# The combinatorics of weight systems and characteristic polynomials of
isolated quasihomogeneous singularities

Claus Hertling and Makiko Mase  Claus Hertling
Lehrstuhl für algebraische Geometrie, Universität Mannheim, B6 26, 68159
Mannheim, Germany<EMAIL_ADDRESS>Makiko Mase
Lehrstuhl für algebraische Geometrie, Universität Mannheim, B6 26, 68159
Mannheim, Germany<EMAIL_ADDRESS>
(Date: August 02, 2021)

###### Abstract.

A paper of the first author and Zilke proposed seven combinatorial problems
around formulas for the characteristic polynomial and the exponents of an
isolated quasihomogeneous singularity. The most important of them was a
conjecture on the characteristic polynomial. Here the conjecture is proved,
and some of the other problems are solved, too. In the cases where also an old
conjecture of Orlik on the integral monodromy holds, this has implications on
the automorphism group of the Milnor lattice. The combinatorics used in the
proof of the conjecture consists of tuples of orders on sets $\\{0,1,...,n\\}$
with special properties and may be of independent interest.

###### Key words and phrases:

quasihomogeneous singularity, weight system, monodromy, characteristic
polynomial, combinatorial problems, Orlik blocks

###### 2010 Mathematics Subject Classification:

32S40, 05C25, 06A05, 11H56

This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) – 242588615

###### Contents

  1. 1 Introduction
  2. 2 Compatibility of a finite set of natural numbers with a tuple of excellent orders
  3. 3 Tensor product and Thom-Sebastiani sum
  4. 4 Maps on the group ring generated by unit roots
  5. 5 Basic properties of weight systems and induced objects
  6. 6 Isolated quasihomogeneous singularities

##  1\. Introduction

The paper [HZ19] proposed seven combinatorial problems around formulas for the
characteristic polynomial and the exponents of an isolated quasihomogeneous
singularity. Problem 6 was a conjecture on the characteristic polynomial. It
was an amendment to an old conjecture of Orlik on the integral monodromy
(which is recalled below). Here the conjecture on the characteristic
polynomial is proved, and also some of the other combinatorial problems are
solved.

The combinatorics which is used in the proof of the conjecture was found in
[HM20-2] and was used there for a partial proof of Orlik’s conjecture. It is
described in detail in section 2. It may be of independent interest.

In this introduction the conjecture and some background are explained. This
contains problem 6 in [HZ19]. Problem 7 in [HZ19] is treated in section 3.
Sections 4 sets some notations and notions. Section 5 solves problem 1.
Section 6 discuss the combinatorics of the weight systems and of the
characteristic polynomial of an isolated quasihomogeneous singularity. It
solves problem 4 and discusses the problems 2, 3 and 5.

We start with the notion of an Orlik block and a result on its automorphisms.
Then we introduce isolated quasihomogenous singularities. Finally we give
Theorem 1.4, which solves problem 6 in [HZ19], and Orlik’s conjecture.

######  Definition 1.1.

Let $M\subset{\mathbb{N}}=\\{1,2,3,...\\}$ be a finite nonempty subset. Its
Orlik block is a pair $(H_{M},h_{M})$ with $H_{M}$ a ${\mathbb{Z}}$-lattice of
rank $\sum_{m\in M}\varphi(m)$ and $h_{M}:H_{M}\to H_{M}$ an automorphism with
characteristic polynomial $\prod_{m\in M}\Phi_{m}$ ($\Phi_{m}$ is the $m$-th
cyclotomic polynomial) and with a cyclic generator $e_{1}\in M$, i.e.

$\displaystyle H_{M}=\bigoplus_{j=1}^{rkM}{\mathbb{Z}}\cdot
h_{M}^{j-1}(e_{1}).$ (1.1)

$(H_{M},h_{M})$ is unique up to isomorphism.
$\operatorname{Aut}_{S^{1}}(H_{M},h_{M})$ denotes the group of all
automorphisms of $H_{M}$ which commute with $h_{M}$ and which have all
eigenvalues in $S^{1}$.

######  Definition 1.2.

A finite set $M\subset{\mathbb{N}}$ is enriched as follows to a directed graph
$(M,E(M))$ with set of vertices $M$ and set of oriented edges $E(M)\subset
M\times M$. An edge goes from $m_{1}\in M$ to $m_{2}\in M$ if
$\frac{m_{1}}{m_{2}}$ is a power of a prime number $p$. Then it is called a
$p$-edge.

The main result in [He20] is as follows.

######  Theorem 1.3.

[He20, Theorem 1.2] Let $(H_{M},h_{M})$ be the Orlik block of a finite
nonempty subset $M\subset{\mathbb{N}}$. Then
$\operatorname{Aut}_{S^{1}}(H_{M},h_{M})=\\{\pm
h_{M}^{k}\,|\,k\in{\mathbb{Z}}\\}$ if and only if condition (I) or condition
(II) in Definition 2.6 (c) and (d) are satisfied. They are conditions on the
graph $(M,E(M))$.

A weight system ${\bf w}=(w_{1},...,w_{n};1)$ with
$w_{i}\in{\mathbb{Q}}_{>0}\cap(0,1)$ equips any monomial ${\bf x}^{\bf
j}=x_{1}^{j_{1}}...x_{n}^{j_{n}}$ with a weighted degree $\deg_{\bf w}{\bf
x}^{\bf j}:=\sum_{i=1}^{n}w_{i}j_{i}$. A polynomial
$f\in{\mathbb{C}}[x_{1},...,x_{n}]$ is an isolated quasihomogeneous
singularity if for some weight system ${\bf w}$ each monomial in $f$ has
weighted degree 1 and if the functions $\frac{\partial f}{\partial
x_{1}},...,\frac{\partial f}{\partial x_{n}}$ vanish simultaneously only at
$0\in{\mathbb{C}}^{n}$. Then the Milnor lattice
$H_{Mil}:=H_{n-1}(f^{-1}(1),{\mathbb{Z}})$ (resp. the reduced homology in the
case $n=1$) is a ${\mathbb{Z}}$-lattice of some rank $\mu\in{\mathbb{N}}$
[Mi68], which is called Milnor number. It comes equipped with a natural
automorphism $h_{Mil}:H_{Mil}\to H_{Mil}$ of finite order, the monodromy. Its
characteristic polynomial has the shape

$\displaystyle p_{ch,h_{Mil}}$
$\displaystyle=\prod_{m\in{\mathbb{N}}}\Phi_{m}^{\psi(m)}$

for a function $\psi:{\mathbb{N}}\to{\mathbb{N}}_{0}$ with finite support
$M_{1}:=\\{n\in{\mathbb{N}}\,|\,\psi(n)\neq 0\\}$. Denote
$l_{\psi}:=\max(\psi(m)\,|\,m\in{\mathbb{N}})$ and for $j=1,...,l_{\psi}$

$\displaystyle M_{j}:=\\{m\in{\mathbb{N}}\,|\,\psi(m)\geq j\\}.$

The tuple $(M_{1},...,M_{l_{\psi}})$ is called standard covering of $\psi$. It
satisfies (and is determined by this)

$\displaystyle M_{1}\supset M_{2}\supset...\supset
M_{l_{\psi}}\neq\emptyset,\quad\psi(m)=|\\{j\in\\{1,...,l_{\psi}\\}\,|\,m\in
M_{j}\\}|.$

The most important result in this paper is Theorem 1.4.

######  Theorem 1.4.

(a) In the case of an isolated quasihomogeneous singularity, each set $M_{j}$
satisfies condition (I) in Definition 2.6 (c).

(b) $\operatorname{Aut}_{S^{1}}(H_{M_{j}},h_{M_{j}})=\\{\pm
h_{M_{j}}^{k}\,|\,k\in{\mathbb{Z}}\\}.$

Proof: Part (b) follows from part (a) and Theorem 1.3. Part (a) follows from
Lemma 2.7 and Theorem 6.6. Lemma 2.7 says that a finite set
$M\subset{\mathbb{N}}$ which is compatible with a tuple of excellent orders
(Definition 2.4 (d)) satisfies condition (I). Theorem 6.6 says that in the
case of an isolated quasihomogeneous hypersurface singularity each set $M_{j}$
is compatible with a certain tuple of excellent orders. It was proved in
[HM20-2, Theorem 12.1]. It is cited in section 6. $\Box$

Problem 6 in [HZ19] asked precisely for part (a) of Theorem 1.4. The notion of
an excellent order and the notion of compatibility of a finite set
$M\subset{\mathbb{N}}$ with a tuple of excellent orders were first used in
[HM20-2, section 9]. We found them when we searched for a partial proof of
Orlik’s conjecture. They are recalled in section 2.

Theorem 1.4 is independent of Orlik’s conjecture. But it is useful only in the
cases where Orlik’s conjecture holds.

######  Conjecture 1.5.

(Orlik’s conjecture, [Or72, conjecture 3.1]) For any isolated quasihomogeneous
singularity, there is an isomorphism

$\displaystyle(H_{Mil},h_{Mil})\cong\bigoplus_{j=1}^{l_{\psi}}(H_{M_{j}},h_{M_{j}}).$

Our partial positive results [HM20-2, Theorem 1.3] on this conjecture are
recalled below in Theorem 3.4 and right after it.

In those isolated quasihomogeneous singularities, where Orlik’s conjecture
holds, $(H_{Mil},h_{Mil})$ decomposes (not uniquely in general) into Orlik
blocks $(H_{M_{j}},h_{M_{j}})$, and Theorem 1.4 says that the group of
automorphisms with eigenvalues in $S^{1}$ of each of these blocks is the
smallest possible, it is $\\{\pm h_{M_{j}}^{k}\,|\,k\in{\mathbb{Z}}\\}$. This
is very helpful also in determining the group of all automorphisms of the
triple $(H_{Mil},h_{Mil},L_{Mil})$ where $L_{Mil}:H_{Mil}\times
H_{Mil}\to{\mathbb{Z}}$ is the Seifert form. This group is important for
period maps and Torelli conjectures for singularities (e.g. [He11]).

Problem 7 in [HZ19] is a strengthening of problem 6 and is also solved by
Lemma 2.7 and the results in [HM20-2, section 9]. This is explained in section
3. The other problems 1–5 in [HZ19] are discussed in the sections 5 and 6.

######  Notations 1.6.

${\mathbb{N}}=\\{1,2,3,..\\}$, ${\mathbb{N}}_{0}=\\{0,1,2,..\\}$. Always
$m\in{\mathbb{N}}$ and $n\in{\mathbb{N}}$. For $I\subset{\mathbb{R}}$,
${\mathbb{Z}}_{I}:={\mathbb{Z}}\cap I$. The support of a map
$\psi:{\mathbb{N}}\to{\mathbb{Q}}$ is the set
$\operatorname{supp}(\psi):=\\{m\in{\mathbb{N}}\,|\,\psi(m)\neq 0\\}$. We will
consider only maps $\psi:{\mathbb{N}}\to{\mathbb{Q}}$ with finite support.
Most often, $\psi$ will be a map $\psi:{\mathbb{N}}\to{\mathbb{N}}_{0}$ with
finite support.

The set of all prime numbers is ${\mathcal{P}}\subset{\mathbb{N}}$. For
$p\in{\mathcal{P}}$ and $m\in{\mathbb{N}}$, $v_{p}(m)\in{\mathbb{N}}_{0}$ is
the unique number with $m=\prod_{q\in{\mathcal{P}}}q^{v_{q}(m)}$. Also the
projection

$\pi_{p}:{\mathbb{N}}\to\\{n\in{\mathbb{N}}\,|\,p\not|n\\},\quad m\mapsto
m\cdot p^{-v_{p}(m)},$

will be used.

##  2\. Compatibility of a finite set of natural numbers with a tuple of
excellent orders

This section proposes and discusses a condition for a finite set
$M\subset{\mathbb{N}}$ of natural numbers and a condition for a map
$\psi:{\mathbb{N}}\to{\mathbb{N}}_{0}$ with finite support. They are given in
Definition 2.4 (d) and (e). They have a number of good properties, which will
be given below in Lemma 2.5, Lemma 2.7, and the Theorems 3.2, 3.4 and 3.6.
This is prepared by several definitions. The Definitions 2.1, 2.3 and 2.4,
Lemma 2.5, and the Theorems 3.2, 3.4 and 3.6 are recalled from [HM20-2,
section 9]. Definition 2.6 is recalled from [He20]. Lemma 2.7 is new. It
connects the condition in Definition 2.4 (d) for a finite set
$M\subset{\mathbb{N}}$ with condition (I) (in Theorem 1.3 and Definition 2.6
(c)).

######  Definition 2.1.

[HM20-2, 9.1] (a) An excellent order $\succ$ on a set
${\mathbb{Z}}_{[0,s(\succ)]}$ for some bound $s(\succ)\in{\mathbb{N}}_{0}$ is
a strict order (so transitive and for all $a,b\in{\mathbb{Z}}_{[0,s(\succ)]}$
either $a=b$ or $a\succ b$ or $b\succ a$) which is determined by the set

$\displaystyle S(\succ)$ $\displaystyle:=$
$\displaystyle\\{k\in{\mathbb{Z}}_{[0,s(\succ)]}\,|\,k\succ 0\\}$ (2.1)

in the following way:

$\displaystyle\left.\begin{array}[]{l}\succ\textup{ equals }>\textup{ on
}S(\succ)\cup\\{0\\},\\\ \succ\textup{ equals }<\textup{ on
}{\mathbb{Z}}_{[0,s(\succ)]}-S(\succ).\end{array}\right\\}$ (2.4)

($S(\succ)=\emptyset$ is allowed.) The maximal element of
${\mathbb{Z}}_{[0,s(\succ)]}$ with respect to $\succ$ is called
$s^{+}(\succ)$, so $s^{+}(\succ)\succ k$ for any other element
$k\in{\mathbb{Z}}_{[0,s(\succ)]}$.

(b) The trivial excellent order is $\succ_{0}$ with $s(\succ_{0}):=0$, so it
is the empty order on ${\mathbb{Z}}_{[0,s(\succ_{0})]}=\\{0\\}$ (and, of
course $S(\succ_{0})=\emptyset$).

(c) The tensor product of two excellent orders $\succ_{1}$ and $\succ_{2}$ is
the excellent order $\succ_{1}\otimes\succ_{2}$ with

$\displaystyle s(\succ_{1}\otimes\succ_{2})$ $\displaystyle:=$
$\displaystyle\max(s(\succ_{1}),s(\succ_{2}))\quad\textup{and}$ (2.5)
$\displaystyle S(\succ_{1}\otimes\succ_{2})$ $\displaystyle:=$
$\displaystyle(S(\succ_{1})\cup S(\succ_{2}))-(S(\succ_{1})\cap
S(\succ_{2})).$ (2.6)

######  Examples 2.2.

[HM20-2, 9.2] (i) The excellent order $\succ_{1}$ with $s(\succ_{1})=7$ and
$S(\succ_{1})=\\{6,4,1\\}$ is given by

$\displaystyle
s^{+}(\succ_{1})=6\succ_{1}4\succ_{1}1\succ_{1}0\succ_{1}2\succ_{1}3\succ_{1}5\succ_{1}7.$

(ii) The excellent order $\succ_{2}$ with $s(\succ_{2})=6$ and
$S(\succ_{2})=\\{6,5,2,1\\}$ is given by

$\displaystyle
s^{+}(\succ_{2})=6\succ_{2}5\succ_{2}2\succ_{2}1\succ_{2}0\succ_{2}3\succ_{2}4.$

(iii) The excellent order $\succ_{3}:=(\succ_{1}\otimes\succ_{2})$ for
$\succ_{1}$ and $\succ_{2}$ in (i) and (ii) satisfies $s(\succ_{3})=7$,
$S(\succ_{3})=\\{5,4,2\\}$ and is given by

$\displaystyle
5\succ_{3}4\succ_{3}2\succ_{3}0\succ_{3}1\succ_{3}3\succ_{3}6\succ 7.$

(iv) For any excellent order $\succ$, the tensor product with the trivial
excellent order is $\succ$ itself, $\succ\otimes\succ_{0}=\succ$.

######  Definition 2.3.

[HM20-2, 9.3] (a) A path in a finite directed graph $(V,E)$ (so $V$ is a
finite non-empty set and $E\subset V\times V$) is a tuple $(v_{1},...,v_{l})$
for some $l\in{\mathbb{Z}}_{[2,\infty)}$ with $v_{j}\in V$ and
$(v_{j},v_{j+1})\in E$ for $j\in{\mathbb{Z}}_{[1,l-1]}$. It is a path from
$v_{1}$ to $v_{l}$, so with source $v_{1}$ and target $v_{l}$.

(b) A finite directed graph $(V,E)$ has a center $v_{V}\in V$ if it has no
path from any vertex to itself and if it has at least one path from $v_{V}$ to
any other vertex $v\in V$. (The center is unique, which justifies the notation
$v_{V}$.)

(c) Consider a tuple $(\succ_{p})_{p\in P}$ of excellent orders for a finite
set $P\subset{\mathcal{P}}$ of prime numbers. It defines a finite directed
graph $(V,E_{V})$ with center $v_{V}$ as follows. Its set
$V=V((\succ_{p})_{p\in P})$ of vertices is the quadrant in ${\mathbb{N}}$

$\displaystyle V$ $\displaystyle:=$ $\displaystyle\\{\prod_{p\in
P}p^{k_{p}}\,|\,k_{p}\in{\mathbb{Z}}_{[0,s(\succ_{p})]}\\}.$ (2.7)

Its set of edges $E_{V}=E((\succ_{p})_{p\in P})$ is the set

$\displaystyle E_{V}$ $\displaystyle:=$ $\displaystyle\bigcup_{p\in
P}E_{V,p}\textup{ with }$ (2.8) $\displaystyle E_{V,p}$ $\displaystyle:=$
$\displaystyle\\{(m_{a},m_{b})\in V\times
V\,|\,\pi_{p}(m_{a})=\pi_{p}(m_{b}),v_{p}(m_{a})\succ_{p}v_{p}(m_{b})\\}.$

The edges in $E_{V,p}$ are called $p$-edges. So, the underlying undirected
graph coincides with the undirected graph which underlies the directed graph
$(V,E(V))$ in Definition 1.2. But the directions of edges may have changed.
The graph $(V,E_{V})$ is obviously centered with center

$\displaystyle v_{V}$ $\displaystyle=$ $\displaystyle\prod_{p\in
P}p^{s^{+}(\succ_{p})}.$ (2.9)

######  Definition 2.4.

[HM20-2, 9.4] (a) Let $\succ$ be an excellent order on the set
${\mathbb{Z}}_{[0,s(\succ)]}$. A set $K\subset{\mathbb{N}}_{0}$ is subset
compatible with $\succ$ if a bound $k_{K}\in{\mathbb{Z}}_{[0,s(\succ)]}$ with

$\displaystyle K$ $\displaystyle=$
$\displaystyle\\{k\in{\mathbb{Z}}_{[0,s(\succ)]}\,|\,k\succ k_{K}\\}$ (2.10)

exists or if $K={\mathbb{Z}}_{[0,s(\succ)]}$. (The bound $k_{K}=s^{+}(\succ)$
gives $K=\emptyset$, which is allowed.)

(b) For a finite non-empty set $M\subset{\mathbb{N}}$, let

$\displaystyle{\mathcal{P}}(M)$ $\displaystyle:=$
$\displaystyle\\{p\in{\mathcal{P}}\,|\,M\neq\pi_{p}(M)\\}$ $\displaystyle=$
$\displaystyle\\{p\in{\mathcal{P}}\,|\,\exists\ m\in M\textup{ with
}v_{p}(m)>0\\}$

be the set of prime numbers which turn up as factors of some numbers in $M$.

(c) For a finite non-empty set $M\subset{\mathbb{N}}$, a prime number
$p\in{\mathcal{P}}$, and a number $m_{0}\in\pi_{p}(M)$, define the finite set
$K_{M,p,m_{0}}\subset{\mathbb{N}}_{0}$ by

$\displaystyle\pi_{p}^{-1}(m_{0})=\\{m_{0}\cdot p^{k}\,|\,k\in
K_{M,p,m_{0}}\\}.$ (2.12)

(d) A finite non-empty set $M\subset{\mathbb{N}}$ is compatible with a tuple
$(\succ_{p})_{p\in P}$ of excellent orders for a finite set
$P\supset{\mathcal{P}}(M)$ of prime numbers if

$\displaystyle M$ $\displaystyle\subset$ $\displaystyle V((\succ_{p})_{p\in
P})$ (2.13)

and if for any prime number $p\in{\mathcal{P}}(M)$ and any
$m_{0}\in\pi_{p}(M)$ the set $K_{M,p,m_{0}}$ is subset compatible with
$\succ_{p}$. (So, here the excellent orders $\succ_{p}$ for
$p\in{\mathcal{P}}-{\mathcal{P}}(M)$ are irrelevant. But considering
$P\supset{\mathcal{P}}(M)$ instead of $P={\mathcal{P}}(M)$ will be useful.)

(e) A map $\psi:{\mathbb{N}}\to{\mathbb{N}}_{0}$ with finite support
$\operatorname{supp}(\psi)\subset{\mathbb{N}}$ is compatible with a tuple
$(\succ_{p})_{p\in P}$ of excellent orders for a finite set
$P\supset{\mathcal{P}}(\operatorname{supp}(\psi))$ of prime numbers if

$\displaystyle\operatorname{supp}(\psi)$ $\displaystyle\subset$ $\displaystyle
V((\succ_{p})_{p\in P})$ (2.14)

and if for any edge $(m_{a},m_{b})\in E_{V}$

$\displaystyle\psi(m_{a})$ $\displaystyle\geq$ $\displaystyle\psi(m_{b}).$
(2.15)

(f) A covering of a map $\psi:{\mathbb{N}}\to{\mathbb{N}}_{0}$ with finite
support is a tuple $(M_{1},...,M_{l})$ ($l\in{\mathbb{N}}_{0}$) of finite non-
empty sets $M_{j}\subset{\mathbb{N}}$ with

$\displaystyle\psi(m)$ $\displaystyle=$
$\displaystyle|\\{j\in\\{1,...,l\\}\,|\,m\in M_{j}\\}|\textup{ for any
}m\in{\mathbb{N}}.$ (2.16)

Here obviously $l\geq\max(\psi(m)\,|\,m\in{\mathbb{N}}_{0})=:l_{\psi}$. In the
case $\operatorname{supp}(\psi)=\emptyset$ we have $l=0$ and an empty tuple.
The standard covering of $\psi$ is the tuple
$(M_{1}^{(st)},...,M_{l_{\psi}}^{(st)})$ with

$\displaystyle M_{j}^{(st)}=\\{m\in\operatorname{supp}(\psi)\,|\,\psi(m)\geq
j\\}\textup{ for }j\in\\{1,...,l_{\psi}\\}.$ (2.17)

It is the unique covering with $M_{1}\supset...\supset M_{l}$, and it
satisfies $M_{1}^{(st)}=\operatorname{supp}(\psi)$.

(g) Let $\psi:{\mathbb{N}}\to{\mathbb{N}}_{0}$ have finite support, let
$P\supset{\mathcal{P}}(\operatorname{supp}(\psi))$ be a finite set of prime
numbers, and let $(\succ_{p})_{p\in P}$ be a tuple of excellent orders with
(2.14). A covering $(M_{1},...,M_{l})$ of $\psi$ is called compatible with
$(\succ_{p})_{p\in P}$ if each set $M_{j}$ is compatible with
$(\succ_{p})_{p\in P}$.

The following lemma expresses the compatibility conditions in Definition 2.4
(d) and (e) in a different way, and it shows their relationship. The proof is
not difficult. It is given in [HM20-2].

######  Lemma 2.5.

[HM20-2, 9.5] (a) Let $M\subset{\mathbb{N}}$ be a finite non-empty set, let
$P\supset{\mathcal{P}}(M)$ be a finite set of prime numbers, and let
$(\succ)_{p\in P}$ be a tuple of excellent orders with (2.13). (Recall the
definition of $(V,E_{V},v_{V})$ in Definition 2.3 (c).) The following three
conditions are equivalent:

  * (i)

$M$ is compatible with $(\succ)_{p\in P}$.

  * (ii)

$(M,E_{V}\cap M\times M)$ is a directed graph with center $v_{V}$ (so
$v_{V}\in M)$, and if $M$ contains the target of a path in $(V,E_{V})$, it
contains all vertices in this path.

  * (iii)

If $m_{b}\in M$ and $(m_{a},m_{b})\in E_{V}$ then $m_{a}\in M$.

(b) Let $\psi:{\mathbb{N}}\to{\mathbb{N}}_{0}$ be a map with finite support,
let $P\supset{\mathcal{P}}(\operatorname{supp}(\psi))$ be a finite set of
prime numbers, and let $(\succ_{p})_{p\in P}$ be a tuple of excellent orders
with (2.14). The following three conditions are equivalent.

  * (i)

$\psi$ is compatible with $(\succ_{p})_{p\in P}$.

  * (ii)

$\psi$ has a covering $(M_{1},...,M_{l})$ which is compatible with
$(\succ_{p})_{p\in P}$.

  * (iii)

The standard covering of $\psi$ is compatible with $(\succ_{p})_{p\in P}$.

Lemma 2.5 (a) allows to prove in Lemma 2.7 that any finite non-empty set
$M\subset{\mathbb{N}}$ which is compatible with a tuple $(\succ_{p})_{p\in P}$
of excellent orders satisfies condition (I) in Definition 2.6 (c) and in
Theorem 1.3. Before, we recall some necessary definitions from [He20].

######  Definition 2.6.

[He20, Definition 1.1 and Theorem 1.2] Let $M\subset{\mathbb{N}}$ be a finite
non-empty set. Recall the definition of the graph $(M,E(M))$ in Definition
1.2.

(a) For any prime number $p$, the components of the graph $(M,E(M)-E_{p}(M))$
are called the $p$-planes of the graph. A $p$-plane is called a highest
$p$-plane if no $p$-edge ends at a vertex of the $p$-plane. A $p$-edge from
$m_{1}\in M$ to $m_{2}\in M$ is called a highest $p$-edge if no $p$-edge ends
at $m_{1}$.

(b) Two properties $(T_{p})$ and $(S_{p})$ are defined for any prime number
$p$:

$\displaystyle(T_{p})$ $\displaystyle:$ $\displaystyle\textup{The graph
}(M,E(M))\textup{ has only one highest }p\textup{-plane}.\hskip 28.45274pt$
(2.18) $\displaystyle(S_{p})$ $\displaystyle:$ $\displaystyle\textup{The graph
}(M,E(M)-\\{\textup{highest }p\textup{-edges}\\})$ (2.19)
$\displaystyle\textup{ has only 1 or 2 components}.$

(c) Condition (I) says: The graph $(M,E(M))$ is connected. It satisfies
$(S_{2})$. It satisfies $(T_{p})$ for any prime number $p\geq 3$.

(d) Condition (II) says: The graph $(M,E(M))$ has two components $M_{1}$ and
$M_{2}$. They are 2-planes. They satisfy $(T_{p})$ for any prime number $p\geq
3$. Furthermore,

$\displaystyle\gcd(\operatorname{lcm}(M_{1}),\operatorname{lcm}(M_{2}))$
$\displaystyle\in$ $\displaystyle\\{1,2\\},$ (2.20) $\displaystyle
v_{2}(\operatorname{lcm}(M_{2}))$ $\displaystyle>$ $\displaystyle
v_{2}(\operatorname{lcm}(M_{1}))\in\\{0;1\\}.$ (2.21)

Lemma 2.7 is new. Part (b) is used in the proof of Theorem 1.4.

######  Lemma 2.7.

Let $M\subset{\mathbb{N}}$ be a finite non-empty set.

(a) $(S_{p})\Longrightarrow(T_{p})$ for any prime number $p$ if the graph
$(M,E(M))$ is connected.

(b) Let $(\succ_{p})_{p\in P}$ be a tuple of excellent orders with which $M$
is compatible. Then the graph $(M,E(M))$ is connected. $M$ satisfies $(S_{p})$
for any prime number $p$. $M$ satisfies condition (I).

Proof: (a) Suppose that the graph $(M,E(M))$ is connected and satisfies
$(S_{p})$, but has at least two highest $p$-planes. Any highest $p$-plane is a
component of the graph $(M,E(M)-\\{\textup{highest }p\textup{-edges}\\})$.
Therefore this graph consists of exactly two highest $p$-planes. But then the
graph $(M,E(M))$ has no $p$-edges and has the same two components, so it is
not connected, a contradiction.

(b) By Lemma 2.5 (a), $(M,E_{V}\cap M\times M)$ is a directed subgraph of
$(V,E_{V})$ with center $v_{V}$. Therefore it is connected. As the underlying
undirected graphs of $(V,E(V))$ and $(V,E_{V})$ coincide, also the graph
$(M,E(M))$ is connected.

Fix a prime number $p$. We will prove $(S_{p})$ in the next three paragraphs.
Let $m_{1}\in M$ be arbitrary. There is a path in $(V,E_{V})$ from
$\pi_{p}(v_{V})\cdot p^{v_{p}(m_{1})}$ to $m_{1}$ which consists completely of
edges which are not $p$-edges. Because of Lemma 2.5 (a) (i)$\Rightarrow$(ii),
it is a path in $(M,E_{V}\cap M\times M)$. Therefore the set $\\{m\in
M\,|\,v_{p}(m)=v_{p}(m_{1})\\}$ is a single $p$-plane in the graph $(M,E(M))$.
Furthermore, the set $\\{\pi_{p}(v_{V})\cdot p^{k}\,|\,k\in
K_{M,p,\pi_{p}(v_{V})}\\}$ intersects each $p$-plane in one element.

Especially, there is only one highest $p$-plane, and it intersects the set
above in $\widetilde{v}_{V}:=\pi_{p}(v_{V})\cdot p^{k^{(max)}}$ where
$k^{(max)}:=\max(v_{p}(m)\,|\,m\in M)$. This highest $p$-plane is one
component of the graph $(M,E(M)-\\{\textup{highest }p\textup{-edges}\\})$.

The edges in $E(M)$ with both vertices in the set $\\{\pi_{p}(v_{V})\cdot
p^{k}\,|\,k\in K_{M,p,\pi_{p}(v_{V})}-\\{k^{(max)}\\}\\}$ are all $p$-edges,
but not highest $p$-edges. Therefore the second component (if $M$ does not
consist only of one $p$-plane) is the union of all other $p$-planes. The
property $(S_{p})$ holds.

Because $(M,E(M))$ is connected and because of part (a), $(M,E(M))$ satisfies
condition (I). $\Box$

##  3\. Tensor product and Thom-Sebastiani sum

The following is a reformulation of Problem 7 in [HZ19]:

(a) Find a natural condition for a map $\psi:{\mathbb{N}}\to{\mathbb{N}}_{0}$
with finite support which implies that each set
$M_{j}^{st}\subset{\mathbb{N}}$ in the standard covering of $\psi$ satisfies
condition (I) and which is stable under tensor product.

(b) Prove that in the case of an isolated quasihomogeneous singularity the map
$\psi:{\mathbb{N}}\to{\mathbb{N}}_{0}$ satisfies this condition, where
$p_{ch,h_{Mil}}=\prod_{m\in{\mathbb{N}}}\Phi_{m}^{\psi(m)}$ is the
characteristic polynomial of the monodromy.

Problem 7 is motivated by the facts (which are discussed in [HZ19]) that
condition (I) is not stable under tensor product and that it is too difficult
to prove Theorem 1.4 directly.

Here we propose as solution the natural condition that a tuple of excellent
orders exists, with which $\psi$ is compatible (Definition 2.4 (e)). The first
half of part (a) is true because of Lemma 2.7 (b) and Lemma 2.5 (b). Part (b)
is true because of Theorem 6.6, which is [HM20-2, Theorem 12.1]. The second
half of part (a) is also true. It follows from Theorem 3.2 below, which is a
main result of [HM20-2]. Before, Definition 3.1 explains what is meant here by
tensor product.

######  Definition 3.1.

(a) Let $f(t)=\prod_{i=1}^{n}(t-\kappa_{i})\in{\mathbb{C}}[t]$ and
$g(t)=\prod_{j=1}^{n}(t-\lambda_{j})\in{\mathbb{C}}[t]$ be unitary polynomials
of degrees $n,m\geq 1$. Their tensor product is the unitary polynomial

$\displaystyle(f\otimes g)(t)$ $\displaystyle:=$
$\displaystyle\prod_{i=1}^{n}\prod_{j=1}^{m}(t-\kappa_{i}\lambda_{j})\in{\mathbb{C}}[t]$
(3.1)

of degree $nm$.

(b) Let $\psi_{1}:{\mathbb{N}}\to{\mathbb{N}}_{0}$ and
$\psi_{2}:{\mathbb{N}}\to{\mathbb{N}}_{0}$ be two maps with finite supports
$M=\operatorname{supp}(\psi_{1})$ and $N=\operatorname{supp}(\psi_{2})$. Their
tensor product is the map
$\psi_{1}\otimes\psi_{2}:{\mathbb{N}}\to{\mathbb{N}}_{0}$ with

$\displaystyle\Bigl{(}\prod_{m\in
M}\Phi_{m}^{\psi_{1}(m)}\Bigr{)}\otimes\Bigl{(}\prod_{n\in
N}\Phi_{n}^{\psi_{2}(n)}\Bigr{)}=\prod_{l\in{\mathbb{N}}}\Phi_{l}^{(\psi_{1}\otimes\psi_{2})(l)}.$
(3.2)

It has also finite support.

The beginning of section 7 in [HM20-2] (namely Definition 7.1, Lemma 7.2 and
Definition 7.3) gives more information on $\psi_{1}\otimes\psi_{2}$. The next
Theorem 3.2 is a main result of [HM20-2]. The long and difficult proof is
built up from several intermediate results in the sections 7–9 in [HM20-2].
Its part (a) solves the second half of part (a) of Problem 7 above.

######  Theorem 3.2.

[HM20-2, Theorem 9.10] Let $\psi_{1}:{\mathbb{N}}\to{\mathbb{N}}_{0}$ and
$\psi_{2}:{\mathbb{N}}\to{\mathbb{N}}_{0}$ be two maps with finite supports
$M=\operatorname{supp}(\psi_{1})$ and $N=\operatorname{supp}(\psi_{2})$. Write
$\psi_{3}:=\psi_{1}\otimes\psi_{2}$. Denote
$L:=\operatorname{supp}(\psi_{3})$. It satisfies
${\mathcal{P}}(L)\subset{\mathcal{P}}(M)\cup{\mathcal{P}}(N)$.

Let $P\supset{\mathcal{P}}(M)\cup{\mathcal{P}}(N)$ be a finite set of prime
numbers. Let $(\succ_{p}^{M})_{p\in P}$ and $(\succ_{p}^{N})_{p\in P}$ be two
tuples of excellent orders such that $\psi_{1}$ is compatible with
$(\succ_{p}^{M})_{p\in P}$ and $\psi_{2}$ is compatible with
$(\succ_{p}^{N})_{p\in P}$. Write
$\succ_{p}^{L}:=(\succ_{p}^{M}\otimes\succ_{p}^{N})$ for any $p\in P$.

(a) $\psi_{3}$ is compatible with the tuple $(\succ_{p}^{L})_{p\in P}$ of
excellent orders.

(b) Let $(M_{1}^{(st)},...,M_{l_{\psi_{1}}}^{(st)})$,
$(N_{1}^{(st)},...,N_{l_{\psi_{2}}}^{(st)})$ and
$(L_{1}^{(st)},...,L_{l_{\psi_{3}}}^{(st)})$ be the standard coverings of
$\psi_{1}$, $\psi_{2}$ and $\psi_{3}$. Then

$\displaystyle\Bigl{(}\bigoplus_{i=1}^{l_{\psi_{1}}}{\rm
Or}(M_{i}^{(st)})\Bigr{)}\otimes\Bigl{(}\bigoplus_{j=1}^{l_{\psi_{2}}}{\rm
Or}(N_{j}^{(st)})\Bigr{)}\cong\bigoplus_{k=1}^{l_{\psi_{3}}}{\rm
Or}(L_{k}^{(st)}),$ (3.3)

so the tensor product of the sums of Orlik blocks on the left hand side admits
a standard decomposition into Orlik blocks.

Thom and Sebastiani proved a result which specializes to the case of isolated
quasihomogeneous singularities as follows.

######  Theorem 3.3.

[ST71] Let $f=f(x_{1},...,x_{n_{f}})$ and
$g=g(x_{n_{f}+1},...,x_{n_{f}+n_{g}})$ be two isolated quasihomogeneous
singularities in different variables. Then their Thom-Sebastiani sum $f+g$ is
an isolated quasihomogeneous singularity, and there is a canonical isomorphism

$\displaystyle(H_{Mil},h_{Mil})(f+g)\cong(H_{Mil},h_{Mil})(f)\otimes(H_{Mil},h_{Mil})(g).$
(3.4)

This theorem, Theorem 6.6 and part (b) of Theorem 3.2 imply the following main
result of [HM20-2].

######  Theorem 3.4.

[HM20-2, Theorem 1.3 (c)] Let $f=f(x_{1},...,x_{n_{f}})$ and
$g=g(x_{n_{f}+1},...,x_{n_{f}+n_{g}})$ be two isolated quasihomogeneous
singularities in different variables, which both satisfy Orlik’s conjecture.
Then also $f+g$ satisfies Orlik’s conjecture.

Together with the proofs of Orlik’s conjecture for the chain type
singularities [HM20-2, Theorem 1.3 (a)] and for the cycle type singularities
[HM20-1, Theorem 1.3], it shows Orlik’s conjecture for a large family of
isolated quasihomogeneous singularities, namely for all invertible
polynomials.

######  Remark 3.5.

The search for Theorem 3.2 led us in the following way to the excellent orders
and the compatibility of a finite set $M\subset{\mathbb{N}}$ with a tuple of
excellent orders. We found a condition sdiOb-sufficient [HM20-2, Definition
7.3 (d)] (short for standard decomposition into Orlik blocks) for a pair
$(M,N)$ of finite subsets $M,N\subset{\mathbb{N}}$ which implies (and probably
is equivalent to) the property that the tensor product
$(H_{M},h_{M})\otimes(H_{N},h_{N})$ of two Orlik blocks allows a standard
decomposition into Orlik blocks [HM20-2, Theorem 7.4]. Lemma 9.12 in [HM20-2]
says especially that the following conditions (i) and (ii) are equivalent.
Here $M\subset{\mathbb{N}}$ is a finite non-empty set.

  * (i)

For any $n_{N}\in{\mathbb{N}}$, the pair
$(M,\\{n\in{\mathbb{N}}\,|\,n|n_{N}\\})$ is sdiOb-sufficient.

  * (ii)

A tuple $(\succ_{p})_{p\in{\mathcal{P}}(M)}$ of excellent orders exists such
that $M$ is compatible with it.

Another interesting result from [HM20-2] which complements Theorem 3.2 (and
which, in fact, is used in its proof) is the following.

######  Theorem 3.6.

[HM20-2, Theorem 9.6] Let $(\succ_{p})_{p\in P}$ be at tuple of excellent
orders for a finite set $P$ of prime numbers, and let
$\psi:{\mathbb{N}}\to{\mathbb{N}}_{0}$ be a map with finite support which is
compatible with $(\succ_{p})_{p\in P}$. Let
$(M_{1}^{(1)},...,M_{l_{1}}^{(1)}))$ and $(M_{1}^{(2)},...,M_{l_{2}}^{(2)})$
be two coverings of $\psi$ which are both compatible with $(\succ_{p})_{p\in
P}$. Then the corresponding sums of Orlik blocks are isomorphic,

$\displaystyle\bigoplus_{i=1}^{l_{1}}{\rm Or}(M_{i}^{(1)})$
$\displaystyle\cong$ $\displaystyle\bigoplus_{j=1}^{l_{2}}{\rm
Or}(M_{j}^{(2)}),$ (3.5)

and $l_{1}=l_{2}=l_{\psi}(:=\max(\psi(m)\,|\,m\in{\mathbb{N}}))$.

##  4\. Maps on the group ring generated by unit roots

This section collects notations and classical formulas which will be needed in
the later sections.

######  Notations 4.1.

(a) Denote by $\mu({\mathbb{C}})\subset S^{1}$ the group of all unit roots.
Denote by ${\mathbb{Q}}[\mu({\mathbb{C}})]$ and
${\mathbb{Z}}[\mu({\mathbb{C}})]$ the group rings of elements
$\sum_{j=1}^{l}b_{j}[\zeta_{j}]$ where $\zeta_{j}\in\mu({\mathbb{C}})$ and
where $b_{j}\in{\mathbb{Q}}$ respectively ${\mathbb{Z}}$, with multiplication
$[\zeta_{1}][\zeta_{2}]=[\zeta_{1}\zeta_{2}]$. The unit element is $[1]$. The
trace and the degree of an element
$b=\sum_{j=1}^{l}b_{j}[\zeta_{j}]\in{\mathbb{Q}}[\mu({\mathbb{C}})]$ are

$\displaystyle\operatorname{tr}(b):=\sum_{j=1}^{l}b_{j}\zeta_{j}\in{\mathbb{C}},\quad\deg(b):=\sum_{j=1}^{l}b_{j}\in{\mathbb{Q}}.$

The trace map
$\operatorname{tr}:{\mathbb{Q}}[\mu({\mathbb{C}})]\to{\mathbb{C}}$ and the
degree map $\deg:{\mathbb{Q}}[\mu({\mathbb{C}})]\to{\mathbb{Q}}$ are ring
homomorphisms. For $k\in{\mathbb{N}}_{0}$, the $k$-th Lefschetz number
$L_{k}(b)$ of an element $b=\sum_{j=1}^{l}b_{j}[\zeta_{j}]$ is

$\displaystyle
L_{k}(b):=\sum_{j=1}b_{j}\zeta_{j}^{k}=\operatorname{tr}\left(\sum_{j=1}b_{j}[\zeta_{j}^{k}]\right)\in{\mathbb{C}}.$
(4.1)

(b) The divisor of a unitary polynomial $f=\prod_{i=1}^{n}(t-\kappa_{i})$ with
$\kappa_{i}\in\mu({\mathbb{C}})$ is

$\displaystyle\operatorname{div}f:=\sum_{i=1}^{n}[\kappa_{i}]\in{\mathbb{Z}}[\mu({\mathbb{C}})].$

The divisor of the constant polynomial 1 is $\operatorname{div}1=0$.

(c) The order $\operatorname{ord}(\zeta)\in{\mathbb{N}}$ of a unit root
$\zeta\in\mu({\mathbb{C}})$ is the minimal number $m\in{\mathbb{N}}$ with
$\zeta^{m}=1$. For $m\in{\mathbb{N}}$, the $m$-th cyclotomic polynomial is

$\displaystyle\Phi_{m}:=\prod_{\zeta:\,\operatorname{ord}(\zeta)=m}(t-\zeta).$

Define

$\displaystyle\Lambda_{m}:=\operatorname{div}(t^{m}-1),\quad\Psi_{m}:=\operatorname{div}\Phi_{m}.$
(4.2)

Then $\lambda_{1}=\Psi_{1}=[1]$. The Möbius function $\mu_{Moeb}$ is [Ai79]

$\displaystyle\mu_{Moeb}:{\mathbb{N}}$ $\displaystyle\to$
$\displaystyle\\{0,1,-1\\},$ $\displaystyle m$ $\displaystyle\mapsto$
$\displaystyle\left\\{\begin{array}[]{ll}(-1)^{r}&\textup{if
}m=p_{1}\cdot...\cdot p_{r}\textup{ with }p_{1},...,p_{r}\\\
&\textup{different prime numbers},\\\ 0&\textup{else}\end{array}\right.$

(here $r=0$ is allowed, so $\mu_{Moeb}(1)=1$).

The next lemma collects well known facts.

######  Lemma 4.2.

(a) Let $f=\prod_{i=1}^{n}(t-\kappa_{i})$ and
$g=\prod_{j=1}^{m}(t-\lambda_{j})$ be unitary polynomials with $\kappa_{i}$
and $\lambda_{j}\in\mu({\mathbb{C}})$. The tensor product $(f\otimes
g)(t)=\prod_{i=1}^{n}\prod_{j=1}^{m}(t-\kappa_{i}\lambda_{j})$ in (3.1)
satisfies

$\displaystyle\operatorname{div}(f\otimes
g)=(\operatorname{div}f)\cdot(\operatorname{div}g).$

(b) The cyclotomic polynomial $\Phi_{m}$ is in ${\mathbb{Z}}[t]$, it has
degree $\varphi(m)$, and it is irreducible in ${\mathbb{Z}}[t]$ and
${\mathbb{Q}}[t]$.

$\displaystyle t^{n}-1$ $\displaystyle=$
$\displaystyle\prod_{m|n}\Phi_{m}(t),\hskip
56.9055pt\Lambda_{n}=\sum_{m|n}\Psi_{m},$ (4.4) $\displaystyle\Phi_{m}$
$\displaystyle=$
$\displaystyle\prod_{m|n}(t^{n}-1)^{\mu_{Moeb}(\frac{m}{n})},\quad\Psi_{m}=\sum_{n|m}\mu_{Moeb}(\frac{m}{n})\cdot\Lambda_{n}.$
(4.5)

The traces of $\Lambda_{m}$ and $\Psi_{m}$ are

$\displaystyle\operatorname{tr}\Lambda_{m}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}1&\textup{if }m=1\\\ 0&\textup{if
}m\geq 2,\end{array}\right.$ (4.8) $\displaystyle\operatorname{tr}\Psi_{m}$
$\displaystyle=$ $\displaystyle\mu_{Moeb}(m).$ (4.9)

The product $\Lambda_{m}\Lambda_{n}$ is

$\displaystyle\Lambda_{m}\Lambda_{n}$ $\displaystyle=$
$\displaystyle\gcd(m,n)\cdot\Lambda_{\operatorname{lcm}(m,n)}.$ (4.10)

For the product $\Psi_{m}\Psi_{n}$ see [HZ19, (2.19)–(2.21)] or [HM20-2, Lemma
7.2].

(c) Let $\psi:{\mathbb{N}}\to{\mathbb{Q}}$ be a map with finite support, and
consider the element

$\displaystyle
b:=\sum_{m\in{\mathbb{N}}}\psi(m)\cdot\Psi_{m}\in{\mathbb{Q}}[\mu({\mathbb{C}})].$
(4.11)

There is a unique map $\chi:{\mathbb{N}}\to{\mathbb{Q}}$ with finite support
and

$\displaystyle b=\sum_{n\in{\mathbb{N}}}\chi(n)\cdot\Lambda_{n}.$ (4.12)

The four data $\psi$, $b$, $\chi$ and $(L_{k}(b))_{k\in{\mathbb{N}}_{0}}$
determine one another, by (4.11)–(4.12) and

$\displaystyle\psi(m)=\sum_{n:\,m|n}\chi(n),\quad\chi(n)=\sum_{m:\,n|m}\psi(m)\cdot\mu_{Moeb}(\frac{m}{n}),$
(4.13) $\displaystyle L_{k}(b)=\sum_{n:\,n|k}n\chi(n),\quad
n\chi(n)=\sum_{k|n}L_{k}(b)\cdot\mu_{Moeb}(\frac{n}{k}).$ (4.14)

Denote $d_{\chi}:=\operatorname{lcm}(n\,|\,n\in\operatorname{supp}(\chi))$.
The Lefschetz numbers $L_{k}(b)$ are rational. They are determined by the
values $L_{k}(b)$ for $k\in{\mathbb{N}}$ with $k|d_{\chi}$, because of the
(extended) periodicity property

$\displaystyle L_{k}(b)=L_{\gcd(k,d_{\chi})}(b)\qquad\textup{for any
}k\in{\mathbb{N}}.$ (4.15)

Proof: (a) Trivial. (b) The statements on $\Phi_{m}$ are classical. The
formulas (4.5) follow from the formulas (4.4) by Moebius inversion [Ai79]. The
trace of $\Lambda_{m}$ is obvious, the trace of $\Psi_{m}$ follows with (4.5).
The product $\Lambda_{m}\Lambda_{n}$ is obvious.

(c) Formula (4.13) is a consequence of (4.4) and (4.5). The first formula in
(4.14) can be seen as follows,

$\displaystyle L_{k}(\Lambda_{n})$ $\displaystyle=$
$\displaystyle\operatorname{tr}\left(\sum_{a=0}^{n-1}[e^{2\pi iak/n}]\right)$
$\displaystyle=$
$\displaystyle\operatorname{tr}\left(\gcd(k,n)\Lambda_{n/\gcd(k,n)}\right)=\left\\{\begin{array}[]{ll}n&\textup{
if }n|k,\\\ 0&\textup{ if }n\not|k,\end{array}\right.$ $\displaystyle
L_{k}(b)$ $\displaystyle=$ $\displaystyle\sum_{n\in{\mathbb{N}}}\chi(n)\cdot
L_{k}(\Lambda_{n})=\sum_{n:\,n|k}n\chi(n).$

The first formula in (4.14) gives the second formula in (4.14) by Moebius
inversion [Ai79]. It also implies $L_{k}(b)\in{\mathbb{Q}}$ and the
periodicity property (4.15). $\Box$

##  5\. Basic properties of weight systems and induced objects

In this section, problem 1 in [HZ19] will be solved as part of Lemma 5.4. It
is elementary. Before, notations from [HZ19, section 3] are recalled. In this
section, we fix a number $n\in{\mathbb{N}}$ and denote $N:=\\{1,2,...,n\\}$.

######  Definition 5.1.

(a) A weight system is a tuple
$(v_{1},...,v_{n};d)\in({\mathbb{Q}}_{>0})^{n+1}$ with $v_{i}<d$. Another
weight system is equivalent, if the second one has the form
$q\cdot(v_{1},...,v_{n};d)$ for some $q\in{\mathbb{Q}}_{>0}$. A weight system
is integer if $(v_{1},...,v_{n};d)\in{\mathbb{N}}^{n+1}$. It is reduced if it
is integer and $\gcd(v_{1},...,v_{n},d)=1$. It is normalized if $d=1$.

Any equivalence class contains a unique reduced weight system and a unique
normalized weight system. From now on, the letters $(v_{1},...,v_{n};d)$ will
be reserved for an integer weight system, and the letters
$(w_{1},...,w_{n};1)$ will be the equivalent normalized weight system, i.e.
$w_{i}=v_{i}/d$.

(b) Let $(v_{1},...,v_{n};d)$ be an integer weight system (not necessarily
reduced). For $J\subset N$ and $k\in{\mathbb{N}}_{0}$ define

$\displaystyle({\mathbb{Z}}^{J})_{k}$ $\displaystyle:=$
$\displaystyle\\{\alpha\in{\mathbb{Z}}^{n}\,|\,\alpha_{i}=0\textup{ for
}i\notin J,\ \sum_{i\in J}\alpha_{i}=k\\},$ (5.1)
$\displaystyle({\mathbb{N}}_{0}^{J})_{k}$ $\displaystyle:=$
$\displaystyle{\mathbb{N}}_{0}^{n}\cap({\mathbb{Z}}^{J})_{k}.$

(c) Two conditions for an integer weight system $(v_{1},...,v_{n};d)$ (or,
equivalently, for the equivalent normalized weight system
$(\frac{v_{1}}{d},...,\frac{v_{n}}{d};1)$) are defined as follows:

$\displaystyle\overline{(C2)}:$ $\displaystyle\forall\ J\subset N\textup{ with
}J\neq\emptyset\quad\exists\ K\subset N$ $\displaystyle\textup{with
}|K|=|J|\textup{ and }\forall\ k\in K\
({\mathbb{Z}}^{J})_{d-v_{k}}\neq\emptyset.$ $\displaystyle(C2):$
$\displaystyle\forall\ J\subset N\textup{ with }J\neq\emptyset\quad\exists\
K\subset N$ $\displaystyle\textup{with }|K|=|J|\textup{ and }\forall\ k\in K\
({\mathbb{N}}_{0}^{J})_{d-v_{k}}\neq\emptyset.$

######  Remarks 5.2.

(i) Of course $(C2)$ implies $\overline{(C2)}$. Lemma 2.1 in [HK12] gives four
conditions $(C1)$, $(C1)^{\prime}$, $(C2)^{\prime}$ and $(C3)$ which are
equivalent to $(C2)$. They all have versions $\overline{(C1)}$,
$\overline{(C1)^{\prime}}$, $\overline{(C2)^{\prime}}$ and $\overline{(C3)}$,
which are equivalent to $\overline{(C2)}$. Theorem 6.1 below shows the
relevance of $(C2)$.

(ii) Let $(v_{1},...,v_{n};d)$ be an integer weight system. For $J\subset N$
with $J\neq\emptyset$ define the semigroup

$\displaystyle SG(J)$ $\displaystyle:=$ $\displaystyle\sum_{j\in
J}{\mathbb{N}}_{0}\cdot v_{j}\subset{\mathbb{N}}_{0},$

and observe $\sum_{j\in J}{\mathbb{Z}}\cdot
v_{j}={\mathbb{Z}}\cdot\gcd(v_{j}\,|\,j\in J)$. Therefore

$\displaystyle({\mathbb{N}}_{0}^{J})_{k}\neq\emptyset$ $\displaystyle\iff$
$\displaystyle k\in SG(J),$ (5.2)
$\displaystyle({\mathbb{Z}}^{J})_{k}\neq\emptyset$ $\displaystyle\iff$
$\displaystyle\gcd(v_{j}\,|\,j\in J)|k.$ (5.3)

$\overline{(C2)}$ is equivalent to the condition

$\displaystyle(GCD)$ $\displaystyle\forall\ J\subset N\ \gcd(v_{j}\,|\,j\in
J)\textup{ divides at least }|J|$ $\displaystyle\textup{of the numbers
}d-v_{k}\textup{ for }k\in N.$

######  Definition 5.3.

Let $({\bf v},d)=(v_{1},...,v_{n},d)\in{\mathbb{N}}^{n+1}$ be an integer
weight system.

(a) Define unique numbers $s_{1},...,s_{n},t_{1},...,t_{n}\in{\mathbb{N}}$ by

$\displaystyle\frac{v_{i}}{d}=\frac{s_{i}}{t_{i}}\quad\textup{and}\quad\gcd(s_{i},t_{i})=1.$
(5.4)

They depend only on the normalized weight system ${\bf
w}=(w_{1},...w_{n})=(\frac{v_{1}}{d},...,\frac{v_{n}}{d})$. Define

$\displaystyle d_{\bf w}:=\operatorname{lcm}(t_{j}\,|\,j\in N).$ (5.5)

Of course $d_{\bf w}|d$. If $({\bf v},d)$ is reduced and
$\gcd(v_{1},...,v_{n})|d$ (which holds for example if $\overline{(C2)}$
holds), then $\gcd(v_{1},...,v_{n})=1$ and then $d_{\bf w}=d$.

(b) For $k\in{\mathbb{N}}$ define

$\displaystyle M(k)$ $\displaystyle:=$ $\displaystyle\\{j\in
N\,|\,t_{j}|k\\},$ (5.6) $\displaystyle\textup{and }\mu(k)$ $\displaystyle:=$
$\displaystyle\prod_{j\in M(k)}(\frac{1}{w_{j}}-1)=\prod_{j\in
M(k)}\frac{d-v_{j}}{v_{j}}\in{\mathbb{Q}}_{>0}$ (5.7)

(the empty product is by definition $1$).

(c) Define a quotient of polynomials

$\displaystyle\rho_{({\bf v},d)}(t)$ $\displaystyle:=$ $\displaystyle
t^{v_{1}+...+v_{n}}\cdot\prod_{j=1}^{n}\frac{t^{d-v_{j}}-1}{t^{v_{j}}-1}\in{\mathbb{Q}}(t)$
(5.8)

and an element of ${\mathbb{Q}}[\mu({\mathbb{C}})]$

$\displaystyle D_{\bf w}$ $\displaystyle:=$
$\displaystyle\prod_{j=1}^{n}\left(\frac{1}{s_{j}}\Lambda_{t_{j}}-\Lambda_{1}\right)\in{\mathbb{Q}}[\mu({\mathbb{C}})].$
(5.9)

######  Lemma 5.4.

([HZ19, Lemma 3.7] for (a)–(c)) Let $({\bf
v};d)=(v_{1},...,v_{n};d)\in{\mathbb{N}}^{n+1}$ be an integer weight system.

(a) Then $M(k)=M(\gcd(k,d_{\bf w}))$ and $\mu(k)=\mu(\gcd(k,d_{\bf w}))$.

(b) The Lefschetz numbers $L_{k}(D_{\bf w})$ are

$\displaystyle L_{k}(D_{\bf w})$ $\displaystyle=$
$\displaystyle(-1)^{n-|M(k)|}\mu(k)$ (5.10) $\displaystyle=$ $\displaystyle
L_{\gcd(k,d_{\bf w})}(D_{\bf w}).$ (5.11)

(c) $({\bf v};d)$ satisfies $\overline{(C2)}\iff\rho_{({\bf
v};d)}\in{\mathbb{Z}}[t]$.

(d) Suppose that $({\bf v};d)$ satisfies $\overline{(C2)}$. Write

$\displaystyle\rho_{({\bf v};d)}=\sum_{\alpha\in
d^{-1}{\mathbb{Z}}}\sigma(\alpha)\cdot t^{d\cdot\alpha}$

with a unique map $\sigma:d^{-1}{\mathbb{Z}}\to{\mathbb{Z}}$ with finite
support. Then

$\displaystyle D_{\bf w}$ $\displaystyle=$ $\displaystyle\sum_{\alpha\in
d^{-1}{\mathbb{Z}}}\sigma(\alpha)\cdot[e^{2\pi i\alpha}].$ (5.12)

and $\mu(k)\in{\mathbb{N}}$.

Proof: (a) All $t_{j}$ divide $d_{\bf w}$. Therefore $t_{j}|k\iff
t_{j}|\gcd(k,d_{\bf w})$.

(b) For completeness sake, the calculation in [HZ19] is copied.

$\displaystyle L_{k}(D_{\bf w})$ $\displaystyle=$
$\displaystyle\operatorname{tr}\left(\prod_{j=1}^{n}\left(\frac{\gcd(k,t_{j})}{s_{j}}\Lambda_{t_{j}/\gcd(k,t_{j})}-\Lambda_{1}\right)\right)$
$\displaystyle=$
$\displaystyle\prod_{j=1}^{n}\left(\frac{\gcd(k,t_{j})}{s_{j}}\cdot\operatorname{tr}\left(\Lambda_{t_{j}/\gcd(k,t_{j})}\right)-1\right)$
$\displaystyle=$ $\displaystyle\prod_{j\in
M(k)}\left(\frac{t_{j}}{s_{j}}-1\right)\cdot\prod_{j\notin M(k)}(-1)$
$\displaystyle=$ $\displaystyle(-1)^{n-|M(k)|}\cdot\mu(k).$

Formula (5.11) follows from part (a).

(c) The conditions $\overline{(C2)}$ and $(GCD)$ in Remark 5.2 are equivalent.
Condition $(GCD)$ says that any cyclotomic polynomial in the denominator of
$\rho_{({\bf v};d)}$ turns up with at least the same multiplicity in the
numerator. Therefore $(GCD)$ is equivalent to $\rho_{({\bf
v};d)}\in{\mathbb{Z}}[t]$.

(d) The formulas (5.10) and (5.12) imply $\mu(k)\in{\mathbb{N}}$. For (5.12),
it is because of Lemma 4.2 (c) sufficient to show

$\displaystyle L_{k}(D_{\bf w})$ $\displaystyle=$ $\displaystyle
L_{k}\left(\sum_{\alpha\in d^{-1}{\mathbb{Z}}}\sigma(\alpha)\cdot[e^{2\pi
i\alpha}]\right)\quad\textup{for all }k\in{\mathbb{N}}_{0}.$ (5.13)

The right hand side of (5.13) is

$\displaystyle\operatorname{tr}\left(\sum_{\alpha\in
d^{-1}{\mathbb{Z}}}\sigma(\alpha)\cdot[e^{2\pi
ik\alpha}]\right)=\sum_{\alpha\in d^{-1}{\mathbb{Z}}}\sigma(\alpha)\cdot
e^{2\pi ik\alpha}=\rho_{({\bf v};d)}(e^{2\pi ik/d})$ $\displaystyle=$
$\displaystyle\lim_{t\mapsto e^{2\pi ik/d}}\left(\prod_{j\in
M(k)}\frac{t^{d}-t^{v_{j}}}{t^{v_{j}}-1}\cdot\prod_{j\notin
M(k)}\frac{t^{d}-t^{v_{j}}}{t^{v_{j}}-1}\right)$ $\displaystyle=$
$\displaystyle\left(\prod_{j\in
M(k)}\frac{d-v_{j}}{v_{j}}\right)\cdot(-1)^{n-|M(k)|}=(-1)^{n-|M(k)|}\mu(k)=L_{k}(D_{\bf
w}).$

Here we used $j\notin M(k)\iff\frac{k}{d}v_{j}\notin{\mathbb{Z}}\iff(e^{2\pi
ik/d})^{v_{j}}\neq 1$. $\Box$

######  Remark 5.5.

Problem 1 in [HZ19] asked whether (5.12) holds. In [HZ19], the first author
had missed the calculation in the proof of part (d).

##  6\. Isolated quasihomogeneous singularities

Finally we come to the discussion of weight systems which allow isolated
quasihomogeneous singularities and of associated objects. The following
Theorems 6.1 and 6.4 are basic. As in section 5, $n\in{\mathbb{N}}$ and
$N=\\{1,...,n\\}$ are fixed.

######  Theorem 6.1.

(Kouchnirenko [Ko76, Remarque 1.13 (ii)]) Let $({\bf
v};d)=(v_{1},...,v_{n};d)$ be an integer weight system. The following
conditions are equivalent.

  * (IS3):

There exists a quasihomogeneous polynomial $f$ with weight system $({\bf
v};d)$ and an isolated singularity at 0.

  * (IS3)’:

A generic quasihomogeneous polynomial with weight system $({\bf v};d)$ has an
isolated singularity at 0.

  * $(C2)$:

The weight system $({\bf v};d)$ satisfies condition $(C2)$.

######  Remarks 6.2.

(i) The reference [Ko76, Remarque 1.13 (ii)] does not give a detailed proof,
but the reference [Ko77] (in russian) does. The theorem was rediscovered and
reproved several times, in [OR76], [KS92] and [Fl00]. More general results are
given in [Sh79] (without proof) and [Wa96] (with proof). The paper [HK12]
discussed the history of Theorem 6.1, but it missed the reference [Fl00].

(ii) Lemma 2.1 in [HK12] shows combinatorially the equivalence of (C2) with
several other conditions (C1), (C1)’, (C2)’ and (C3). Some of the references
above prove the equivalence of (IS3) with some other of these conditions.

(iii) The implication (IS3)$\Rightarrow$(C1) was already shown in [Sa71].

######  Remark 6.3.

The paper [HK12] made good use of the part of condition (C2) which concerns
subsets $J\subset N$ with $|J|=1$. Problem 3 in [HZ19] asked about making good
use of the conditions in (C2) for sets $J\subset N$ with $|J|\geq 2$. Of
course, the proofs of Theorem 6.1 in the references in Remark 6.2 (i) make
such good use, as they need the full condition (C2). We do not have other
solutions of problem 3 in [HZ19].

######  Theorem 6.4.

Let $({\bf v};d)=(v_{1},...,v_{n};d)$ be an integer weight system which
satisfies condition (C2), and let ${\bf
w}=(\frac{v_{1}}{d},...,\frac{v_{n}}{d};1)$ be the equivalent normalized
weight system.

(a) (Milnor and Orlik [MO70]) $D_{\bf w}$ is the divisor of the characteristic
polynomial of the monodromy on the Milnor lattice of an isolated
quasihomogeneous singularity with weight system $({\bf v};d)$. Especially
$D_{\bf w}\in{\mathbb{N}}_{0}[\mu({\mathbb{C}})]$.

(b) (Many people, e.g. [AGV85]) $t^{-v_{1}-...-v_{n}}\rho_{({\bf v};d)}$ is
the generating function of the weighted degrees of the Jacobi algebra

${\mathbb{C}}[x_{1},...,x_{n}]/\Bigl{(}\frac{\partial f}{\partial
x_{1}},...,\frac{\partial f}{\partial x_{n}}\Bigr{)}$

of an isolated quasihomogeneous singularity $f$ with weight system $({\bf
v};d)$. Especially $\rho_{({\bf v};d)}\in{\mathbb{N}}_{0}[t]$.

######  Remarks 6.5.

(i) By Lemma 5.4 (d), $\rho_{({\bf v};d)}$ and $D_{\bf w}$ are related by

$\displaystyle\rho_{({\bf v};d)}=\sum_{\alpha\in
d^{-1}{\mathbb{Z}}}\sigma(\alpha)\cdot t^{d\cdot\alpha},\qquad D_{\bf
w}=\sum_{\alpha\in d^{-1}{\mathbb{Z}}}\sigma(\alpha)\cdot[e^{2\pi i\alpha}],$
(6.1)

where $\sigma:d^{-1}{\mathbb{Z}}\to{\mathbb{N}}_{0}$ has finite support.

(ii) If we write $\rho_{({\bf v};d)}=\sum_{i=1}^{\mu}t^{d\cdot\alpha_{i}}$ for
suitable $\alpha_{i}\in{\mathbb{Q}}$, then these numbers
$(\alpha_{1},...,\alpha_{\mu})$ are the exponents of the singularity, and
$e^{2\pi i\alpha_{1}},...,e^{2\pi i\alpha_{\mu}}$ are the eigenvalues of the
monodromy.

(iii) Problem 2 in [HZ19] asked for a combinatorial proof of the fact
$\rho_{({\bf v};d)}\in{\mathbb{N}}_{0}[t]$ for a weight system $({\bf v};d)$
with condition (C2). (The other two parts of Problem 2 in [HZ19] would follow
from this and from Lemma 5.4 (d).) Theorem 6.4 (b), which was rediscovered by
many people, contains this statement. The classical proof uses that the tuple
of partial derivatives $(\frac{\partial f}{\partial x_{1}},...,\frac{\partial
f}{\partial x_{n}})$ is a regular sequence. We do not know a proof which does
not use the Jacobi algebra and this regular sequence. So we do not have a
solution to problem 2 in [HZ19].

The following theorem was proved in [HM20-2] as a step in a partial proof of
Orlik’s conjecture 1.5. It is also a step in the proof of Theorem 1.4. Its
proof in [HM20-2] uses the Theorems 6.1 and 6.4 (a) and especially that
certain subsystems ${\bf w}^{(p)}$ of ${\bf w}$ (for certain prime numbers
$p$) also satisfy condition (C2) and thus that $D_{{\bf
w}^{(p)}}\in{\mathbb{N}}_{0}[\mu({\mathbb{C}})]$ holds.

######  Theorem 6.6.

Consider a normalized weight system ${\bf w}=(w_{1},...,w_{n};1)$ which
satisfies condition (C2), and consider the numbers
$s_{i},t_{i}\in{\mathbb{N}}$ with $w_{i}=\frac{s_{i}}{t_{i}}$ and
$\gcd(s_{i},t_{i})=1$. Write $D_{\bf w}=\sum_{m\in M_{\bf w}}\psi_{\bf
w}(m)\cdot\Psi_{m}$ where the map $\psi_{\bf w}:M_{\bf w}\to{\mathbb{N}}_{0}$
has finite support $M_{\bf w}\subset{\mathbb{N}}$. The map $\psi_{\bf w}$ is
compatible with the tuple $(\succ_{p}^{\bf w})_{p\in{\mathcal{P}}(M_{\bf w})}$
of excellent orders which is defined as follows,

$\displaystyle s(\succ_{p}^{\bf w})$ $\displaystyle:=$
$\displaystyle\max(v_{p}(m)\,|\,m\in M_{\bf w}),$ (6.2) $\displaystyle
S(\succ_{p}^{\bf w})$ $\displaystyle:=$
$\displaystyle\\{k\in{\mathbb{Z}}_{[1,s(\succ_{p})]}\,|\,|\\{j\in
N\,|\,p^{k}|t_{j}\\}|\textup{ is odd.}\\}$ (6.3)

Finally, we solve problem 4 in [HZ19] in Remark 6.8 and discuss problem 5 in
[HZ19] in Remark 6.9.

######  Remark 6.7.

An isolated quasihomogeneous singularity $f\in{\mathbb{C}}[x_{1},...,x_{n}]$
has multiplicity $\geq 3$ if all monomials in it have degree $\geq 3$ (the
standard degree, not the weighted degree). A result in [Sa71] says first that
the weight system ${\bf w}$ of any isolated quasihomogeneous singularity with
multiplicity $\geq 3$ is unique and satisfies $w_{i}\in(0,\frac{1}{2})$ for
all $i$, and second that any quasihomogeneous singularity can be transformed
by a coordinate change to a Thom-Sebastiani sum
$f(x_{1},...,x_{k})+x_{k+1}^{2}+...+x_{n}^{2}$ of an isolated quasihomogeneous
singularity $f$ of multiplicity $\geq 3$ and the $A_{1}$-singularity
$x_{k+1}^{2}+...+x_{n}^{2}$. Because of Theorem 3.3, one can often restrict to
(isolated quasihomogeneous) singularities of multiplicity $\geq 3$.

######  Remark 6.8.

Let $f$ be an isolated quasihomogeneous singularity of multiplicity $\geq 3$
with weight system ${\bf w}=(w_{1},...,w_{n};1)$. If $n\leq 3$,
$(C2)\iff\overline{(C2)}$. This was first proved in [Sa87, Theorem 3]. A short
combinatorial proof is given in [HK12, Lemma 2.5]. For $n\geq 4$, $(C2)$ is
stronger than $\overline{(C2)}$. Though only one weight system ${\bf w}$ with
$n=4$ which satisfies $\overline{(C2)}$, but not $(C2)$, is published, the
weight system $(58,33,24,1;265)$. It is due to Ivlev [AGV85]. It is also
discussed in [HZ19, Example 5.1]. There problem 4 is posed. It asks for other
weight systems ${\bf w}$ with $n=4$ which satisfy $\overline{(C2)}$, but not
$(C2)$.

We used the software PARI/GP [PARI19] in order to check all reduced integer
weight systems $({\bf v};d)=(v_{1},v_{2},v_{3},v_{4};d)$ with
$\frac{d}{2}>v_{1}\geq v_{2}\geq v_{3}\geq v_{4}$ and $d\leq 360$ (reduced
means $\gcd(v_{1},v_{2},v_{3},v_{4},d)=1$). We found 654077 such weight
systems which satisfy $\overline{(C2)}$. Only 103 of them do not satisfy
$(C2)$. Up to $d=200$, there are 185013 weight systems as above with
$\overline{(C2)}$, and only 23 of them do not satisfy $(C2)$. Table 1 lists
these 23 weight systems $(v_{1},v_{2},v_{3},v_{4};d)$ in the lexicographic
ordering with respect to $(d,v_{1},v_{2},v_{3},v_{4})$. It also gives the
Milnor number $\mu$, the number $L$ of the weight system in the lexicographic
ordering with respect to $(d,v_{1},v_{2},v_{3},v_{4})$ of all weight systems
as above which satisfy $\overline{(C2)}$, and a tuple
$[a_{1},a_{2},a_{3},a_{4},a_{5},a_{6}]\in\\{0,1,2,3,4\\}^{6}$ which says the
following. Write

$(J_{1},J_{2},J_{3},J_{4},J_{5},J_{6})=(\\{1,2\\},\\{1,3\\},\\{1,4\\},\\{2,3\\},\\{2,4\\},\\{3,4\\}).$

Then $a_{i}=|\\{k\in
N\,|\,({\mathbb{N}}_{0}^{J_{i}})_{d-v_{k}}\neq\emptyset\\}|$. That $(C2)$ is
not satisfied implies that for at least one $i\in\\{1,2,3,4,5,6\\}$ the number
$a_{i}$ is $\leq 1$.

$\begin{array}[]{r|r|r|r|c}(v_{1},v_{2},v_{3},v_{4})&d&\mu&L&[a_{1},a_{2},a_{3},a_{4},a_{5},a_{6}]\\\
\hline\cr(27,16,10,1)&81&4615&22598&[2,2,4,1,4,4]\\\
(29,14,12,3)&87&1825&26464&[2,2,4,1,4,2]\\\
(31,18,10,3)&93&2075&31738&[2,2,4,1,2,4]\\\
(35,26,8,1)&105&7663&42215&[2,2,4,1,4,4]\\\
(41,30,8,3)&123&3565&61217&[2,2,4,1,2,4]\\\
(43,34,8,1)&137&14523&78151&[2,3,4,1,4,4]\\\
(49,22,15,12)&147&1125&92880&[2,2,2,3,1,2]\\\
(43,20,14,9)&149&2385&94336&[3,2,4,1,3,4]\\\
(51,28,13,10)&153&1375&100171&[2,2,2,3,1,4]\\\
(49,38,8,3)&155&6201&102969&[2,3,4,1,4,4]\\\
(53,26,12,3)&159&6517&108079&[2,2,4,1,4,2]\\\
(43,32,10,1)&161&26727&112336&[3,2,4,1,4,4]\\\
(55,21,18,16)&165&1043&117928&[2,2,2,2,3,1]\\\
(59,42,9,8)&177&2535&138768&[2,2,2,2,1,4]\\\
(57,36,10,1)&181&26970&149200&[2,3,4,1,4,4]\\\
(49,36,10,3)&183&11591&151842&[3,2,4,1,2,4]\\\
(61,20,18,3)&183&8965&152011&[2,2,4,1,4,2]\\\
(61,22,16,7)&183&3841&152019&[2,2,2,1,4,4]\\\
(57,26,14,3)&185&10176&154800&[2,3,2,1,4,4]\\\
(61,38,10,3)&193&10230&169630&[2,3,4,1,4,4]\\\
(65,46,11,8)&195&2533&172715&[2,2,2,3,1,4]\\\
(59,22,18,1)&199&38010&180617&[3,2,4,1,4,4]\\\
(61,23,22,16)&199&1593&180630&[3,2,3,4,4,1]\end{array}$ Table 1. All reduced
weight systems $(v_{1},v_{2},v_{3},v_{4};d)$ with $d\leq 200$,
$\frac{d}{2}>v_{1}\geq v_{2}\geq v_{3}\geq v_{4}$, and condition
$\overline{(C2)}$, but not condition $(C2)$.

The table does not contain Ivlev’s example $(58,33,24,1;265)$ because
$265>200$. Ivlev’s example has Milnor number 66516. All examples in Table 1
have smaller Milnor numbers. We checked also with PARI/GP for all 103 reduced
weight systems with $d\leq 360$ and $\overline{(C2)}$, but not $(C2)$, that
$\rho_{({\bf v};d)}\in{\mathbb{N}}_{0}[t]$ holds. It is not clear whether
there are weight systems with larger $d$ which satisfy $\overline{(C2)}$ and
$\rho_{({\bf v};d)}\in{\mathbb{Z}}[t]-{\mathbb{N}}_{0}[t]$.

Arnold [AGV85] associated to each reduced weight system $(v_{1},...,v_{n};d)$
which satisfies $\overline{(C2)}$ one type or several types in the following
way. A type is a conjugacy class with respect to the symmetric group $S_{n}$
of a (not necessarily bijective) map $\sigma:N\to N$. Here $\sigma$ is a map
which satisfies $v_{j}|(d-v_{\sigma(j)})$ for $j\in N$. For each $j\in N$ such
a $\sigma(j)$ exists because of condition $\overline{(C2)}$ for sets $J\subset
N$ with $|J|=1$. The number of types is $3$ for $n=2$, $7$ for $n=3$ [AGV85],
$19$ for $n=4$ [YS79], $47$ for $n=5$ and $128$ for $n=6$ [HK12, Examples
3.2]. In [YS79] the types for $n=4$ are numbered $I,II,...,XIX$ (this is
reproduced in [HK12, Example 3.2 (iii)]). At most 9 of the 19 types allow
weight systems which satisfy $\overline{(C2)}$, but not $(C2)$, the weight
systems $V,VIII,XI,XII,XIII,XV,XVI,XVII$ and $XIX$. The other 10 types allow
only weight systems of Thom-Sebastiani sums of cycle type singularities and
chain type singularities. The 23 weight systems in table 1 realize 7 of the 9
types, they do not realize $XIII$ and $XIX$.

######  Remark 6.9.

(i) K. Saito conjectured [Sa88, (3.13) and (4.2)] that the weight system ${\bf
w}=(w_{1},...,w_{n};1)$ of an isolated quasihomogeneous singularity with
multiplicity $\geq 3$ satisfies $\psi_{\bf w}(d_{\bf w})>0$. Here $d=d_{\bf
w}$ for the equivalent reduced integer weight system $(v_{1},...,v_{n};d)$,
because condition $(C2)$ for the set $J=N$ implies $v_{j}|d$ for any $j$.
Counter-examples with $n=4$ to this conjecture were given in [HZ19, Examples
5.5]. The tables in [HK11] show that for $n=4$ and $\mu\leq 500$ the only
counter-examples are those in [HK12, Examples 5.5]. Problem 5 (b) asked for an
answer to the question whether in the case $n=4$ and $\mu\geq 501$ the only
counter-examples are those in [HZ19, Examples 5.5]. We did not settle this.

(ii) With the software PARI/GP [PARI19] we checked that for $n=5$ and $d\leq
200$ only 10 counter-examples to the conjecture in part (i) exist. They are
all obvious generalizations of the examples in [HZ19, Examples 5.5]. They are
given in the following table. They are Thom-Sebastiani sums of singularities
of type $A_{k}$, $D_{k}$ (with odd $k$), and/or a chain type singularity with
one of the three normalized weight systems

$K_{1}:\ \bigl{(}\frac{1}{4},\frac{3}{20},\frac{17}{40}\bigr{)},\quad K_{2}:\
\bigl{(}\frac{1}{4},\frac{3}{20},\frac{25}{56}\bigr{)},\quad K_{3}:\
\bigl{(}\frac{1}{6},\frac{5}{18},\frac{13}{36}\bigr{)}.$

(See [HZ19, ch. 4 and 5] for formulas around the singularities $A_{k}$,
$D_{k}$ and the chain type singularities.) The number $L$ has the same meaning
as in Remark 6.8.

$\begin{array}[]{l|r|r|r|r}\textup{Thom-Sebastiani
sum}&(v_{1},v_{2},v_{3},v_{4},v_{5})&d&\mu&L\\\ \hline\cr D_{13}\otimes
K_{1}&(55,51,30,18,10)&120&299&305495\\\ D_{13}\otimes D_{21}\otimes
A_{3}&(57,55,30,10,6)&120&819&307654\\\ D_{13}\otimes
K_{2}&(77,75,42,18,14)&168&403&740415\\\ D_{11}\otimes D_{29}\otimes
A_{3}&(81,77,42,14,6)&168&1131&744420\\\ D_{11}\otimes
K_{3}&(81,65,50,30,18)&180&253&903983\\\ D_{11}\otimes D_{19}\otimes
A_{5}&(85,81,30,18,10)&180&1045&907794\\\ D_{11}\otimes D_{19}\otimes
A_{2}&(85,81,60,18,10)&180&418&907798\\\ D_{31}\otimes
K_{3}&(87,65,50,30,6)&180&713&909833\\\ D_{19}\otimes D_{31}\otimes
A_{5}&(87,85,30,10,6)&180&2945&909911\\\ D_{19}\otimes D_{31}\otimes
A_{2}&(87,85,60,10,6)&180&1178&909915\end{array}$ Table 2. All reduced weight
systems $(v_{1},v_{2},v_{3},v_{4},v_{5};d)$ with $d\leq 200$,
$\frac{d}{2}>v_{1}\geq...\geq v_{5}$, condition $\overline{(C2)}$, and
$\psi_{\bf w}(d_{\bf w})=0$.

We calculated with PARI/GP also the number of all reduced weight systems
$(v_{1},v_{2},v_{3},v_{4},v_{5};d)$ with $\overline{(C2)}$ and $d\leq 200$ and
$\frac{d}{2}>v_{1}\geq...\geq v_{5}$. It is 1176435.

(iii) K. Saito posed also the weaker conjecture [Sa88, (3.13) and (4.2)] that
the weight system ${\bf w}$ of any isolated quasihomogeneous singularity
satisfies $\psi_{\bf w}(d_{\bf w})>0$ or $\psi_{\bf w}(d_{\bf w}/2)>0$.
Problem 5 (a) in [HZ19] rephrases this conjecture. If the only counter-
examples to the stronger conjecture in part (i) are the obvious
generalizations of the examples 5.5 in [HZ19], the weaker conjecture is true.
But this is not clear.

## References

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[cor1, cor2]Corresponding authors

# Measuring the continuous research impact of a researcher: The $K_{z}$ index

Kiran Sharma<EMAIL_ADDRESS>Ziya Uddin<EMAIL_ADDRESS>School
of Engineering & Technology, BML Munjal University, Gurugram, Haryana-122413,
India

###### Abstract

The ongoing discussion regarding the utilization of individual research
performance for academic hiring, funding allocation, and resource distribution
has prompted the need for improved metrics. While traditional measures such as
total publications, citations count, and the $h$-index provide a general
overview of research impact, they fall short of capturing the continuous
contribution of researchers over time. To address this limitation, we propose
the implementation of the $K_{z}$ index, which takes into account both
publication impact and age. In this study, we calculated $K_{z}$ scores for
376 research profiles. $K_{z}$ reveals that the researchers with the same
$h$-index can exhibit different $K_{z}$ scores, and vice versa. Furthermore,
we observed instances where researchers with lower citation counts obtained
higher $K_{z}$ scores, and vice versa. Interestingly, the $K_{z}$ metric
follows a log-normal distribution. It highlights its potential as a valuable
tool for ranking researchers and facilitating informed decision-making
processes. By measuring the continuous research impact, we enable fair
evaluations, enhance decision-making processes, and provide focused career
advancement support and funding opportunities.

###### keywords:

$h$-index Research impact Research evaluation Citation analysis Science policy
Scientometrics

Traditional metrics such as total publications, citations count, and the
$h$-index provide an overall measure of research impact but fail to capture
the continuous contribution of researchers. Therefore, there is a need for a
robust tool to measure the continuous research impact.

The proposed $K_{z}$ index is introduced as a solution, taking into account
both the impact and age of publications.

Even if two or more researchers have identical total publications, citations
count, and $h$-index, it is unlikely that they share the same $K_{z}$ scores
This characteristic of the $K_{z}$ index makes it a valuable ranking tool.

The $K_{z}$ index enables the identification of both star contributors and
those with lower impact in the realm of research.

By measuring the continuous research impact, a more comprehensive assessment
can be achieved, leading to fair evaluations towards career progression
support and research funding.

##  1 Introduction

Research impact is a crucial factor when evaluating the contributions of
researchers (Egghe, 2010). It plays a vital role in assessing the quality,
significance, and reach of their work, which is instrumental in academic
promotions, grant allocations, award selections, and overall career
progression. Existing indices like the $h$-index and citation count are
commonly used to measure research impact (Bornmann and Daniel, 2005, 2009);
however, it’s important to recognize that citations may not provide a
comprehensive representation of impact, especially in fields where citation
practices differ or in emerging research domains with limited citation
opportunities. Therefore, a more nuanced approach is necessary to capture the
full extent of the research impact, considering multiple dimensions beyond
traditional metrics.

The $h$-index has been subject to criticism due to its limitations in
providing a comprehensive view of scientific impact (Costas and Bordons,
2007). Initially introduced in 2005 by Hirsch, the $h$-index is calculated
based on the number of papers that have received at least $h$ citations from
other papers (Hirsch, 2005). Since its introduction, the $h$-index has gained
significant popularity in academia and has been commonly employed to evaluate
the academic success of scientists in various areas, including hiring
decisions, promotions, and grant acceptances. Despite efforts by researchers
to propose alternative variants of the $h$-index (Egghe, 2006; Jin et al.,
2007; Zhang, 2009; Alonso et al., 2010; Khurana and Sharma, 2022), the
traditional $h$-index remains widely used as a performance metric in the
assessment of scientists because of its simplicity.

To overcome the limitations of $h$-index, Egghe in 2006 proposed $g$-index
which is determined by the distribution of citations across their
publications. It is determined by sorting the articles in decreasing order
based on the number of citations they have received. The $g$-index is defined
as the largest number $g$ for which the top $g$ articles collectively
accumulate at least $g^{2}$ citations (Egghe, 2006). This means that a
researcher with a $g$-index of 10 has published at least 10 articles that
collectively have received at least $(10^{2}=100)$ citations. It’s important
to note that unlike the $h$-index, the citations contributing to the $g$-index
can be generated by only a small number of articles. For example, a researcher
with 10 papers, where 5 papers have no citations and the remaining five have
350, 35, 10, 2, and 2 citations respectively, would have a $g$-index of 10 but
an $h$-index of 3 (as only three papers have at least three citations each).

Further, after recognizing the limitations of the $h$-index (Ding et al.,
2020), researchers have proposed various complementary measures to provide a
more comprehensive assessment of research impact such as $R$-index (Jin et
al., 2007), $e$-index (Zhang, 2009), $h^{\prime}$-index (Zhang, 2013). In the
study by Khurana et al. (2022) (Khurana and Sharma, 2022), an enhancement to
the $h$-index is proposed to capture the impact of the highly cited paper.
They introduced $h_{c}$ which is based on the weight assigned to the highly
cited paper. $h_{c}$ has a greater impact on researchers with lower $h$-index
values, particularly by highlighting the significance of their highly cited
paper. However, the effect of $h_{c}$ on established researchers with higher
$h$-index values was found to be negligible. It is worth noting that the
$h_{c}$ focuses on the first highly cited paper and does not consider the
impact of subsequent highly cited papers. This limitation again highlights the
need for a more comprehensive measure that takes into account all the
important factors contributing to research impact (Martin, 1996).

The another measure named, $L$-sequence, introduced by Liu et al. (Liu and
Yang, 2014), computes the $h$-index sequence for cumulative publications while
taking into account the yearly citation performance. In this approach, the $L$
number is calculated based on the $h$-index concept for a specific year.
Consequently, the impact of the most highly cited paper in that year may be
overlooked, and papers with less than $L$ citations are also not considered.
Although the concept captures the yearly citation performance of all papers,
it does not effectively capture the continuous impact of each individual
paper. Also gathering data for the $L$-sequence can be challenging, as it
requires delving into the citation history of each paper for every year.

Quantifying research impact is a multifaceted endeavor (Batista et al., 2006).
There is no universally accepted metric or methodology for measuring
continuous research impact, and different stakeholders may prioritize
different indicators, such as publications, citations, patents, or societal
impact. Measuring the continuous research impact of a researcher is crucial
for granular assessment, differentiation among researchers, funding decisions,
identification of emerging talent, etc. Determining an inclusive and
comprehensive approach that captures the diverse dimensions of research impact
remains a challenge.

###  1.1 Research Objective

The primary objective of this study is to introduce a reliable metric that can
effectively capture the continuous research impact of a researcher. The aim of
the proposed metric is to differentiate between two researchers who possess
identical research parameters. In order to accomplish the stated objective, a
newly introduced measure called the $K_{z}$-index has been proposed.

##  2 $K_{z}$-index

The proposed $K_{z}$ index serves as a tool to measure the research impact of
a researcher. It aims to capture the continuous and evolving contributions
made by the researcher over time, considering factors such as total
publications, citation count, and publication age.

###  2.1 Definition of $K_{z}$-index

To measure the continuous research impact of a researcher, $K_{z}$ takes into
account two important factors of research:

  1. 1.

Impact ($k$): The impact of a paper is determined by considering two factors:
the number of citations ($C$) it has received and its $h$-index.

The impact of the paper is calculated by using the following equation;

$C\leq(h+1)^{k},$ (1)

where $k\in\mathbb{R}^{+}$.

  2. 2.

Age ($\Delta t$): $\Delta t$ represents the publication age in relation to the
current year and can be determined through the following computation.

$\Delta t=C_{y}-P_{y}$ (2)

where $C_{y}$ represents the current year and $P_{y}$ represents the
publication year.

Now, from Eq.1 and Eq.2, $K_{z}$ can be calculated for every researcher $i$ as

$K_{z}=\sum_{i=1}^{N}\frac{k_{i}}{\Delta t_{i}}$ (3)

where $N$ is number of publications and $N>0$.

Equation 3 highlights the sig nificance of $K_{z}$ metric by incorporating
essential research indicators, including total citations, year of publication,
number of publications, publication age, and $h$-index. This comprehensive
approach ensures that all significant aspects of a researcher’s work are
considered, resulting in a more robust and holistic assessment of their
research impact.

###  2.2 Advantages of $K_{z}$

Measuring the continuous research impact of a researcher is crucial for
several reasons:

  1. 1.

Granular assessment: Traditional matrices such as the citations count,
$h$-index, etc. present an overall impact on a researcher and do not have the
capability to capture the ongoing progress and advancement of their work,
whereas $K_{z}$ can acquire a more nuanced and thorough comprehension of a
researcher’s contributions as they evolve over time.

  2. 2.

Differentiation among researchers: Even if two researchers possess the same
$h$-index, their patterns of impact over time may vary significantly.
Analyzing their continuous research impact can uncover disparities in
productivity and can provide a more comprehensive understanding of their
individual profiles. Hence, $K_{z}$ allows for a more nuanced differentiation
among researchers.

  3. 3.

Evaluation of long-term impact: Researchers may experience fluctuations in
their productivity and impact over their careers. Measuring continuous
research impact enables the evaluation of long-term contributions. $K_{z}$ has
the capability of highlighting researchers who consistently generate
influential work and have a lasting impact on their field.

  4. 4.

Career progression and funding decisions: Many academic institutions, funding
agencies, and hiring committees rely on research performance metrics to make
decisions. $K_{z}$ can provide more informed evaluations of researchers,
enabling fairer assessments and enhancing the recognition of sustained
excellence.

  5. 5.

Identification of emerging talent: Continuous research impact measurement can
help identify early-career researchers with promising trajectories. By
recognizing their continuous growth and impact, further opportunities can be
provided to nurture their potential.

##  3 Case studies of $K_{z}$

We conducted four case studies to explore the significance of $K_{z}$. Each
case study involved two researchers, namely $R1$ and $R2$. The number of
publications was kept constant across all cases, while the focus was on
comparing the $h$-index and total citations ($TC$) of two researchers.

  1. 1.

Case I - Identical $h$-index and total citations: Table 1 represents the first
case study where we assumed that both researchers $R1$ and $R2$ have the same
$h$-index and total citations count. However, despite sharing these
characteristics, researcher $R2$ obtained a higher $K_{z}$ score than $R1$.
This difference in $Kz$ scores can be attributed to the impact of the
publication year, which played a dominant role in determining the continuous
research impact of each researcher. It highlights the significance of
considering the temporal aspect of research contributions when assessing the
research impact on individuals.

  2. 2.

Case II - Identical $h$-index and different total citations: In this case
(Table 2), both researchers $R1$ and $R2$ have an equal number of publications
and $h$-index, but they differ in their total citations count. Researcher $R1$
has one highly cited paper, while researcher $R2$ has multiple highly cited
papers. Despite $R1$ having a higher total number of citations compared to
$R2$, $R2$ obtains a higher $K_{z}$ score. This indicates that the impact of
having multiple highly cited papers outweighs the effect of a single highly
cited paper in determining the continuous research impact.

  3. 3.

Case III(a) - Different $h$-index and total citations: In this case (Table 3),
both researchers have an equal number of publications but differ in their
$h$-index, number of high impact papers, and total citations. Researcher $R1$
has a higher $h$-index but lower total citation count compared to $R2$.
However, despite $R1$ having a lower total citation count, they obtain the
highest $K_{z}$ score. This highlights the importance of considering the
continuous research impact captured by $K_{z}$, which takes into account not
only the number of citations but also the publication age and impact of
publications.

  4. 4.

Case III(b) - Different $h$-index and Total Citations:: In this case (Table
4), we again considered two researchers with an equal number of publications
but different $h$-index, high impact papers, and total citations. Researchers
$R1$ had a higher $h$-index and total citation count compared to researcher
$R2$. Surprisingly, despite these differences, it was researcher $R2$ who
obtained the highest $K_{z}$ score. This finding suggests that the $K_{z}$
score takes into account factors beyond just $h$-index and total citations,
emphasizing the importance of considering the continuous impact and temporal
aspects of research contributions.

Table 1: Two researchers with identical $h$-index and total citations. Case I | Researcher 1, $h=4$ | Researcher 2, $h=4$
---|---|---
S. No | $P_{y}$ | $C$ | $k$ | $\Delta t$ | $k^{\prime}$ | $P_{y}$ | $C$ | $k$ | $\Delta t$ | $k^{\prime}$
1 | 2014 | 40 | 2.2921 | 9 | 0.255 | 2014 | 2 | 0.4307 | 9 | 0.048
2 | 2015 | 30 | 2.1133 | 8 | 0.264 | 2015 | 3 | 0.6827 | 8 | 0.085
3 | 2016 | 0 | 0 | 7 | 0 | 2016 | 3 | 0.6827 | 7 | 0.098
4 | 2017 | 3 | 0.6827 | 6 | 0.114 | 2016 | 40 | 2.2921 | 7 | 0.327
5 | 2018 | 24 | 1.9747 | 5 | 0.395 | 2017 | 1 | 0 | 6 | 0
6 | 2019 | 1 | 0 | 4 | 0 | 2018 | 30 | 2.1133 | 5 | 0.423
7 | 2020 | 1 | 0 | 3 | 0 | 2019 | 22 | 1.9206 | 4 | 0.48
8 | 2021 | 1 | 0 | 2 | 0 | 2020 | 0 | 0 | 3 | 0
9 | 2022 | 0 | 0 | 1 | 0 | 2021 | 1 | 0 | 2 | 0
10 | 2022 | 10 | 1.4307 | 1 | 1.431 | 2022 | 8 | 1.2921 | 1 | 1.292
|  $TC=110$, $Kz=2.459$ |  $TC=110$, $Kz=2.753$
Table 2: Two researchers with identical $h$-index and different total citations. Case II | Researcher 1, $h=4$ | Researcher 2, $h=4$
---|---|---
S. No | $P_{y}$ | $C$ | $k$ | $\Delta t$ | $k^{\prime}$ | $P_{y}$ | $C$ | $k$ | $\Delta t$ | $k^{\prime}$
1 | 2014 | 1000 | 4.2921 | 9 | 0.477 | 2014 | 500 | 3.8614 | 9 | 0.429
2 | 2015 | 4 | 0.8614 | 8 | 0.108 | 2015 | 300 | 3.544 | 8 | 0.443
3 | 2016 | 0 | 0 | 7 | 0 | 2016 | 100 | 2.8614 | 7 | 0.409
4 | 2017 | 4 | 0.8614 | 6 | 0.144 | 2016 | 0 | 0 | 7 | 0
5 | 2018 | 5 | 1.0001 | 5 | 0.2 | 2017 | 2 | 0.4307 | 6 | 0.072
6 | 2019 | 1 | 0 | 4 | 0 | 2018 | 50 | 2.4307 | 5 | 0.486
7 | 2020 | 1 | 0 | 3 | 0 | 2019 | 1 | 0 | 4 | 0
8 | 2021 | 1 | 0 | 2 | 0 | 2020 | 3 | 0.6827 | 3 | 0.228
9 | 2022 | 0 | 0 | 1 | 0 | 2021 | 1 | 0 | 2 | 0
10 | 2022 | 0 | 0 | 1 | 0 | 2022 | 0 | 0 | 1 | 0
|  $TC=1016$, $K_{z}=0.929$ |  $TC=957$, $K_{z}=2.067$
Table 3: Two researchers with different $h$-index and total citations where R1 has higher $h$-index and lower total citations than R2. Case III | Researcher 1, $h=5$ | Researcher 2, $h=3$
---|---|---
S. No | $P_{y}$ | $C$ | $k$ | $\Delta t$ | $k^{\prime}$ | $P_{y}$ | $C$ | $k$ | $\Delta t$ | $k^{\prime}$
1 | 2014 | 90 | 2.5114 | 9 | 0.279 | 2014 | 250 | 3.9829 | 9 | 0.443
2 | 2015 | 80 | 2.4457 | 8 | 0.306 | 2015 | 2 | 0.5001 | 8 | 0.063
3 | 2016 | 20 | 1.672 | 7 | 0.239 | 2016 | 2 | 0.5001 | 7 | 0.071
4 | 2017 | 3 | 0.6132 | 6 | 0.102 | 2016 | 82 | 3.1788 | 7 | 0.454
5 | 2018 | 24 | 1.7738 | 5 | 0.355 | 2017 | 2 | 0.5001 | 6 | 0.083
6 | 2019 | 2 | 0.3869 | 4 | 0.097 | 2018 | 110 | 3.3907 | 5 | 0.678
7 | 2020 | 3 | 0.6132 | 3 | 0.204 | 2019 | 1 | 0 | 4 | 0
8 | 2021 | 3 | 0.6132 | 2 | 0.307 | 2020 | 2 | 0.5001 | 3 | 0.167
9 | 2022 | 2 | 0.3869 | 1 | 0.387 | 2021 | 2 | 0.5001 | 2 | 0.25
10 | 2022 | 23 | 1.75 | 1 | 1.75 | 2022 | 0 | 0 | 1 | 0
|  $TC=250$, $K_{z}=4.026$ |  $TC=453$,$K_{z}=2.209$
Table 4: Two researchers with different $h$-index and total citations where R1 has higher $h$-index and total citations than R2. Case IV | Researcher 1, $h=6$ | Researcher 2, $h=4$
---|---|---
S. No | $P_{y}$ | $C$ | $k$ | $\Delta t$ | $k^{\prime}$ | $P_{y}$ | $C$ | $k$ | $\Delta t$ | $k^{\prime}$
1 | 2014 | 200 | 2.7228 | 9 | 0.303 | 2014 | 2 | 0.4307 | 9 | 0.048
2 | 2015 | 150 | 2.575 | 8 | 0.322 | 2015 | 2 | 0.4307 | 8 | 0.054
3 | 2016 | 5 | 0.8271 | 7 | 0.118 | 2016 | 3 | 0.6827 | 7 | 0.098
4 | 2017 | 10 | 1.1833 | 6 | 0.197 | 2016 | 1 | 0 | 7 | 0
5 | 2018 | 35 | 1.8271 | 5 | 0.365 | 2017 | 280 | 3.5011 | 6 | 0.584
6 | 2019 | 1 | 0 | 4 | 0 | 2018 | 2 | 0.4307 | 5 | 0.086
7 | 2020 | 33 | 1.7969 | 3 | 0.599 | 2019 | 40 | 2.2921 | 4 | 0.573
8 | 2021 | 1 | 0 | 2 | 0 | 2020 | 70 | 2.6398 | 3 | 0.88
9 | 2022 | 2 | 0.3563 | 1 | 0.356 | 2021 | 2 | 0.4307 | 2 | 0.215
10 | 2022 | 32 | 1.7811 | 1 | 1.781 | 2022 | 50 | 2.4307 | 1 | 2.431
|  $TC=469$, $K_{z}=4.041$ |  $TC=452$, $K_{z}=4.969$

##  4 Empirical study

To calculate the continuous research impact ($K_{z}$) of researchers, the
research profiles of 376 individuals affiliated with Monash University,
Australia were obtained. Monash University is a public research institution
located in Australia, and information about the researchers can be found on
their webpage at https://research.monash.edu/en/persons/. The webpage provides
the researcher’s research ID and Orcid ID, which facilitated the extraction of
their publication details and citations from the Web of Science database. From
a pool of 6316 researchers’ profiles, we selected 376 profiles across
different disciplines, ensuring a range of $h$-index values ($1\leq h\leq
112$). The choice of databases was made based on data availability. For each
researcher ID, information regarding the publication year and the
corresponding citations received were extracted. For each researcher, the
$h$-index, $g$-index, and $K_{z}$ were computed. Additionally, the overall
research age or career length of the researcher was determined by subtracting
the year of his/her first publication from the current year.

###  4.1 Comparison of $K_{z}$ with $h$-index and career length

Figure 1: Scattered plot of $K_{z}$ versus career length. Each dot correspond
to a researcher. The horizontal dashed line represents the median of the axis
and vertical dashed lines divides the plots in three zones based on the
researcher’s career length.

By using equation 3, we calculated the $K_{z}$ score of 376 researchers. In
Figure 1, a scatter plot depicting the relationship between $K_{z}$ and career
length. Each dot on the plot represents an individual researcher. The
horizontal dashed line represents the median of the axis, while vertical
dashed lines are used to divide the plot into three zones based on the length
of the researchers’ careers: early career ($\leq 10$ years), mid career ($>10$
and $\leq 20$ years) and advanced career ($>20$ years). This visualization
clearly differentiate between the star performer and average performer at
different career stages.

Table 5 provides examples of researchers who have the same $h$-index values of
25 and 30. It also includes the computation of the $g$-index, which
demonstrates that researchers with the same $h$-index can have different $g$
values due to variations in their total citation counts. Therefore, it is
possible for a researcher with a lower citation count to have a higher
$g$-index, and vice versa. The presence of the same $h$-index highlights its
limitation in differentiating the top-performing researcher from others
whereas $K_{z}$ significantly differentiates the impactful researcher from
others. This distinction highlights the varying impact among researchers.
Similarly, Table 6 showcases profiles of researchers with the same career age,
yet their $K_{z}$ scores differ. $K_{z}$ clearly differentiates the impactful
researcher from others where researchers are of the same career length. The
same observation applies to total publication and citation counts.

In Table 7, we examined 11 comparative cases of researchers with identical
$h$-index and career length. Among these cases, one noteworthy instance is
$S1$, where two researchers share the same career length of 8 years and
$h$-index of 12. However, the researcher with higher total publications and
citations count has a higher $K_{z}$ score than the other. Whereas, in case
$S3$, two researchers have a career length of 13 years and $h$-index of 19,
the one with lower total publications but higher citation counts, compared to
the other researcher, has a higher $K_{z}$ score. On the other hand, in case
$S7$, two researchers have a career length of 17 years and $h$-index of 13,
the one with higher total publications but lower citation counts, compared to
the other researcher, has a higher $K_{z}$ score. Hence, this indicates that
the $K_{z}$ metric considers all relevant research indicators such as total
publications, citation count, $h$-index, and publication age to capture the
continuous impact of an individual. It is not safe to assume that a higher
$K_{z}$ score is solely determined by either higher total publications or
higher citation counts. Additionally, it cannot be concluded that a person
with a higher $h$-index will always have a higher $K_{z}$ score. The $K_{z}$
metric takes a comprehensive approach in evaluating research impact,
considering multiple factors simultaneously.

Table 5: Comparative analysis among researchers having identical $h$-index. WoS Researcher ID |  | Career
---
Length (yrs)
Publications |  | Total
---
Citations
$h$-index |  $g$-index | $K_{z}$
B-6419-2008 | 17 | 44 | 2415 | 25 | 44 | 5.24
H-6054-2014 | 19 | 38 | 3433 | 25 | 37 | 6.76
D-5776-2019 | 26 | 68 | 1984 | 25 | 45 | 6.828
J-1532-2014 | 18 | 59 | 2982 | 25 | 46 | 7.896
N-8153-2014 | 20 | 78 | 4217 | 25 | 65 | 9.156
E-6623-2015 | 14 | 59 | 1530 | 25 | 38 | 10.618
A-3854-2010 | 21 | 86 | 2034 | 25 | 44 | 11.224
K-5277-2012 | 24 | 73 | 3783 | 25 | 64 | 11.912
B-8486-2008 | 29 | 79 | 2851 | 30 | 54 | 4.487
G-1412-2012 | 34 | 69 | 2816 | 30 | 56 | 5.517
H-3196-2013 | 13 | 94 | 2538 | 30 | 49 | 8.684
F-2273-2010 | 16 | 102 | 2627 | 30 | 48 | 10.446
I-1956-2014 | 23 | 123 | 3797 | 30 | 60 | 11.05
I-1738-2013 | 19 | 105 | 3306 | 30 | 57 | 11.309
D-4239-2011 | 25 | 133 | 3343 | 30 | 59 | 12.475
H-4935-2013 | 15 | 100 | 2945 | 30 | 52 | 18.97
Table 6: Comparative analysis among researchers having identical career length. WoS Researcher ID |  | Career
---
Length (yrs)
Publications |  | Total
---
Citations
$h$-index |  $g$-index | $K_{z}$
K-5514-2018 | 10 | 9 | 32 | 4 | 6 | 1.043
P-7354-2019 | 10 | 8 | 171 | 6 | 8 | 1.69
I-9365-2017 | 10 | 20 | 287 | 10 | 17 | 3.823
G-3877-2013 | 10 | 75 | 1189 | 18 | 34 | 9.813
L-4481-2018 | 10 | 90 | 6012 | 28 | 83 | 22.385
N-4364-2019 | 20 | 23 | 757 | 14 | 23 | 1.905
A-4190-2009 | 20 | 32 | 832 | 14 | 29 | 3.795
B-7556-2008 | 20 | 60 | 7144 | 27 | 54 | 6.847
C-9764-2013 | 20 | 122 | 5917 | 42 | 77 | 10.995
I-1587-2014 | 20 | 107 | 1127 | 18 | 30 | 12.88
C-4319-2011 | 20 | 170 | 5080 | 39 | 62 | 19.088
H-9193-2014 | 30 | 26 | 181 | 8 | 14 | 2.939
P-8366-2016 | 30 | 98 | 5701 | 40 | 77 | 6.378
B-9553-2008 | 30 | 91 | 6784 | 45 | 85 | 10.524
H-5706-2014 | 30 | 171 | 4559 | 35 | 60 | 15.996
A-5452-2008 | 30 | 283 | 26495 | 89 | 158 | 25.657
I-6251-2012 | 30 | 280 | 58171 | 68 | 244 | 29.05
Table 7: Comparative analysis among researchers having identical research career length (yrs) and $h$-index.  S.No | WoS Researcher ID |  | Career
---
Length (yrs)
Publications |  | Total
---
Citations
$h$-index |  $g$-index | $K_{z}$
S1 | F-9424-2013 | 8 | 37 | 1595 | 12 | 5 | 7.041
O-7942-2018 | 8 | 34 | 454 | 12 | 7 | 5.291
S2 | AAE-7279-2019 | 12 | 47 | 1529 | 15 | 18 | 11.122
I-9929-2012 | 12 | 37 | 1236 | 15 | 26 | 4.321
S3 | L-4989-2018 | 13 | 84 | 1875 | 19 | 16 | 20.182
M-7607-2014 | 13 | 106 | 1130 | 19 | 29 | 8.26
S4 | E-6431-2011 | 14 | 16 | 508 | 8 | 16 | 4.057
N-1676-2017 | 14 | 14 | 726 | 8 | 22 | 2.771
S5 | A-7222-2013 | 14 | 28 | 608 | 14 | 21 | 6.299
L-1320-2019 | 14 | 23 | 875 | 14 | 21 | 3.264
S6 | K-7419-2014 | 15 | 52 | 482 | 11 | 16 | 2.845
G-1470-2011 | 15 | 36 | 351 | 11 | 13 | 4.741
S7 | O-9174-2014 | 17 | 36 | 708 | 13 | 22 | 4.444
J-5651-2016 | 17 | 16 | 857 | 13 | 22 | 2.173
S8 | Q-9068-2018 | 18 | 47 | 2034 | 21 | 36 | 7.279
H-4554-2014 | 18 | 53 | 1462 | 21 | 26 | 8.99
S9 | F-6776-2014 | 18 | 159 | 1843 | 23 | 28 | 15.62
H-8387-2012 | 18 | 78 | 1798 | 23 | 34 | 8.635
S10 | F-4112-2014 | 22 | 18 | 617 | 13 | 38 | 2.402
C-6296-2014 | 22 | 35 | 1456 | 13 | 38 | 4.842
S11 | C-2440-2013 | 28 | 38 | 6087 | 27 | 60 | 2.401
N-5018-2017 | 28 | 87 | 2588 | 27 | 62 | 7.02

###  4.2 Probability distribution of $K_{z}$

Figure 2 presents a graphical representation of the plot for $\log(K_{z})$,
which exhibits a mean value of $\mu$ and a standard deviation of $\sigma$.
This plot is compared to the normal distribution with the same mean and
standard deviation. The overlapping nature of the two plots suggests that the
variable $K_{z}$ follows a log-normal distribution. To confirm this
observation, a “Goodness of Fit" test was conducted using the $\chi^{2}$
distribution. The objective of the Goodness of Fit Test was to assess the
suitability of the null hypothesis that states “the distribution of
$\log(K_{z})$ conforms well to a normal distribution.” The test was executed
in the following manner:

The logarithm of the values of $K_{z}$ was computed, and these values were
then classified into seven distinct classes, taking into account the mean
($\mu=0.78787$) and standard deviation ($\sigma=0.37448$). Subsequently, the
observed frequencies ($O_{i}$) for each class were determined. To obtain the
expected frequencies ($E_{i}$), the entire dataset consisting of 376
observations was subjected to calculations based on the normal distribution.
The specific calculations and their results are provided in Table 8.

The $\chi^{2}$ value was computed using the formula
$\chi^{2}=\sum\frac{(O_{i}-E_{i})^{2}}{E_{i}}$ and yielded a value of $7.466$.
As the calculated $\chi^{2}$ value is smaller than the critical value
$\chi^{2}_{(6,0.05)}=12.592$, we cannot reject the null hypothesis at a
significance level of $0.05$. Therefore, we can conclude that $\log(K_{z})$ is
a suitable fit for the normal distribution.

Figure 2: Distribution of $\log(K_{z})$ (dashed red) versus normal
distribution (solid black) with same $\mu$ and $\sigma$.

####  4.2.1 Identification of top contributors and low contributors

In the case of a normal distribution, the middle 50% of the data is
encompassed within a range of $+0.67$ and $-0.67$ standard scores from the
mean. Consequently, researchers in the top 25% satisfy the condition
$K_{z}\geq e^{(\mu+0.67\sigma)}$, while researchers in the bottom 25% satisfy
the condition $K_{z}\leq e^{(\mu-0.67\sigma)}$. Similarly, using the
properties of normal distribution, the $\alpha\%$ of top and bottom performers
can be identifies. Unlike previous indices such as the $h$, $g$, $e$, $h_{c}$,
etc., the $K_{z}$-index allows for the identification of both top and bottom
contributors. This categorization based on $K_{z}$ scores can be beneficial
for universities, scientific communities, and research funding agencies in
identifying significant contributors.

Table 8: Goodness of fit test. Classes |  | Observed
---
Frequencies $(O_{i})$
| Expected frequencies
---
$(E_{i})$ for$\mathcal{N}(\mu,\sigma)$
$\log(K_{z})<\mu-1.5\sigma$ | 14 | 25
$\mu-1.5\sigma\leq\log(K_{z})<\mu-\sigma$ | 34 | 35
$\mu-\sigma\leq\log(K_{z})<\mu-0.5\sigma$ | 57 | 56
$\mu-0.5\sigma\leq\log(K_{z})<\mu+0.5\sigma$ | 157 | 144
$\mu+0.5\sigma\leq\log(K_{z})<\mu+\sigma$ | 57 | 56
$\mu+\sigma\leq\log(K_{z})<\mu+1.5\sigma$ | 29 | 35
$\log(K_{z})\geq\mu-1.5\sigma$ | 28 | 25
Total | 376 |

##  5 Discussion and conclusion

In this study, we have discussed various research indicators, including total
publications, citations count, $h$-index, $g$-index, etc., commonly used to
measure the impact of research. While total publications, citation count, and
$h$-index are commonly used indicators to assess research impact, they have
some limitations when considered individually.

  1. 1.

Total publications: Relying solely on the number of publications can be
misleading, as it does not consider the quality or impact of those
publications. Quantity alone does not reflect the significance or influence of
a researcher’s work.

  2. 2.

Citations count: While citation count is a useful indicator of the influence
and visibility of a researcher’s work, it can be influenced by factors such as
the field of study, publication age, and citation practices within the
research community. Additionally, self-citations can artificially inflate
citation counts and impact assessments.

  3. 3.

$h$-index: The $h$-index takes into account both the number of publications
and their corresponding citations. However, it does not differentiate between
highly cited publications and those with fewer citations. A researcher with a
few highly influential papers can have the same $h$-index as someone with many
moderately cited papers. Additionally $h$-index ignores all the papers which
are cited less than the $h$.

  4. 4.

Temporal considerations: Individual metrics may not capture the continuous
progress and development of a researcher’s work over time. They provide a
snapshot of impact at a specific moment and may not reflect the long-term
contributions or evolving research trajectory.

To overcome these limitations and capture the dynamic nature of research
impact, it is essential to consider multiple indicators and employ
comprehensive assessment approaches like the $K_{z}$ metric, which
incorporates various factors to provide a more nuanced understanding of
research impact. $K_{z}$ is filed independent as well as takes into account
the temporal aspect of the work. Unlike other research indicators, $K_{z}$
takes into account not only the total publications and citations count but the
age of the publications too. Our results demonstrate how $K_{z}$ can
effectively differentiate between two potential researchers who may have the
same $h$-index, citations count, or career length. By incorporating $K_{z}$
into the evaluation process, we can better assess the research dynamics of an
individual and gain insights into their continuous impact over time.

To conclude, $K_{z}$ holds the potential to serve as a superior measure for
capturing the impact of individuals, institutions, or journals. Its
comprehensive consideration of various research indicators allows a more
nuanced assessment of research impact. Further$K_{z}$ can be utilized as a
ranking method to evaluate and rank researchers within an institution based on
their research impact. Similarly, institutions and journals can be compared
and ranked according to their research impact. This information can be
valuable in decision-making processes, as funding agencies, research award
committees and hiring bodies can leverage the power of $K_{z}$ to rank
potential candidates within a specific field. It provides a standardized tool
to assess and compare the impact of research entities, facilitating more
informed decisions and promoting recognition based on research excellence.

There are some challenges associated with computing the $K_{z}$ metric too.
Some of the potential challenges include:

  1. 1.

Data availability and accuracy: Obtaining accurate and comprehensive data from
various sources can be a challenge. Different databases may have variations in
the coverage of publications and citations, potentially leading to incomplete
or inconsistent data.

  2. 2.

Data quality and reliability: The accuracy and reliability of the data sources
used for computing $K_{z}$ are crucial as inaccurate or incomplete data can
result in misleading or flawed assessments of research impact.

  3. 3.

Self-citation manipulation: The issue of self-citation manipulation, where
researchers excessively cite their own work to inflate their impact metrics,
can pose a challenge as detecting such manipulations requires careful scrutiny
and data filtering techniques.

As discussed, it can be inferred that the $K_{z}$ index is a comprehensive
mathematical function that considers multiple factors to assess the impact of
a researcher. These factors include the researcher’s total publications, the
citation count of each paper, the researcher’s $h$-index, and the age of
publication. The $K_{z}$ index recognizes influential papers which often
receive citations at a faster rate, indicating a greater impact, and therefore
assigns them higher weight in impact evaluation. By considering these aspects,
the $K_{z}$ index tends to yield higher values in cases where a researcher has
made significant contributions that have garnered substantial citations.

## Acknowlegement

We acknowledge the suggestions provided by Dr. Satyam Mukherjee.

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