposite $\lambda$-ring map
$\sigma_{t}=\lambda_{-t}^{-1}$, with $\sigma_{t}(a):=\sum_{n\geqslant
0}\sigma^{n}(a)t^{n}$ is a $\lambda$-ring homomorphism
$\sigma_{t}:A\rightarrow W(A)$. In particular,

$\Lambda_{\sigma_{t}}(\sigma_{t}(a))=\bm{\sigma_{u}}(\sigma_{t}(a))$

###### Proof.

Since $A$ is a $\lambda$-ring, $\lambda_{t}:A\rightarrow\Lambda(A)$ is a
$\lambda$-ring map. The Artin-Hasse exponential $\iota:\Lambda(A)\rightarrow
W(A),P(t)\mapsto P(-t)^{-1}$ is a $\lambda$-ring map, and
$\iota\circ\lambda_{t}=\sigma_{t}$. ∎

The following examples will be useful:

  * •

The ring $A=\mathbb{Z}$ is a $\lambda$-ring with $\lambda_{t}(a)=(1+t)^{a}$.
Here the map $\sigma_{t}$ is given by

$\sigma_{t}(a)=(1-t)^{-a},\quad\quad\sigma^{n}(a)=\left(\begin{array}[]{c}n+a+1\\\
a\end{array}\right).$

  * •

For a $\lambda$-ring $A$, the polynomial ring $A[z]$ can be given a natural
$\lambda$-ring structure by setting $\lambda_{t}(z)=(1+zt)$. Here the map
$\sigma_{t}$ is given by

$\sigma_{t}(z)=(1-zt)^{-1}=[z],\quad\quad\sigma^{n}(z)=z^{n}.$

Note that for $A=\mathbb{Z}$ this means that
$\sigma_{t}(az)=(1-zt)^{-a}=a[z]$.

  * •

For arbitrary $A$, consider the canonical $\lambda$-ring structure on $W(A)$.
The opposite $\lambda$-ring structure on $W(A)$ is a $\lambda$-ring map
$\bm{\sigma_{u}}:W(A)\rightarrow W(W(A))$. On Teichmüller elements $[a]\in
W(A)$ we have $\bm{\sigma_{u}}([a])=([1]-[a]u)^{-1}$. We denote this by
$[[\;[a]\;]]$, the _double Teichmüller element._

##  3\. Varieties over Finite Fields

Let $X$ be a variety over a finite field $\mathbb{F}_{q}$ (reduced scheme of
finite type over $\operatorname{Spec}\mathbb{F}_{q}$). The Hasse-Weil zeta
function (or simply the zeta function) of $X$ is

$Z(X,t)=\prod_{x\in|X|}(1-t^{\deg x})^{-1}$ (2)

a power series over $\mathbb{Z}$, where the product ranges over closed points
$x\in|X|$. This may also be expressed as

$Z(X,t)=\exp\left[\sum_{r=1}^{\infty}N_{r}(X)\frac{t^{r}}{r}\right]$ (3)

where $N_{r}(X)=\\#X(\mathbb{F}_{q^{r}})$ the number of points over
$\mathbb{F}_{q^{r}}$.

Due to the work of Grothendieck and others on the Weil conjectures, the zeta
function can be written in terms the action of the Frobenius on (compactly
supported) étale cohomology:

$Z(X,t)=\frac{\prod_{i}P_{2i+1}(t)}{\prod_{i}P_{2i}(t)},\quad
P_{i}(t)=\prod_{j}(1-\alpha_{ij}t)^{-1}$ (4)

where $\alpha_{ij}$ are the inverse eigenvalues of the Frobenius action $\Phi$
on $H_{et,c}^{i}(\overline{X};\mathbb{Q}_{\ell})$. These are so-called Weil
$q$-numbers $|\alpha_{i,j}|=q^{r/2}$ for some $r\leqslant i$ (Deligne) and
$Z(X,t)$ is a rational function (Dwork).

We wish to consider the zeta function $Z(X,t)$ as an element in the big Witt
ring $W(\mathbb{Z})$. The following is shown in [12, Thm 2.1]:

######  Proposition 3.1.

For $X$ and $Y\in\operatorname{Var}_{\mathbb{F}_{q}}$, the zeta function
$X\times_{\mathbb{F}_{q}}Y$ is given by the Witt product

$Z(X\times_{\mathbb{F}_{q}}Y,t)=Z(X,t)*_{W}Z(Y,t).$

Moreover, the Frobenius operator $\operatorname{Fr}_{r}$ on $W(\mathbb{Z})$
corresponds to base change $X_{\mathbb{F}_{q^{r}}}$, so that
$\operatorname{Fr}_{r}(Z(X/\mathbb{F}_{q},t))=Z(X/\mathbb{F}_{q^{r}},t).$

###### Sketch.

Notice that from (3), $\operatorname{gh}_{n}(Z(X,t))=N_{n}(X)$. As this is
multiplicative, the zeta function takes products in the Witt ring. Finally,
recall that
$\operatorname{gh}_{n}\circ\operatorname{Fr}_{r}=\operatorname{gh}_{nr}$ and
note that
$\operatorname{gh}_{n}(X_{\mathbb{F}_{q^{r}}})=N_{nr}(X)=\operatorname{gh}_{nr}(X/\mathbb{F}_{q})$.
∎

The zeta function can be written in the Witt ring $W(\mathbb{Z})$ as

$Z(X,t)=\sum_{i,j}(-1)^{i+1}[\alpha_{ij}]$

where the sum is taking place in the Witt ring and $[\alpha]$ is the
Teichmüller element $(1-\alpha t)^{-1}$. This directly follows from the
presentation in (4) as an alternating product of polynomials
$P_{i}=(1-\alpha_{ij}t)^{-1}$ and Proposition 3.1 above. Although each
Teichmüller element $[\alpha_{ij}]$ is in $W(\overline{\mathbb{Q}}_{\ell})$,
the sum nevertheless lies in the subring $W(\mathbb{Z})$.

We now restate our main result.

######  Theorem 3.2.

Given a quasi-projective variety $X\in\operatorname{Var}_{\mathbb{F}_{q}}$, we
have

$\sum_{n\geqslant
0}Z(\operatorname{Sym}^{n}X,t)u^{n}=\sum_{i,j}(-1)^{i+1}[[\;[\alpha_{ij}]\;]]\in
W(W(\overline{\mathbb{Q}_{\ell}}))$ (5)

This can be viewed as a closed product formula for the generating series for
zeta functions of symmetric powers.

###  3.1. Proof of the Main Theorem

Recall the result on Newton’s identities from Lemma 2.1: in $\Lambda(A)$, for
$P(t)=\sum_{n}a_{n}t^{n}$ and $b_{n}=\operatorname{gh}_{n}(P(t))$, we have

$na_{n}=b_{n}+a_{1}b_{n-1}+\cdots a_{n-1}b_{1}$

Recursively, this gives a way to recover $a_{n}$ from the ghost coordinates
$b_{1},b_{2},\ldots,b_{n}$. That is, $a_{n}$ can be written purely in terms of
$b_{i}$ for $i=1,2,\ldots n$ by replacing each $a_{i}$ in the above relation.
Let us call this relation $\phi$ so that

$a_{n}=\phi(b_{1},b_{2},\ldots,b_{n}).$

######  Lemma 3.3.

For $X$ a quasi-projective variety over $\mathbb{F}_{q}$,

$N_{r}(\operatorname{Sym}^{n}X)=\phi(N_{r}(X),N_{2r}(X),\ldots,N_{nr}(X))$ (6)

###### Proof.

Over $\overline{\mathbb{F}_{q}}$, the symmetric power
$\operatorname{Sym}^{n}X$ parametrizes effective zero cycles. These are
exactly the degree $n$ effective cycles on $X$. Thus the zeta function
$Z(X,t)$ may be written as

$Z(X,t)=\sum_{n}N_{1}(\operatorname{Sym}^{n}X)t^{n}.$ (7)

As $\operatorname{gh}_{r}(Z(X,t))=N_{r}(X)$, this implies that

$N_{1}(\operatorname{Sym}^{n}X)=\phi(N_{1}(X),N_{2}(X),\ldots,N_{n}(X)).$

Apply the Frobenius operator $\operatorname{Fr}_{r}$ for $W(\mathbb{Z})$ to
equation (7). Equation (6) then follows from the observation that on ghost
coordinates,
$\operatorname{gh}_{n}(\operatorname{Fr}_{r}(P))=\operatorname{gh}_{nr}(P)$
for $P\in W(\mathbb{Z})$ and from Proposition 3.1,
$\operatorname{Fr}_{r}Z(X/\mathbb{F}_{q},t)=Z(X/\mathbb{F}_{q^{r}},t)$. ∎

###### Proof of Theorem 1.1.

Denote the ghost map on $W(W(\mathbb{Z}))$ by $\operatorname{gh}^{u}$ and let
$\beta_{n}$ be the $n$th ghost coordinate of the sum double Teichmüller
elements in (5):

$\beta_{n}=\operatorname{gh}^{u}_{n}\left(\sum_{i,j}(-1)^{i+1}[[\,[\alpha_{ij}]\,]]\right)=\sum_{i,j}(-1)^{i+1}[\alpha_{ij}]^{n}\in
W(\mathbb{Z}).$

We wish to show

$Z(\operatorname{Sym}^{n}X,t)=\phi(\beta_{1},\beta_{2},\ldots,\beta_{n}).$

This relation is taking place in $W(\mathbb{Z})$. We now use the ghost map
$\operatorname{gh}^{t}$ on $W(\mathbb{Z})$. It suffices to show for all $r$,

$\operatorname{gh}_{r}^{t}Z(\operatorname{Sym}^{n}X,t)=\phi(\operatorname{gh}_{r}^{t}\beta_{1},\operatorname{gh}_{r}^{t}\beta_{2},\ldots,\operatorname{gh}_{r}^{t}\beta_{n}).$

We have

$\operatorname{gh}^{t}_{r}\beta_{n}=\operatorname{gh}^{t}_{r}\left(\sum_{i,j}(-1)^{i+1}[\alpha_{ij}]^{n}\right)=\sum_{i,j}(-1)^{i+1}\alpha_{ij}^{nr}=N_{nr}(X)\in\mathbb{Z}.$

Moreover,
$\operatorname{gh}_{r}^{t}Z(\operatorname{Sym}^{n}X,t)=N_{r}(\operatorname{Sym}^{n}X)$.
Now recall from Lemma 3.3

$N_{r}(\operatorname{Sym}^{n}X)=\phi(N_{r}(X),N_{2r}(X),\ldots,N_{nr}(X)).$

As this holds for all $r$, the proof follows. ∎

##  4\. Examples

We apply the generating series formula to compute
$Z(\operatorname{Sym}^{n}X,t)$ for various cases of varieties $X$ over finite
fields. These serve to verify the formula and demonstrate the facility of
writing the zeta function in the Witt ring. Note that below, sums of
Teichmüller elements are taking place in the appropriate Witt ring.

###  4.1. Affine and projective space

Consider $n$-dimensional affine space $\mathbb{A}^{n}$ and $n$-dimensional
projective space $\mathbb{P}^{n}$ over $\mathbb{F}_{q}$. Then
$N_{r}(\mathbb{A}^{n})=q^{nr}$ and we have

$Z(\mathbb{A}^{n},t)=\frac{1}{(1-q^{n}t)}$

It is also easy to show that
$N_{r}(\mathbb{P}^{n})=q^{nr}+q^{(n-1)r}+\cdots+q^{r}+1$ so that

$Z(\mathbb{P}^{n},t)=\frac{1}{(1-t)(1-qt)\cdots(1-q^{m}t)}$

In the Witt ring $W(\mathbb{Z})$, these zeta functions are
$Z(\mathbb{A}^{n},t)=[q^{n}]$ and
$Z(\mathbb{P}^{n},t)=[q^{m}]+[q^{m-1}]+\cdots+[q]+[1]$.

Recall that $\operatorname{Sym}^{n}\mathbb{A}^{1}=\mathbb{A}^{n}$ and
$\operatorname{Sym}^{n}\mathbb{P}^{1}=\mathbb{P}^{n}$, and that

$Z(\mathbb{A}^{1},t)=\frac{1}{1-qt}=[q]\quad\textrm{and}\quad
Z(\mathbb{P}^{1},t)=\frac{1}{(1-t)(1-qt)}=[1]+[q]$

The formula provided in Theorem 1.1 predicts that
$Z(\operatorname{Sym}^{n}\mathbb{A}^{1},t)$ is the coefficient of $u^{n}$ in
$[[\;[q]\;]]\in W(W(\mathbb{Z}))$. Similarly, that
$Z(\operatorname{Sym}^{n}\mathbb{P}^{1},t)$ is the coefficient of $u^{n}$ in
$[[\;[1]\;]]+[[\;[q]\;]]\in W(W(\mathbb{Z}))$. We have

$[[\;[q]\;]]=\frac{1}{[1]-[q]u}=[1]+[q]u+[q^{2}]u^{2}+\cdots$

and

$[[\;[1]\;]]+[[\;[q]\;]]=\left(\frac{1}{[1]-[1]u}\right)\left(\frac{1}{[1]-[q]u}\right)=[1]+([1]+[q])u+\cdots$

which agrees with the zeta functions $Z(\mathbb{A}^{n},t)$ and
$Z(\mathbb{P}^{n},t)$ described above:

###  4.2. Elliptic curves

Let $E$ be an elliptic curve over $\mathbb{F}_{q}$. In this case, we have

$Z(E,t)=\frac{(1-\alpha t)(1-\beta t)}{(1-t)(1-qt)}$

where $\alpha+\beta=a\in\mathbb{Z}$ and $\alpha\beta=q$. Written in the Witt
ring, this appears as

$Z(E,t)=[1]-[\alpha]-[\beta]+[q],$

where $[\alpha]=(1-\alpha t)^{-1}$ the Teichmüller element. The formula
provided by Theorem 1.1 is

$Z(\operatorname{Sym}^{n}E,t)=\textrm{coefficient of $u^{n}$
in}\left[\frac{(1-[\alpha]u)(1-[\beta]u))}{(1-[1]u)(1-[q]u))}\right].$

The results in the section in fact work for any curve $C$ over
$\mathbb{F}_{q}$. In this general case, the eigenvalues $\alpha$ and $\beta$
for the action on $H^{1}(\overline{C},\mathbb{Q}_{\ell})$ come in pairs
$\alpha_{i}$ and $\beta_{i}$ for $i=1,2,\ldots,g$ where $g$ is the genus. In
fact, the case of symmetric powers of smooth projective curves was worked out
by MacDonald in [9] in 1962. Here we work in the elliptic curve case for
simplicity.

###  4.3. Symmetric powers of affine and projective space

We use the formula in Theorem 1.1 to compute the zeta functions
$Z(\operatorname{Sym}^{n}\mathbb{A}^{m},t)$ and
$Z(\operatorname{Sym}^{n}\mathbb{P}^{m},t)$. Note that these varieties are no
longer smooth once $m>1$. Recall that

$Z(\mathbb{A}^{m},t)=\frac{1}{(1-q^{m}t)}=[q^{m}]\quad\textrm{and}\quad
Z(\mathbb{P}^{m},t)=\frac{1}{(1-t)(1-[q^{m}])}=[1]+[q^{m}]$

Then by the formula we have for affine space

$Z(\operatorname{Sym}^{n}\mathbb{A}^{m},t)=\textrm{ coefficient of $u^{n}$ in
}[[\;[q^{m}]\;]]=[q^{nm}],$

which agrees with the fact that
$[\operatorname{Sym}^{n}\mathbb{A}^{m}]=[\mathbb{A}^{nm}]$. For projective
space,

$Z(\operatorname{Sym}^{n}\mathbb{P}^{m},t)=\textrm{ coefficient of $u^{n}$ in
}[[\;[1]\;]]+[[\;[q]\;]]+\cdots+[[\;[q^{m}]\;]].$

Note that this coefficient is not $[1]+[q]+\cdots+[q^{nm}]$ as
$\operatorname{Sym}^{n}\mathbb{P}^{m}$ is not $\mathbb{P}^{nm}$. For instance,
consider $n=2$ and $m=2$,

$Z(\operatorname{Sym}^{2}\mathbb{P}^{2},t)=[1]+[q]+2[q^{2}]+[q^{3}]+[q^{4}]$

whereas

$Z(\mathbb{P}^{4},t)=[1]+[q]+[q^{2}]+[q^{3}]+[q^{4}].$

###  4.4. Product of elliptic curves

Consider $X=E_{1}\times E_{2}$ a product of elliptic curves over
$\mathbb{F}_{q}$. We use the formula in Theorem 1.1 to compute
$Z(\operatorname{Sym}^{n}X,t)$ in terms of $Z(E_{1},t)$ and $Z(E_{2},t)$. Let

$Z(E_{1},t)=\frac{(1-\alpha_{1}t)(1-\beta_{1}t)}{(1-t)(1-qt)}\quad\textrm{and}\quad
Z(E_{2},t)=\frac{(1-\alpha_{2}t)(1-\beta_{2}t)}{(1-t)(1-qt)}$

and recall that in the Witt ring, these can be written as

$Z(E_{1},t)=[1]-[\alpha_{1}]-[\beta_{1}]+[q]\quad\textrm{and}\quad
Z(E_{2},t)=[1]-[\alpha_{2}]-[\beta_{2}]+[q].$

Then since the zeta function takes products in the Witt ring, we may compute
$Z(E_{1}\times E_{2},t)$ as

$\displaystyle Z(E_{1}\times E_{2},t)$ $\displaystyle=$
$\displaystyle([1]-[\alpha_{1}]-[\beta_{1}]+[q])*([1]-[\alpha_{2}]-[\beta_{2}]+[q])$
$\displaystyle=$
$\displaystyle[1]-[\alpha_{1}]-[\alpha_{2}]-[\beta_{1}]-[\beta_{2}]+[\alpha_{1}\beta_{2}]+[\alpha_{2}\beta_{1}]+[\alpha_{1}\alpha_{2}]$
$\displaystyle+[\beta_{1}\beta_{2}]-[\alpha_{1}q]-[\alpha_{2}q]-[\beta_{1}q]-[\beta_{2}q]+[q^{2}].$

Thus the motivic zeta function generating series is

$\displaystyle\zeta_{Z}(E_{1}\times E_{2},u)$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}Z(\operatorname{Sym}^{n}(E_{1}\times
E_{2}),t)u^{n}$ $\displaystyle=$
$\displaystyle[[\;[1]\;]]-[[\;[\alpha_{1}]\;]]-[[\;[\alpha_{2}]\;]]-[[\;[\beta_{1}]\;]]-[[\;[\beta_{2}]\;]]$
$\displaystyle+[[\;[\alpha_{1}\beta_{2}]\;]]+[[\;[\alpha_{2}\beta_{1}]\;]]+[[\;[\alpha_{1}\alpha_{2}]\;]]+[[\;[\beta_{1}\beta_{2}]\;]]$
$\displaystyle-[[\;[\alpha_{1}q]\;]]-[[\;[\alpha_{2}q]\;]]-[[\;[\beta_{1}q]\;]]-[[\;[\beta_{2}q]\;]]+[[\;[q^{2}]\;]].$

Note that these equalities are being written in $W(W(A))$. These methods
similarly also work for $n$-fold products of smooth projective curves;
however, the formulas quickly become unwieldy.

##  5\. Lambda-ring valued motivic measures

Let $k$ be a field and $\operatorname{Var}_{k}$ be the category of varieties
over $k$ (reduced schemes of finite type over $\operatorname{Spec}k$). The
_Grothendieck ring of varieties_ $K_{0}(\operatorname{Var}_{k})$ is the
abelian group generated by symbols $[X]$ of isomorphism classes of
$X\in\operatorname{Var}_{k}$ subject to the scissor relation
$[X]=[Y]+[X\setminus Y]$ for $Y$ any closed subvariety of $X$.
$K_{0}(\operatorname{Var}_{k})$ is a commutative ring under the product
$[X]\cdot[Y]=[X\times_{k}Y]$. This is the so-called universal value group of
Euler-Poincaré characteristics on $\operatorname{Var}_{k}$; ring homomorphisms
on $K_{0}(\operatorname{Var}_{k})$ are called _motivic measures_. That is, for
$A$ a commutative ring with identity, we consider
$\mu:K_{0}(\operatorname{Var}_{k})\rightarrow A$. Kapranov in [5] constructs a
_motivic zeta function_ $\zeta_{\mu}(X,t)$ for $X\in\operatorname{Var}_{k}$
using these algebraic invariants $\mu(X)$. For $X$ quasi-projective, it is the
generating series for (the measure of) symmetric powers
$\operatorname{Sym}^{n}X$:

$\zeta_{\mu}(X,t)=\sum_{n=0}^{\infty}\mu(\operatorname{Sym}^{n}X)t^{n}\in
A[[t]].$

As $K_{0}(\operatorname{Var}_{k})$ is additively generated by quasiprojective
$X$, this defines a group homomorphism on $K_{0}(\operatorname{Var}_{k})$
taking values in the group of invertible power series $\Lambda(A)$. If
$\zeta_{\mu}:K_{0}(\operatorname{Var}_{k})\rightarrow W(A)$ is a ring
homomorphism , we say that $\mu$ exponentiates (see [12]).

For example, in the finite field case222In this case, we must pass through the
$\widetilde{K_{0}}(\operatorname{Var}_{\mathbb{F}_{q}})$ Grothendieck ring
modulo _radicial morphisms_ , but all the measures on
$\operatorname{Var}_{\mathbb{F}_{q}}$ we consider factor through
$\widetilde{K_{0}}(\operatorname{Var}_{\mathbb{F}_{q}})$. See Mustata [11, p.
78]. $X\in\operatorname{Var}_{\mathbb{F}_{q}}$, the motivic zeta function
$\zeta_{\mu_{\\#}}$ associated to counting measure
$\mu_{\\#}([X])=\\#X(\mathbb{F}_{q})$ recovers the usual zeta function
$Z(X,t)$ (see Mustata [11, Prop 7.31]). We have seen that
$Z(X\times_{k}Y,t)=Z(X,t)*_{W}Z(Y,t)$; hence $\mu_{\\#}$ can be exponentiated.

We are particularly interested in $A$-valued motivic measures where $A$ is a
$\lambda$-ring. Note that $K_{0}(\operatorname{Var}_{k})$ has a
pre-$\lambda$-ring structure provided by the Kapranov motivic zeta function,
defined such that $\sigma^{n}([X])=[\operatorname{Sym}^{n}X]$ for $X$
quasiprojective. This, however, is not a $\lambda$-ring structure (see Larsen
and Lunts [7]).

######  Definition 5.1.

Let $A$ be a $\lambda$-ring. A motivic measure
$\mu:K_{0}(\operatorname{Var}_{k})\rightarrow A$ is called a $\lambda$-measure
if any of the equivalent conditions hold:

  * •

$\mu$ is a pre-$\lambda$-ring map,
$\mu(\operatorname{Sym}^{n}[X])=\sigma^{n}\mu(X)$.

  * •

The associated motivic zeta function factors as

$\textstyle{K_{0}(\operatorname{Var}_{k})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\zeta_{\mu}}$$\scriptstyle{\mu}$$\textstyle{R\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma_{t}}$$\textstyle{W(R)}$

######  Proposition 5.2 (Ramachandran, Tabuada [13]).

If $\mu$ is a $\lambda$-measure, then $\mu$ exponentiates.

###### Proof.

The composition above is a ring homomorphism. ∎

######  Example 5.3.

The generating series in (1) for the Poincaré polynomial is the Kapranov
motivic zeta function $\zeta_{\mu_{P}}$ associated to the Poincaré polynomial
measure $\mu_{P}(X)=P(X,z)$. Since the ghost coordinates
$\operatorname{gh}_{n}(\zeta_{\mu_{P}}(X,t))=P(X,z^{n})$ are multiplicative,
$\zeta_{P}$ takes values in the Witt ring. MacDonald’s formula (1) can be
written in the Witt ring $W(\mathbb{Z}[z])$ as

$\zeta_{\mu_{P}}(X,t)=\sum_{i=0}^{2m}(-1)^{i}b_{i}(X)[z^{i}].$

The zeta function for the Poincaré polynomial appears as a Poincaré polynomial
written in Teichmüller elements. Similarly for $\chi$, we have

$\zeta_{\chi}(X,t)=\sum_{i=0}^{2m}(-1)^{i}b_{i}(X)[1]=\chi(X)[1],$

an Euler characteristic written in Teichmüller elements. Moreover, both these
motivic zeta functions factor through the $\lambda$-ring structure maps and
thus both $\chi$ and $\mu_{P}$ are $\lambda$-measures:

######  Proposition 5.4.

For the Euler characteristic, $\zeta_{\chi}=\sigma_{t}\circ\chi$, where
$\sigma_{t}$ is the opposite $\lambda$-ring structure map on $\mathbb{Z}$ and
for the Poincaré polynomial $\zeta_{\mu_{P}}=\sigma_{t}^{\prime}\circ\mu_{P}$,
where $\sigma_{t}^{\prime}$ is the opposite $\lambda$-ring structure map on
$\mathbb{Z}[z]$.

###### Proof.

Recall that for $a\in\mathbb{Z}$, $\sigma_{t}(a)=a[1]$ and for
$f(z)\in\mathbb{Z}[z]$, $\sigma_{t}^{\prime}(f(z))=f([z])$. ∎

######  Definition 5.5.

Given a measure that exponentiates
$\mu:K_{0}(\operatorname{Var}_{k})\rightarrow A$, the associated motivic zeta
function $\zeta_{\mu}:K_{0}(\operatorname{Var}_{k})\rightarrow W(A)$ is a ring
homomorphism. The induced motivic measure is called the _zeta function
measure_ $\mu_{Z}(X)=\zeta_{\mu}(X,t)$; it is a $W(A)$-valued motivic measure.

######  Theorem 5.6.

If $\mu:K_{0}(\operatorname{Var}_{k})\rightarrow A$ is a $\lambda$-measure,
then its induced zeta function measure
$\mu_{Z}:K_{0}(\operatorname{Var}_{k})\rightarrow W(A)$ is also a
$\lambda$-measure. This process iterates ad infinitum.

###### Proof.

By Proposition 2.2, the map $\sigma_{t}$ is a $\lambda$-ring map. The
composition above is a $\lambda$-ring map. ∎

######  Remark.

The zeta function $Z(X,t)$ is the Kapranov motivic zeta function associated to
the counting measure $\mu_{\\#}$; however, counting measure is not a
$\lambda$-measure. Recall the unique $\lambda$-ring structure on $\mathbb{Z}$:
for $a\in\mathbb{Z}$,

$\lambda_{t}(a)=(1+t)^{a},\quad\lambda^{n}(a)=\left(\\!\begin{array}[]{c}a\\\
n\end{array}\\!\right),\quad\sigma^{n}(a)=\left(\\!\begin{array}[]{c}a+n-1\\\
n\end{array}\\!\right).$

However, it is easy to verify that
$\mu_{\\#}(\operatorname{Sym}^{n}X)\neq\sigma^{n}(\mu_{\\#}(X))$. For
instance, $X=\mathbb{A}^{1}$,
$\operatorname{Sym}^{n}\mathbb{A}^{1}=\mathbb{A}^{n}$ and
$\mu_{\\#}(\mathbb{A}^{1})=q$ whereas
$\mu_{\\#}(\mathbb{A}^{n})=q^{n}\neq\sigma^{n}(q)$. Thus, the exponentiability
of the induced zeta function measure $\mu_{Z}$ does not follow from the above
Theorem. However,

######  Corollary 5.7 (Corollary to Theorem 1.1).

Let $Z(X,t)$ be the zeta function for
$X\in\operatorname{Var}_{\mathbb{F}_{q}}$ and $\mu_{Z}$ its induced zeta
function measure. Then

$\zeta_{\mu_{Z}}(X,u)=\bm{\sigma_{u}}(Z(X,t))\in W(W(\mathbb{Z}))$

and so $\mu_{Z}$ is a $\lambda$-measure.

###### Proof.

The formula in Theorem 1.1 shows that for
$X\in\operatorname{Var}_{\mathbb{F}_{q}}$

$\zeta_{\mu_{Z}}(X,u)=\sum_{i,j}(-1)^{i+1}[[\;[\alpha_{ij}]\;]].$

Recall that the map $\bm{\sigma_{u}}$ on Teichmüller elements is given by the
double Teichmüller $\bm{\sigma_{u}}([\alpha])=[[\;[\alpha]\;]]$. As
$\bm{\sigma_{u}}$ is a ring homomorphism, we have
$\zeta_{\mu_{Z}}=\bm{\sigma_{u}}(Z(X,t))$. ∎

######  Remark.

The formula for $Z(\operatorname{Sym}^{n}X,t)$ in Theorem 1.1 is analogous to
the MacDonald formulae above, as it is simply the zeta function written in
(double) Teichmüller elements. And similarly, the formula shows that $\mu_{Z}$
is a $\lambda$-measure.

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