# Non-Pólya Fields with Large Pólya Groups Arising from Lehmer Quintics

Nimish Kumar Mahapatra  and  Prem Prakash Pandey  Indian Institute of Science
Education and Research, Berhampur, India<EMAIL_ADDRESS>Indian
Institute of Science Education and Research, Berhampur, India.
<EMAIL_ADDRESS>
###### Abstract.

In this article we construct a new family of quintic non-Pólya fields with
large Pólya groups. We study the upper bound of Pólya numbers of such fields
and show that the Pólya numbers never exceed five times the size of its Pólya
group. Finally we show that such non-Pólya fields are non-monogenic fields of
field index one.

###### Key words and phrases:

Pólya group; Pólya fields; Hilbert Class Field; Genus field.

###### 2020 Mathematics Subject Classification:

11R09; 11R29; 11R34

##  1\. Introduction

Let $K$ be an algebraic number field and $\mathcal{O}_{K}$ be its ring of
integers. Let $Int(\mathcal{O}_{K})=\\{f\in K[X]\mid
f(\mathcal{O}_{K})\subseteq\mathcal{O}_{K}\\}$ be the ring of integer valued
polynomials on $\mathcal{O}_{K}$. Then the number field $K$ is said to be a
Pólya field if the $\mathcal{O}_{K}$-module $Int(\mathcal{O}_{K})$ has a
regular basis, that is, a basis $(f_{n})$ such that for each
$n\in\mathbb{N}\cup\\{0\\}$, degree$(f_{n})=n$ (see [26]). For each
$n\in\mathbb{N}$, the leading coefficients of the polynomials in
$Int(\mathcal{O}_{K})$ of degree $n$ together with zero form a fractional
ideal of $\mathcal{O}_{K}$, denoted by $\mathfrak{J}_{n}(K)$. The following
result establishes a connection between $\mathfrak{J}_{n}(K)$ and the Pólya-
ness of the number field $K$.

######  Proposition 1.1.

[1] A number field $K$ is a Pólya field if and only if $\mathfrak{J}_{n}(K)$
is principal for all integer $n\geq 1$.

It follows immediately from Proposition 1.1 that if the class number of $K$ is
one then $K$ is a Pólya field. However, the converse is not valid in general.
That is, if the class number $h_{K}$ of $K$ is not one then we can not decide
whether $K$ is a Pólya field or not: for instance, every cyclotomic field is a
Pólya field (see [26]).

Let $Cl(K)$ denote the ideal class group of $K$. For each integer $n\geq 1$,
let $[\mathfrak{J}_{n}(K)]$ be the ideal class in $Cl(K)$ corresponding to the
fractional ideal $\mathfrak{J}_{n}(K)$. The Pólya group $Po(K)$ of $K$ is
defined to be the subgroup of $Cl(K)$ generated by the elements
$[\mathfrak{J}_{n}(K)]$ in $Cl(K)$. Therefore, $K$ is a Pólya field if and
only if $Po(K)=\\{1\\}$.

It is an interesting problem to look for explicit families of number fields
that are Pólya /non-Pólya (for example see [9],[10],[19],[23]). In fact, the
classification of Pólya fields of lower degree number fields is of significant
interest. Towards this, Zantema [26] completely characterized quadratic Pólya
fields.

######  Proposition 1.2.

[26] Let $p$ and $q$ be two distinct odd primes. A quadratic field
$\mathbb{Q}(\sqrt{d})$ is a Pólya field if and only if $d$ is of one of the
following forms :

  1. (1)

$d=2$, or $d=-1$, or $d=-2$, or $d=-p$ where $p\equiv 3\pmod{4}$, or $d=p$,

  2. (2)

$d=2p$, or $d=pq$ where $pq\equiv 1\pmod{4}$ and, in both cases, the
fundamental unit has norm $1$ if $p\equiv 1\pmod{4}$.

Leriche [17] completely classified Galois cubic Pólya fields. We quote the
precise result below.

######  Proposition 1.3.

[17] Let $K$ be a cyclic cubic field. Then, $K$ is a Pólya field if and only
if $K=\mathbb{Q}(\theta)$ where $\theta$ is a root of a polynomial $P(X)$ such
that:

$\text{either }P(X)=X^{3}-3X+1\text{ or }P(X)=X^{3}-3pX-pu$

where $p$ is a prime such that $p=\frac{u^{2}+27w^{2}}{4}$ with $u\equiv
2\pmod{3}$ and $w\in\mathbb{N}^{*}$.

In the same article Leriche also classified cyclic quartic and cyclic sextic
Pólya fields. Moreover, he obtained similar classifications for some more
families of bi-quadratic and sextic fields. (See [17, Theorem 5.1, Theorem
6.2] ). Recently, there have been several attempts to provide families of
Pólya/non-Pólya fields in the remaining cases of the bi-quadratic extensions
(see [9],[10],[19],[23]). In [3] the authors constructed a new family of
totally real bi-quadratic field with large Pólya groups.

In this article we characterize the Pólya-ness of a special family of quintic
fields arising from Lehmer qunitics. For each integer $n\in\mathbb{Z}$, the
Lehmer quintic $f_{n}(x)\in\mathbb{Z}[x]$ is defined as

$\displaystyle f_{n}(x)=$ $\displaystyle
x^{5}+n^{2}x^{4}-(2n^{3}+6n^{2}+10n+10)x^{3}+(n^{4}+5n^{3}+11n^{2}+15n+5)x^{2}$
$\displaystyle+(n^{3}+4n^{2}+10n+10)x+1.$

Let $\theta_{n}\in\mathbb{C}$ be a root of $f_{n}(x)=0$. If we set
$K_{n}=\mathbb{Q}(\theta_{n})$ then $[K_{n}:\mathbb{Q}]=5$ and the fields
$K_{n}$ are called Lehmer quintic fields [15]. The main theorem we prove in
this article is the following one:

######  Theorem 1.4.

Let $\\{K_{n}\\}$ be the family of Lehmer quintic fields. If
$m_{n}=n^{4}+5n^{3}+15n^{2}+25n+25$ is cube free, then

  1. (1)

$K_{n}$ is a Pólya field if and only if $m_{n}$ is a prime or $m_{n}=25$.

  2. (2)

We have $Po(K_{n})\simeq(\mathbb{Z}/5\mathbb{Z})^{\omega({m_{n}})-1}$. Here
$\omega(t)$ denotes the number of distinct prime divisors of $t$.

Moreover, there are infinitely many non-Pólya fields in the family $K_{n}$.

Let $G$ be a finite group, and if $m>1$ is an integer then the $m$-rank of $G$
is defined to be the maximal integer $r$ such that
$(\mathbb{Z}/m\mathbb{Z})^{r}$ is a subgroup of $G$. The following folklore
conjecture is widely believed to be true but it is still open.

######  Conjecture.

Let $d>1$ and $m>1$ be two integers. Then the $m$-rank of the class group of
$K$ is unbounded when $K$ runs through the number fields of degree
$[K:\mathbb{Q}]=d$.

It is known that when $m=d$, or more generally when $m\mid d$, this conjecture
follows from the class field theory (see [6]). In the following corollary to
Theorem 1.4 we give an alternative and elementary proof of the conjecture for
the case $m=d=5$.

######  Corollary 1.5.

The 5-rank of the class groups of the non-Pólya Lehmer quintic fields $K_{n}$
are unbounded.

Leriche [18] studied the embedding of a number field inside a Pólya field and
introduced the notion of Pólya numbers. For a number field $K$ the minimum of
degrees $[L:K]$ of extensions $L/K$ such that $L$ is a Pólya field is called
the Pólya number of $K$ and it is denoted by $po_{K}$. We study the Pólya
numbers $po_{K_{n}}$ of non-Pólya fields $K_{n}$ and obtained an upper bound
of $po_{K_{n}}$ in terms of the size of its corresponding Pólya groups
$Po(K_{n})$. Primarily we prove the following theorem.

######  Theorem 1.6.

Let $K_{n}$ be the family of Lehmer quintics fields such that $m_{n}$ is cube-
free and $5\mid n$. Then $po_{K_{n}}\leq 5|Po(K_{n})|$.

Let $K$ be a number field and $\theta\in\mathcal{O}_{K}$ be a primitive
element. The index $[\mathcal{O}_{K}:\mathbb{Z}[\theta]]$ is called the index
of $\theta$ in $K$ and is denoted by $I(\theta)$. The index of the number
field $K$ is defined as
$I(K)=\gcd\\{I(\theta)\mid\theta\in\mathcal{O}_{K}\mbox{ and
}K=\mathbb{Q}(\theta)\\}$. If $I(K)>1$ then the number field $K$ is not
monogenic, that is, $\mathcal{O}_{K}\neq\mathbb{Z}[\theta]$ for any $\theta\in
K$. However the the converse is not true in general. That is, there exists
non-monogenic number fields $K$ with $I(K)=1$. These are basically fields
which are not monogenic, but not for a local reason (see [24] for more
details). In this line we prove the following theorem in this article.

######  Theorem 1.7.

Let $K_{n}$ be the family of Lehmer quintic fields such that $m_{n}$ is cube-
free and $5\mid n$. Then $K_{n}$ is not monogenic and $I(K_{n})=1$.

In Section 2, we develop some preliminaries required to prove Theorem 1.4.
Section 3 contains the proof of Theorem 1.4 and Corollary 1.5. In Section 4,
we study the Pólya numbers of Lehmer quintic fields and prove Theorem 1.6. In
Section 5, we study the monogeneity of the non-Pólya number fields $K_{n}$ and
give the proof of Theorem 1.7. Finally in Section 7, we present some
computations performed by us using SAGE.

##  2\. Preliminaries

In this section we assume that the number field $K$ is a Galois extension of
$\mathbb{Q}$. For any prime number $p$, the ramification index of $p$ in
$K/\mathbb{Q}$ is denoted by $e_{p}$. In [2], Chabert obtained a nice
description for the cardinality of $Po(K)$ for cyclic extensions
$K/\mathbb{Q}$.

######  Proposition 2.1.

[2] Assume that the extension $K/\mathbb{Q}$ is cyclic of degree $n$.

  1. (1)

If $K$ is real and $N(\mathcal{O}_{K}^{\times})=\\{1\\}$, then
$|Po(K)|=\frac{1}{2n}\times\prod_{p}e_{p}$.

  2. (2)

In all other cases, $|Po(K)|=\frac{1}{n}\times\prod_{p}e_{p}$.

As an immediate corollary of Proposition 2.1, we record the following
observation.

######  Corollary 2.2.

If $K/\mathbb{Q}$ is cyclic extension of degree $n$ and $n$ is odd, then
$|Po(K)|=\frac{1}{n}\times\prod_{p}e_{p}$.

For a description of $Po(K)$ as in [16], we recall that an ideal class
$[\mathfrak{a}]$ in $K$ is called an ambiguous ideal class if it is invariant
under the action of $Gal(K/\mathbb{Q})$, that is,

$[\mathfrak{a}]^{\tau}=[\mathfrak{a}]\text{ for all }\tau\in
Gal(K/\mathbb{Q}).$

In [16], Leriche showed that for a Galois number field $K$, $Po(K)$ is the
subgroup of $Cl(K)$ generated by the classes of the ambiguous ideals of $K$.

Now we state some results on Lehmer quintics. In [22], Schoof and Washington
showed that $f_{n}(x)$ is irreducible for all $n\in\mathbb{Z}$ and its
discriminants equals to
$(n^{3}+5n^{2}+10n+7)^{2}(n^{4}+5n^{3}+15n^{2}+25n+25)^{4}$. Let
$\theta_{n}\in\mathbb{C}$ be a root of $f_{n}(x)=0$. If we set
$K_{n}=\mathbb{Q}(\theta_{n})$, then in [22], it was shown that $K_{n}$ is
cyclic field for all $n\in\mathbb{Z}$. We denote the ring of integers of
$K_{n}$ by $\mathcal{O}_{K_{n}}$. Now we recall some results of Jeannin [14]
on the discriminant $d(K_{n})$ of $K_{n}$.

######  Lemma 2.3.

All the prime divisors $p\neq 5$ of $n^{4}+5n^{3}+15n^{2}+25n+25$ satisfy
$p\equiv 1\pmod{5}$.

Next lemma gives the discriminant of the field $K_{n}$.

######  Lemma 2.4.

The discriminant $d(K_{n})$ is given by $d(K_{n})=f(K_{n})^{4}$, where the
conductor $f(K_{n})$ of $K_{n}$ is given by

(2.1)
$f(K_{n})=5^{b}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\prod_{\begin{subarray}{c}p\equiv
1\\!\\!\\!\\!\\!\\!\pmod{5}\\\ v_{p}(n^{4}+5n^{3}+15n^{2}+25n+25)\not\equiv
0\\!\\!\\!\\!\\!\pmod{5}\end{subarray}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!p.$

Here $v_{p}(k)$ denotes the exponent of the highest power of the prime $p$
dividing a non-zero integer $k$ and

(2.2) $b=\begin{cases}0,\text{ if }5\nmid n,\\\ 2,\text{ if }5\mid
n.\end{cases}$

We quote the following result due to Erdös [4] which plays a crucial role in
the proof of our main theorem.

######  Theorem 2.5.

[4] Let $f(x)$ be a polynomial of degree $l\geq 3$ whose coefficients are
integers with the highest common factor 1 and the leading coefficient is
positive. Assume that $f(x)$ is not divisible by the $(l-1)$-th power of a
linear polynomial with integer coefficient. Then there are infinitely many
positive integers $n$ for which $f(n)$ is $(l-1)$-th power free.

Now we state a deep result of on power-free values of polynomials (see [11,
12, 21]).

######  Theorem 2.6.

Let $f(x)\in\mathbb{Z}[x]$ be an irreducible polynomial of degree $d\geq 3$
and assume that $f$ has no fixed $(d-1)$th power prime divisor. Define

$N_{f}^{\prime}(X)=\\#\\{p\leq X:p\text{ prime},f(p)\text{ is
}(d-1)\text{-free}\\}.$

Then, for any $C>1$, we have

$N_{f}^{\prime}(X)=c_{f}^{\prime}\pi(X)+O_{C,f}\left(\frac{X}{(\log
X)^{C}}\right),$

as $x\rightarrow\infty$, where

$c_{f}^{\prime}=\prod_{p}\left(1-\frac{\rho^{\prime}(p^{d-1})}{\phi(p^{d-1})}\right)$

and

$\rho^{\prime}(d)=\\#\\{n\\!\\!\\!\\!\\!\pmod{d}:(d,n)=1,d\mid f(n)\\}.$

Let $f(x)$ be an irreducible polynomial with integral coefficients and
$f(m)>0$ for $m=1,2,\dots$. Let $\omega(m)$ denote the number of distinct
primes dividing $m$. Then for primes $p$, the following result due to
Halberstam [8] determines the distribution of values of $\omega(f(p))$.

######  Theorem 2.7.

[8] Let $f(X)\in\mathbb{Z}[X]$ be any non-constant polynomial. For all but
$o(X/\log X)$ primes $p\leq X$,

$\omega\Big{(}f(p)\Big{)}=\Big{(}1+o(1)\Big{)}{\log\log n}.$

Now we state some results on the number of integral solutions of Diophantine
equation of the type

(2.3) $Y^{m}=f(X).$

When $m=2$ and $f(x)$ is a monic qartic polynomial then the following result
due to Masser [20], gives us the specific bound on the integral solutions to
the curve.

######  Theorem 2.8.

[20] Consider the diophantine equation

$Y^{2}=f(X),$

where $f(X)$ is a polynomial of degree four with integer coefficients. Assume
that $f(X)$ is monic and not a perfect square. Then any integer solution
$(x,y)$ of the above equation satisfies

$|x|\leq 26H(f)^{3},$

where $H(f)$ denotes the maximum of the absolute values of the coefficients of
$f(X)$.

##  3\. Proof of Theorem 1.4

###### Proof.

We consider the set

$\mathcal{P}=\\{n\in\mathbb{Z}:m_{n}=n^{4}+5n^{3}+15n^{2}+25n+25\mbox{ is a
cube-free integer}\\}.$

For $n\in\mathcal{P}$, we have

(3.1) $m_{n}=5^{b}AB^{2}.$

Here $b=0$ if $n$ is not divisible by $5$ and $b=2$ otherwise, and $A,B$ are
square-free natural numbers which are relatively prime and $5\nmid AB$. Using
Lemma 2.3 and (2.1) we have

(3.2) $f(K_{n})=5^{b}AB\mbox{ and }d(K_{n})=(5^{b}AB)^{4}.$

Since $K_{n}/\mathbb{Q}$ is Galois and of degree $5$, we see that for any
prime $p$ the ramification index $e_{p}$ of $p$ in $K_{n}$ is given by

$\displaystyle e_{p}=\begin{cases}5\mbox{ if }p\mid 5AB,\\\ 1\mbox{
otherwise}.\end{cases}$

Thus, we obtain

(3.3) $\prod_{p}e_{p}=5^{\omega(d(K_{n}))}=5^{\omega(m_{n})},$

Now from Corollary 2.2 we get

(3.4) $|Po(K_{n})|=5^{\omega(m_{n})-1}.$

Thus, for $n\in\mathcal{P}$, the Lehmer quintic field $K_{n}$ is a Pólya field
if and only if $m_{n}$ is a prime or square of a prime. We claim that $m_{n}$
is a square of a prime if and only if $m_{n}=25$. This claim proves (1). To
prove the claim, consider the curve

(3.5) $Y^{2}=f(X)=X^{4}+5X^{3}+15X^{2}+25X+25.$

From Theorem 2.8, we see that any integral solution $(x,y)$ of (3.5) satisfies

$|x|\leq 26\times 25^{3}=406250.$

Using a SAGE program we find that for $x\in[-406250,406250]$ the only integral
solution to the curve $Y^{2}=f(X)$ is $(x,y)=(0,5)$. In other words
$m_{n}=n^{4}+5n^{3}+15n^{2}+25n+25$ is not a square for any non-zero integer
$n$ unless $m_{n}=25$. This establishes the claim.

We have $Gal(K_{n}/\mathbb{Q})\simeq\mathbb{Z}/5\mathbb{Z}$. Let $\sigma$ be a
generator of $Gal(K_{n}/\mathbb{Q})$ and $[\mathfrak{I}]\neq[1]$ be an
ambiguous ideal class in $K_{n}$. Then we have,

$\displaystyle[\mathfrak{I}]^{5}$
$\displaystyle=[\mathfrak{I}][\mathfrak{I}][\mathfrak{I}][\mathfrak{I}][\mathfrak{I}]$
$\displaystyle=[\mathfrak{I}][\mathfrak{I}]^{\sigma}[\mathfrak{I}]^{\sigma^{2}}[\mathfrak{I}]^{\sigma^{3}}[\mathfrak{I}]^{\sigma^{4}}$
$\displaystyle=[N(\mathfrak{I})]$ $\displaystyle=[1]$

where $N(\mathfrak{I})\in\mathbb{Q}$ denotes the norm of the ideal
$\mathfrak{I}$. We conclude that the order of any non-trivial ambiguous ideal
class in the class group of $K_{n}$ is $5$. From the structure theorem for the
abelian groups we obtain

$Po(K_{n})\simeq\left(\frac{\mathbb{Z}}{5\mathbb{Z}}\right)^{\omega({m_{n}})-1}.$

This finishes the proof of $(2)$ of Theorem 1.4.

From (3.1), we see that $m_{n}$ can not be a prime whenever $5|n$. Thus
$K_{n}$ is a non-Pólya field whenever $n\neq 0,5|n$ and $n\in\mathcal{P}$.
Next we show that there are infinitely many such $n^{\prime}$s. To do this we
show that there are infinitely many $k$ such that $m_{5k}$ is cube-free. Put

(3.6) $\displaystyle m_{5k}:$
$\displaystyle=(5k)^{4}+5(5k)^{3}+15(5k)^{2}+25(5k)+25$
$\displaystyle=25(25k^{4}+25k^{3}+15k^{2}+5k+1)$ $\displaystyle=25g(k).$

If $h(k)=ak+b$ is a linear polynomial such that $h(k)^{3}\mid g(k)$, then for
$t=-b/a\in\mathbb{Q}$, we have

(3.7) $\displaystyle g(t)$ $\displaystyle=0\implies
25t^{4}+25t^{3}+15t^{2}+5t+1=0,$ (3.8) $\displaystyle g^{\prime}(t)$
$\displaystyle=0\implies 100t^{3}+75t^{2}+30t+5=0,$ (3.9) $\displaystyle
g^{\prime\prime}(t)$ $\displaystyle=0\implies 300t^{2}+150t+30=0.$

This contradicts the fact that $t\in\mathbb{Q}$. Thus, from Theorem 2.5, it
follows that $g(k)$ is cube-free for infinitely many $k$. Since $5\nmid g(k)$
for all $k$, it follows that $m_{5k}$ is cube-free for infinitely many
integers $k$. This proves that $\mathcal{P}$ is an infinite set and completes
the proof of the theorem. ∎

######  Remark.

From the proof of Theorem 1.4 it follows that for any $n\neq 0$, the Lehmer
quintic field $K_{5n}$ is non-Pólya whenever $m_{5n}$ is cube-free. However,
there are non-Pólya fields $K_{5n}$ with $m_{5n}$ not being cube-free (see the
entry for $n=-53$ in the Table 1).

Conjecturally, there are infinitely many $n\in\mathbb{Z}$ such that $m_{n}$ is
prime and thus the family $K_{n}$ should have infinitely many Pólya fields.
Under the assumption that $m_{n}$ is cube-free, Theorem 1.4 asserts that there
are infinitely many Pólya fields in the family $K_{n}$ only if there are
infinitely many primes of the form $m_{n}$.

###### Proof.

(Proof of Corollary 1.5) From the above remark, it is enough to find integers
$n$ such that $m_{5n}$ is cube-free and $\omega(m_{5n})$ goes to infinity as
$n$ goes to infinity. Let $g(k)=(25k^{4}+25k^{3}+15k^{2}+5k+1)$, then from
Theorem 2.6 for a positive proportion of primes $p$ we have $g(p)$ is cube-
free. Consequently $m_{5p}$ is cube-free for a positive proportion of prime
numbers $p$. Now onwards, we only consider those primes $p$ such that $m_{5p}$
is cube-free. There is a positive constant $c$ such that for any large real
number $X$ there are at least $c\frac{X}{\log X}$ many primes. Now from
Theorem 2.7, it follows that $\omega(m_{5p})$ goes to infinity as $p$ goes to
infinity.

∎

##  4\. Pólya Numbers and monogenity of Lehmer Quintic Fields

Recall the classical embedding problem: Is every number field $K$ contained in
a field $L$ with class number one? In 1964, Golod and Shafarevich [5] gave a
negative answer to this question. The corresponding embedding problem for
Pólya fields was confirmed affirmatively by Leriche [18]. Leriche proved that
every number field is contained in a Pólya field, namely its Hilbert class
field.

A minimal Pólya field over $K$ is a field extension $L$ of $K$ which is a
Pólya field and such that no intermediate field $K\subseteq M\subsetneq L$ is
a Pólya field.

######  Definition 4.1.

[18] The Pólya number of $K$ is given by the integer

$po_{K}=\text{min}\\{[L:K]\mid K\subseteq L,\>L\text{ P\'{o}lya field}\\}.$

The genus field (resp. genus field in the narrow sense) of $K$ is the maximal
abelian extension $\Gamma_{K}$(resp. $\Gamma_{K}^{\prime}$) of $K$ which is a
compositum of $K$ with an absolute abelian number field and is unramified over
$K$ at all places (resp. all finite places) of $K$. The genus number of $K$ is
defined to be the degree $g_{K}=[\Gamma_{K}:K]$. If $K$ is abelian then
Leriche showed that the genus field $\Gamma_{K}$ is Pólya and hence

(4.1) $po_{K}\leq g_{K}.$

On the other hand, Zantema proved that both the cyclotomic and real cyclotomic
fields are Pólya fields [26]. Thus for abelian number fields $K$, if $f$ is
the conductor of $K$ and $\phi(f)$ is the value of Euler totient function then
we have

(4.2) $po_{K}\leq\frac{\phi(f)}{[K:\mathbb{Q}]}\quad\text{ and }\quad
po_{K}\leq\frac{\phi(f)}{2[K:\mathbb{Q}]}\text{ if $K$ is real}.$

For the general case, when $K$ is a Galois number field (not necessarily
abelian) with class number $h_{K}$ then we have

(4.3) $po_{K}\leq h_{K}.$

To prove Theorem Theorem 1.6 we need the following result due to Ishida [13]
on the genus number of a cyclic number field of prime degree.

######  Theorem 4.2.

[13] Let $K$ be a cyclic number field of degree $q$, where $q$ is an odd
prime. If $t$ is the number of primes $p$ such that $p$ is totally ramified in
$K$. Then the genus number $g_{K}$ of $K$ is given as follows.

$g_{K}=\begin{cases}q^{t},\text{ if }q\text{ is totally ramified in }$K$,\\\
q^{t-1},\text{ otherwise. }\end{cases}$

###### Proof.

(Proof of Theorem 1.6) We have already seen in the proof of Theorem 1.4 that,
the number of primes $p$ such that $p$ is totally ramified in $K_{n}$ is
$\omega(m_{n})$. Now applying Theorem 4.2 to the family of number fields
$K_{n}$ we get,

(4.4) $\displaystyle g_{K_{n}}=5^{\omega(m_{n})}\implies po_{K_{n}}\leq
5^{\omega(m_{n})}.$

Using (3.4) in (4.4) we obtain

(4.5) $po_{K_{n}}\leq 5|Po(K_{n})|.$

∎

######  Remark.

Generally both $po_{K}$ and $Po(K)$ are mutually independent objects, but here
in the case of non-Pólya Lehmer quintic fields we have an unexpected relation.

To prove Theorem Theorem 1.7 we need the following result of Gras [7] on the
monogenicity of cyclic number fields of prime degree.

######  Proposition 4.3.

[7] If $K$ is a cyclic number field of prime degree $p\geq 5$ then $K$ is
monogenic only if $2p+1$ is prime and it is the maximal real subfield of
cyclotomic field $\mathbb{Q}(\zeta_{2p+1})$.

Lastly, we recall a result of Zylinski [25].

######  Proposition 4.4.

If $K$ is a number field of degree $n$, then $I(K)$ has only prime divisors
$p$ satisfying $p<n$.

###### Proof.

(Proof of Theorem 1.7) Let $K_{n}$ be the family of non-Pólya Lehmer quintics
fields. We know that $Gal(K_{n}/\mathbb{Q})\simeq\mathbb{Z}/5\mathbb{Z}$. From
Theorem 1.4 we have

(4.6) $|Po(K_{n})|\geq 5.$

Since real cyclotomic fields are Pólya fields, $K_{n}$ never occurs as the
maximal real subfield of a cyclotomic field. From Proposition 4.3 it follows
that $K_{n}$ is not monogenic.

Next, we aim to show $I(K_{n})=1$, for all non-zero $n$ that are divisible by
$5$. We recall the relation

(4.7) $d(\theta_{n})=[I(\theta_{n})]^{2}d(K_{n})$

As mentioned earlier,

$d(\theta_{n})=(n^{3}+5n^{2}+10n+7)^{2}(n^{4}+5n^{3}+15n^{2}+25n+25)^{4}.$

From Lemma 2.3, we see that $(n^{4}+5n^{3}+15n^{2}+25n+25)$ is not divisible
by $2$ or $3$. It is easily seen that $(n^{3}+5n^{2}+10n+7)$ is also not
divisible by $2$ or $3$. Consequently, we conclude that $I(\theta_{n})$ is not
divisible by $2$ or $3$. Now, from the result of Zylinski, it follows that
$I(\theta_{n})=1$.

∎

##  5\. Computation

In this section we present our computations showing that there are many non-
Pólya fields in the family $K_{5n}$. For most of
$n\in\\{-60,-59,\ldots,59,60\\}$ we see that $m_{5n}$ is cube-free with only
exception occurring at $n=-53$. For all $n$ in this range the Pólya group is
non-trivial. In fact, for $n=-53$, $m_{5n}$ is not cube-free but the field
$K_{5n}$ is non-Pólya. In the table below $C_{m_{5n}}$ denotes the cube part
of $m_{5n}$.These computations are performed by us using SAGE.

Table 1. Family of non-Pólya fields $n$ | $m_{5n}$ | $C_{m_{5n}}$ | $\\#Po(K_{5n})$
---|---|---|---
-60 | 7966342525 | 1 | $5^{2}$
-59 | 7446286775 | 1 | $5^{2}$
-58 | 6952119275 | 1 | $5^{2}$
-57 | 6482966275 | 1 | $5^{2}$
-56 | 6037969025 | 1 | $5^{3}$
-55 | 5616283775 | 1 | $5^{2}$
-54 | 5217081775 | 1 | $5^{3}$
-53 | 4839549275 | 1331 | 52
-52 | 4482887525 | 1 | $5^{3}$
-51 | 4146312775 | 1 | $5^{4}$
-50 | 3829056275 | 1 | $5^{3}$
-49 | 3530364275 | 1 | $5^{2}$
-48 | 3249498025 | 1 | $5^{1}$
-47 | 2985733775 | 1 | $5^{3}$
-46 | 2738362775 | 1 | $5^{2}$
-45 | 2506691275 | 1 | $5^{2}$
-44 | 2290040525 | 1 | $5^{2}$
-43 | 2087746775 | 1 | $5^{2}$
-42 | 1899161275 | 1 | $5^{2}$
-41 | 1723650275 | 1 | $5^{2}$
-40 | 1560595025 | 1 | $5^{4}$
-39 | 1409391775 | 1 | $5^{2}$
-38 | 1269451775 | 1 | $5^{2}$
-37 | 1140201275 | 1 | $5^{1}$
-36 | 1021081525 | 1 | $5^{2}$
-35 | 911548775 | 1 | $5^{2}$
-34 | 811074275 | 1 | $5^{2}$
-33 | 719144275 | 1 | $5^{1}$
-32 | 635260025 | 1 | $5^{1}$
-31 | 558937775 | 1 | $5^{3}$
-30 | 489708775 | 1 | $5^{2}$
-29 | 427119275 | 1 | $5^{3}$
-28 | 370730525 | 1 | $5^{2}$
-27 | 320118775 | 1 | $5^{2}$
-26 | 274875275 | 1 | $5^{2}$
-25 | 234606275 | 1 | $5^{1}$
-24 | 198933025 | 1 | $5^{3}$
-23 | 167491775 | 1 | $5^{3}$
-22 | 139933775 | 1 | $5^{1}$
-21 | 115925275 | 1 | $5^{3}$
-20 | 95147525 | 1 | $5^{3}$

$n$ | $m_{5n}$ | $C_{m_{5n}}$ | $\\#Po(K_{5n})$
---|---|---|---
1 | 1775 | 1 | $5^{1}$
2 | 16775 | 1 | $5^{2}$
3 | 71275 | 1 | $5^{1}$
4 | 206525 | 1 | $5^{2}$
5 | 478775 | 1 | $5^{2}$
6 | 959275 | 1 | $5^{1}$
7 | 1734275 | 1 | $5^{1}$
8 | 2905025 | 1 | $5^{1}$
9 | 4587775 | 1 | $5^{1}$
10 | 6913775 | 1 | $5^{3}$
11 | 10029275 | 1 | $5^{2}$
12 | 14095525 | 1 | $5^{1}$
13 | 19288775 | 1 | $5^{2}$
14 | 25800275 | 1 | $5^{2}$
15 | 33836275 | 1 | $5^{3}$
16 | 43618025 | 1 | $5^{2}$
17 | 55381775 | 1 | $5^{3}$
18 | 69378775 | 1 | $5^{2}$
19 | 85875275 | 1 | $5^{1}$
20 | 105152525 | 1 | $5^{1}$
21 | 127506775 | 1 | $5^{3}$
22 | 153249275 | 1 | $5^{2}$
23 | 182706275 | 1 | $5^{1}$
24 | 216219025 | 1 | $5^{2}$
25 | 254143775 | 1 | $5^{3}$
26 | 296851775 | 1 | $5^{2}$
27 | 344729275 | 1 | $5^{3}$
28 | 398177525 | 1 | $5^{2}$
29 | 457612775 | 1 | $5^{1}$
30 | 523466275 | 1 | $5^{2}$
31 | 596184275 | 1 | $5^{2}$
32 | 676228025 | 1 | $5^{2}$
33 | 764073775 | 1 | $5^{1}$
34 | 860212775 | 1 | $5^{2}$
35 | 965151275 | 1 | $5^{3}$
36 | 1079410525 | 1 | $5^{2}$
37 | 1203526775 | 1 | $5^{2}$
38 | 1338051275 | 1 | $5^{3}$
39 | 1483550275 | 1 | $5^{2}$
40 | 1640605025 | 1 | $5^{2}$
41 | 1809811775 | 1 | $5^{2}$

(To be continued)

$n$ | $m_{5n}$ | $C_{m_{5n}}$ | $\\#Po(K_{5n})$
---|---|---|---
-19 | 77296775 | 1 | $5^{2}$
-18 | 62084275 | 1 | $5^{3}$
-17 | 49236275 | 1 | $5^{2}$
-16 | 38494025 | 1 | $5^{2}$
-15 | 29613775 | 1 | $5^{1}$
-14 | 22366775 | 1 | $5^{2}$
-13 | 16539275 | 1 | $5^{2}$
-12 | 11932525 | 1 | $5^{2}$
-11 | 8362775 | 1 | $5^{1}$
-10 | 5661275 | 1 | $5^{1}$
-9 | 3674275 | 1 | $5^{3}$
-8 | 2263025 | 1 | $5^{2}$
-7 | 1303775 | 1 | $5^{2}$
-6 | 687775 | 1 | $5^{3}$
-5 | 321275 | 1 | $5^{2}$
-4 | 125525 | 1 | $5^{1}$
-3 | 36775 | 1 | $5^{1}$
-2 | 6275 | 1 | $5^{1}$
-1 | 275 | 1 | $5^{1}$

$n$ | $m_{5n}$ | $C_{m_{5n}}$ | $\\#Po(K_{5n})$
---|---|---|---
42 | 1991781775 | 1 | $5^{3}$
43 | 2187141275 | 1 | $5^{4}$
44 | 2396531525 | 1 | $5^{1}$
45 | 2620608775 | 1 | $5^{2}$
46 | 2860044275 | 1 | $5^{2}$
47 | 3115524275 | 1 | $5^{3}$
48 | 3387750025 | 1 | $5^{3}$
49 | 3677437775 | 1 | $5^{4}$
50 | 3985318775 | 1 | $5^{1}$
51 | 4312139275 | 1 | $5^{1}$
52 | 4658660525 | 1 | $5^{2}$
53 | 5025658775 | 1 | $5^{2}$
54 | 5413925275 | 1 | $5^{2}$
55 | 5824266275 | 1 | $5^{3}$
56 | 6257503025 | 1 | $5^{3}$
57 | 6714471775 | 1 | $5^{3}$
58 | 7196023775 | 1 | $5^{2}$
59 | 7703025275 | 1 | $5^{3}$
60 | 8236357525 | 1 | $5^{2}$

## References

  * [1] P. J. Cahen, J. L. Chabert, Integer-Valued Polynomials, Math. Surveys Monogr., Amer. Math. Soc., Providence, 1997.
  * [2] J. L. Chabert, Factorial groups and Pólya groups in Galoisian extensions of $\mathbb{Q}$, Commutative Ring Theory and Applications, Lect. Notes Pure Appl. Math. 231, Dekker, New York, (2003), 77–86.
  * [3] J. Chattopadhyay and A. Saikia, Totally real bi-quadratic fields with large Pólya groups, Res. Number Theory 8, 30 (2022). https://doi.org/10.1007/s40993-022-00327-8
  * [4] P. Erdös, Arithmetical properties of polynomials, J. London Math. Soc. 28 (1953), 416–425.
  * [5] E. S. Golod, I. R. Shafarevich, On the class field tower, Izv. Akad. Nauk 28 (1964), 261–272.
  * [6] J. Gillibert and A. Levin, A Geometric Approach to Large Class Groups: A Survey, In: Chakraborty, K., Hoque, A., Pandey, P. (eds) Class Groups of Number Fields and Related Topics. Springer, Singapore. https://doi.org/10.1007/978-981-15-1514-9_1
  * [7] M. -N. Gras, Non monogénéité de l’anneau des entires des extensions cycliques de $\mathbb{Q}$ de degré premier $l\geq 5$, J. Number Theory 23 (1986), 347–353.
  * [8] H. Halberstam, On the distribution of additive number-theoritic functions. III, J. London Math. Soc. 31 (1956), 14–27.
  * [9] B. Heidaryan and A. Rajaei, Biquadratic Pólya fields with only one quadratic Pólya subfield, J. Number Theory 143 (2014), 279–285.
  * [10] B. Heidaryan and A. Rajaei, Some non-Pólya biquadratic fields with low ramificattion, Rev. Mat. Iberoam 33 (2017), 1037–1044.
  * [11] H. Helfgott, Power-free values, large deviations and integer points on irrational curves, J. Théor. Nombres Bordeaux 19, (2007), 433–472.
  * [12] H. Helfgott, Power-free values repulsion between points, differing beliefs and the existence of error, Anatomy of integers, CRM Proceedings & Lecture Notes 46, (American Mathematical Society, Providence, RI, 2008) 81–88.
  * [13] M. Ishida, The Genus Fields of Algebraic Number Fields, Springer (1976).
  * [14] S. Jeannin, Nombre de classes et unités des corps de nombres cycliques quintiques d’E. Lehmer, J. Théor. Nombres Bordeaux 8 (1996), 75–92.
  * [15] E. Lehmer, Connection between Gaussian periods and cyclic units, Math. Comp. 50 (1988), 535–541.
  * [16] A. Leriche, Pólya field, Pólya groups and Pólya extensions: A question of capitulation, J. Theor. Nombres Bordeaux 23 (2011), 235–249.
  * [17] A. Leriche, Cubic, quartic and sextic Pólya fields, J. Number Theory 133 (2013), 59–71.
  * [18] A. Leriche, About the embedding of a number field in a Pólya field, J. Number Theory 145, (2014), 210–229.
  * [19] A. Maarefparvar, Pólya group in some real biquadratic fields , J. Number Theory 228, (2021), 1–7.
  * [20] D. W. Masser, Polynomial Bounds for Diophantine Equations, Amer. Math. Monthly 93, (1986), 486–488.
  * [21] T. Reuss, Power-free values of polynomials, Bull. London Math. Soc. 47, (2015), 270–284.
  * [22] R. Schoof, L. C. Washington, Quintic polynomials and real cyclotomic fields with large class numbers Math. Comp. 50 (1988), 543–556.
  * [23] C. W. -W. Tougma, Some questions on biquadric Pólya fields, J. Number Theory 229, (2021), 386–398.
  * [24] Z. Wolske, “Number Fields with Large Minimal Index”, thesis, University of Toronto, 2018.
  * [25] E. von Zylinski, Zur Theorie der außerwesentlichen Diskriminantenteiler algebraischer K̈orper, Math. Ann. 73(2), (1913), 273–274.
  * [26] H. Zantema, Integer valued polynomials over a number field, Manuscripta Math. 40, (1982), 155–203.


§ INTRODUCTION

In multi-robot SLAM, most works are based on sparse landmark SLAM where robots share and fuse sub-maps e.g., <cit.>, <cit.>, <cit.>, <cit.>.
In <cit.>, Nettleton selects informative parts of the map to communicate between robots.
They use a hybrid channel filter (CF) – covariance intersection (CI) algorithm to ensure that previously communicated map data is counted not more than once.
Reece and Roberts <cit.> improve upon this work by replacing the CI with a less conservative fusion rule called Bounded Covariance Inflation.
In <cit.> Julier and Uhlmann use different variations of the CI algorithm <cit.> to solve the Full Covariance SLAM problem.
Their approach suggests building three types of maps: main, relative, and local, using different instantiations of a Kalman filter (KF) and then fusing them together using CI to account for unknown dependencies between the estimates.
In <cit.>, Tchiev and Indelman consider the problem of semantic distributed SLAM with sparse landmarks, where robots fuse belief over both the map and classes of objects.
To avoid double counting previously communicated data and simplify bookkeeping, they assume the communication network topology is undirected and acyclic.

While for sparse landmark SLAM it is possible to share parts of the map, i.e. coordinates of landmarks, this approach is less feasible when considering dense maps <cit.>, <cit.>  and metric-semantic SLAM algorithms <cit.>.
For that reason, recent approaches either use a centralized server to merge maps <cit.>, <cit.>, or choose to  collaboratively optimize the robots' trajectories (instead of the map) using pose graph optimization (PGO) techniques <cit.>. The improved trajectory estimate is then fed back to correct the local map estimate.

The problem of collaborative tracking and SLAM has gotten much less attention in the literature than collaborative SLAM.
In <cit.> Wang formulates the SLAM and detection and tracking of moving objects (DATMO) problem in Bayesian terms.
For a single robot problem, they suggest splitting the SLAM and DATMO problems into two separate solvers but assume that data regarding the moving object (target) does not carry any information concerning the map and robot pose.
Moratuwage <cit.> use random finite sets (RFS) for collaborative multi-vehicle SLAM with moving object tracking.
Their work solves for sparse landmark representation of the map and is not fully decentralized, as the posterior pdf is based on all measurements and trajectories.

In this paper, we present a new decentralized graph-based approach to the multi-robot SLAM and tracking problem.
We leverage factor graphs to split between different parts of the problem for efficient data sharing between robots in the network while enabling robots to use different local sparse landmark/dense/metric-semantic SLAM algorithms.
The paper is organized as follows: Sec. <ref> defines the decentralized heterogeneous multi-robot SLAM and tracking problem, Sec. <ref> presents our factor graph-based technical approach, and Sec. <ref> describes our planned simulation study and a roadmap for future hardware experiments.

§ PROBLEM STATEMENT

Consider a network of $n_r=|N_r|$ robots in an unknown joint environment.
Each robot reasons about the environment and tries to gain situational awareness using its local, potentially different, SLAM algorithm.
In addition, the robots are tasked with jointly inferring (estimating) the unknown state of a subset of $n_t=|N_t|$ static or dynamic targets at time step $k$.
In a probabilistic approach, the uncertainty over full system multi-robot SLAM and multi-target tracking problem can be described by,

\begin{equation}
\label{eq:fullSLAM}
\end{equation}

Here, $p(\chi|Z_k)$ is the pdf over the full system set of random variables (rvs), conditioned on the data set $Z_k=\bigcup_{i\in N_r} Z^i_k$ gathered by all robots.
$X_{k:0}$ is the set of rvs describing the uncertain poses of all $n_r$ robots at time steps $0$ to $k$, $T_k$ is the set of rvs describing the uncertain position of all $n_t$ targets at time step $k$, and $M$ is a set of rvs representing the map.
Note that we defined $M$ rather ambiguously, so as to enable heterogeneity in the system. This allows each robot to hold a different representation of the environment, e.g. sparse landmarks, point clouds, and labeled objects.

§.§ Decentralized SLAM and Tracking:

In a decentralized formulation of the problem, each robot $i$ is tasked with inferring a subset of the full set of rvs $\chi^i \subseteq \chi$: its own pose $x^i_{k:0}\subset X_{k:0}$, local map $M^i\subseteq M$, and the states $T^i$ of a subset $N_t^i\subseteq N_t$ of the targets.
The local inference task of any robot $i\in N_r$ can be described by,

\begin{equation}
\label{eq:decSLAM}
\end{equation}

In decentralized SLAM and tracking, each robot maintains a pdf (<ref>) based on the data it gathers from local sensors and the data it receives from neighboring robots through peer-to-peer communication.
Since each robot maintains a different, but overlapping pdf with its neighbors, this is an instance of a heterogeneous fusion problem <cit.>.
In heterogeneous fusion, we split the robot's set of rvs $\chi^i$ to
common rvs $\chi^i_C=\bigcup_{j\in N_r^i}^{}\chi^{ij}_C$, local rvs $\chi^i_L$ which are not monitored by any other robot in the network, and non-mutual rvs $\chi^{i\setminus j}=\chi^i_L\cup\{\chi^i_C\setminus \chi^{ij}_C \}$.
In our proposed SLAM and tracking, the common rvs are the common target states $T^{ij}_C$, the local rvs are the local map, and the robot's pose states; and $T^{i\setminus j}$ is the subset of targets monitored by robot $i$, and possibly other robots, but not $j$.

An edge $e_{lm}\in E$ in the graph only connects a factor node l to a variable node m.
The joint distribution over the graph is then proportional to the global function $f(V)$:

\begin{equation}
p(V)\propto f(V)=\prod_{l}f_l(V_l),
\label{eq:factorization}
\end{equation}

where $f_l(V_l)$ is a function of only those variables $v_m\in V_l$ connected to the factor l.
That enables the graph to explicitly express the conditional independence structure of the problem and makes factor graphs an attractive representation for decentralized inference problems <cit.>.

§ TECHNICAL APPROACH

Factor graph representation of a 2-robot SLAM and target tracking application. Both robots $i$ and $j$ are building local maps $M^i$ and $M^j$ while cooperatively tracking targets in common $T^{ij}_C$ (observed at time step 3). The data  fused regarding common targets will indirectly update the map and the estimate of non-mutual targets $T^{i\setminus j}$.

The key part of our approach is exploiting the probabilistic conditional independence structure of the problem to: (i) enable robots to use different SLAM algorithms and only share `relevant' parts of their pdfs with each other for fusion, and (ii) enable inner-loop robot architecture that separates the SLAM solution, which includes the local map and ego pose, from the target tracking solution.
First, let us look at the example factor graph in Fig. <ref> showing the dependency structure of a two-robot problem.

The scenario depicted in the graph (Fig. <ref>) shows two robots $i$ and $j$, tasked with a SLAM problem – estimating their pose $x^i_{k:1} \ (x^j_{k:1})$ and the local map $M^i \ (M^j)$.
At the same time, the two robots are tasked with estimating the state of a set of common targets $T_C^{ij}$ and a set of targets, only tracked by robot $i$, $T^{i\setminus j}$.
From the graph, we see that non-mutual variables (pose, map, exclusive targets) of each robot are conditionally independent given the common target variables $T^{ij}_{C}$.
With this conditional independence structure, we can use the heterogeneous fusion rule developed in <cit.>, to perform a peer-to-peer fusion of data regarding common targets,

\begin{equation}
\begin{split}
&p_f^i(\chi^i|Z^{i,+}_k)\propto \\
&p^i(x^i_{k:0}, M^i,T^{i\backslash j}|T^{ij}_{C,k},Z^{i,-}_k)\cdot
\frac{p^i(T^{ij}_{C,k}|Z^{i,-}_k)p^j(T^{ij}_{C,k}|Z^{j,-}_k)}{p^{ij}_c(T^{ij}_{C,k}|Z^{i,-}_k \cap Z^{j,-}_k)}.
\end{split}
\label{eq:Heterogeneous_fusion}
\end{equation}

Here $Z^{i,-}_k$ and $Z^{i,+}_k$ are the data sets available for agent $i$ at time $k$ prior and post fusion, respectively.

Second, we can again exploit conditional independence, now in the inner-robot level, to separate the map $M^i$ from the targets $T^i$.
Looking at the two left boxes (purple and green) in Fig. <ref>, the map and tracking variables are conditionally independent given the ego poses in which the targets were observed, i.e. in this example $M^i \perp T^i|x_3^i$.
With this conditional independence, drawing inspiration from the NET-DDF architecture suggested in <cit.>, we can separate between the SLAM and tracking solutions.
As shown in Fig. <ref>, we suggest using two modules - a tracking module, to handle the robot's tracking assignment, and a SLAM module, to solve for the robot's pose and the local map.
This architecture contains a filter and a channel filter (CF) <cit.> to explicitly track the data transferred between the local filter, responsible for tracking, and the SLAM module.
The architecture expresses the following factorization of (<ref>) and data fusion between the two modules,

\begin{equation}
\begin{split}
&p^i(\chi^i|Z^i_k)=\\    &\frac{\overbrace{p(M^i|x^i_{k:0},Z^{i,s}_k)p(x^i_{k:0}|Z^{i,s}_k)}^{\textrm{SLAM Module}}\cdot \overbrace{p(x^i_{K}|Z^{i,t}_k)p(T^i_{k}|x^i_{K},Z^{i,t}_k)}^{\textrm{Tracking Module}}}
{\underbrace{p(x^i_{K}|Z^{i,s}_k\cap Z^{i,t}_k)}_{\textrm{CF}}}.
\end{split}
\label{eq:decSLAMfactorized}
\end{equation}

Where $Z_k^{i,s}$ and $Z_k^{i,t}$ are the data sets gathered by the SLAM and tracking modules, respectively, and $K$ describes a vector of time steps in which the targets were observed.

The pipeline then works in the following way:

* Initialization: Each robot $i$ instantiates (i) a local SLAM engine (purple in Fig. <ref>), (ii) a tracking filter (green in Fig. <ref>) to estimate the set of $N_t^i$ targets, represented by a factor graph, (iii) a stack of CFs, one for the SLAM-Tracking data tracking (pink in Fig. <ref>) and another one for each communication link with neighboring robots (not shown), to track common data regarding common targets <cit.>, <cit.>.

* Run SLAM: Each robot $i$ uses its' local SLAM algorithm to maintain a map of the environment and an ego pose estimate (Fig <ref>(a)).

* Request pose estimate: When the robot observes a target $t\in T^i$, the tracking module `asks' the SLAM module for a pose estimate.

* Send pose estimate: The SLAM module sends a factor (or a set of factors) proportional to the marginal pose estimate, e.g., the blue factor in Fig. <ref>(a) and (c).

* Receive pose estimate: After removing common data by the CF, a new factor is integrated into both the CF and the tracking module's graphs.

* Peer-to-peer fusion: The tracking module can now communicate with a neighboring robot over common targets according to Fig. <ref>, and using the following version of the heterogeneous fusion rule (<ref>),

\begin{equation}
\begin{split}
p_f^i(x^i_{K}, T^i&|Z^{i,t,+}_k)\propto
p^i(x^i_{K},T^{i\backslash j}|T^{ij}_{C,k},Z^{i,t,-}_k)\\ \times &\frac{p^i(T^{ij}_{C,k}|Z^{i,t,-}_k)p^j(T^{ij}_{C,k}|Z^{j,t,-}_k)}{p^{ij}_c(T^{ij}_{C,k}|Z^{i,t,-}_k \cap Z^{j,t,-}_k)}.
\end{split}
\label{eq:Heterogeneous_fusion_tracking}
\end{equation}

* Send pose back to SLAM: The tracking module removes common data using the CF and sends the SLAM module marginal ego pose factors (orange factor in <ref>(b) and (d)), as it now has new data to send, both from the robot-target measurement and indirectly from fusion with a neighboring robot $j$.

* Add factors to SLAM: SLAM module adds new data from the tracking module to the SLAM graph (orange factor in <ref>(b) and (d)).

* Repeat: Process repeats recursively.

§ SIMULATION PLAN AND EXPERIMENTAL ROADMAP

To test our approach, we will simulate $n_r\geq 2$ robots moving in a joint Gazebo environment <cit.>, tracking $n_t\geq 5$ moving targets.
The robots estimate the $2D$ position and velocity of each target $t^t_k=[X^t_k,\dot{X}^t_k,Y^t_k,\dot{Y}_k^t]^T$.
Targets are assumed to be moving according to the following linear kinematics model,
\begin{equation}
\begin{split}
&t^t_{k+1}=Ft^t_{k}+Gu^t_k+\omega_k, \ \ \omega_k \sim \mathcal{N}(0,0.08\cdot I_{n_x\times n_x}),\\
&F=\begin{bmatrix}1 & \Delta t &0 &0\\0 &1 &0 &0\\ 0 &0 &1 & \Delta t\\0& 0 &0 &1 \end{bmatrix}, \quad
G=\begin{bmatrix}\frac{1}{2}\Delta t^2 &0\\\Delta t&0\\0 &\frac{1}{2}\Delta t^2\\0 &\Delta t \end{bmatrix}.
\end{split}
\label{eq:dynamicEq}
\end{equation}

Where $\Delta t$ is the time step, $u_k^t$ is the control input of target $t$, and $\omega_k$ is zero-mean additive white Gaussian (AWGN) noise.
When a target $t$ is within robot $i$'s range, it can take a relative position measurement,

\begin{equation}
\begin{split}
y^{i,t}_{k} &= \begin{bmatrix}
X^i_k-X^t_k \\
\end{bmatrix} +v^{i}_k, \ \ v^{i}_k \sim \mathcal{N}(0,R^{i}_k),
\end{split}
\label{eq:meas_model}
\end{equation}

where $v^i_k$ is zero-mean AWGN noise. For now, it is assumed that the target data association problem is solved separately.

To test our system, we need a dataset that includes data describing (i) multiple robots in a joint environment, and (ii) different measurement modalities, e.g., LiDAR and camera.
To the best of our knowledge, such a dataset does not exist. For example, <cit.> contains several sequences that can be used in parallel to simulate collaborative SLAM, but all sequences contain only visual-inertial data.
On the other hand, <cit.> provides data from multiple sensors such as IMU, LiDAR, and Camera, but it only includes one sequence per scenario, which is not applicable for a multi-robot scenario.
Thus, to rigorously build and test our system, our plan is to start with Gazebo-based simulations before moving to hardware experiments.
We designed a simple Gazebo environment, including Clearpath Jackal unmanned ground vehicles (UGV) that can simulate the SLAM and tracking robots, as well as the targets.
The preliminary study aims to test the architecture of the system, we will use LIO-SAM <cit.>, a factor graph-based lidar-inertial odometry system, as the SLAM engine.
When the tracking module requests a pose estimate from the SLAM module (see <ref>), we query the LIO-SAM factor graph for the marginal distribution over the requested time steps, represented by the information vector and matrix.
These are then integrated as a new factor to the tracking module factor graph and CF, as shown in Fig. <ref>(a) and (c).
When the SLAM module receives a new pose factor back from the tracking algorithm, as shown in Fig. <ref>(b) and (d), it adds it to the LIO-SAM graph as a GPS or loop closure type factors <cit.>.

When successful, the rest of the test plan is as follows: (i) Gazebo simulation with two different SLAM algorithms, e.g., LIO-SAM <cit.> - LiDAR-inertial, and Kimera <cit.> - visual-inertial. (ii) hardware experiments using 2-3 Clearpath Huskey UGVs and varying numbers of maneuvering and static targets in an outdoor environment.

§ SUMMARY

The primary goal of this paper is to report progress toward a heterogeneous robotic system, where robots are able to collaborate despite differences in their onboard algorithms.
We use factor graphs to analyze and exploit the conditional independence structure inherent in the decentralized multi-robot heterogeneous SLAM and tracking problem.
We are then able to design a system architecture that separates the SLAM and tracking solutions of the robot and between the SLAM systems used by neighboring robots.
While this paper focuses on a SLAM and target tracking problem, this approach can be extended to other robotic applications such as cooperative localization and navigation <cit.>, and terrain height mapping <cit.>.