fdsfd ARTICLES 2023 to appear 1 January 5, 2023 April 17, 2023 January 1, 2022

Parametric “Non-nested” Discriminants for Multiplicities of Univariate
Polynomials

<EMAIL_ADDRESS><EMAIL_ADDRESS>
Hoon Hong

Hoon Hong, Jing Yang

# Parametric “Non-nested” Discriminants
for Multiplicities of Univariate Polynomials444This paper has been accepted
for publication in SCIENCE CHINA Mathematics.

Hong Hong  Jing Yang  Department of Mathematics, North Carolina State
University, Box 8205, Raleigh, NC 27695, USA  SMS–HCIC–School of Mathematics
and Physics, Center for Applied Mathematics of Guangxi,
Guangxi Minzu University, Nanning 530006, China

###### Abstract

We consider the problem of complex root classification, i.e., finding the
conditions on the coefficients of a univariate polynomial for all possible
multiplicity structures on its complex roots. It is well known that such
conditions can be written as conjunctions of several polynomial equations and
one inequation in the coefficients. Those polynomials in the coefficients are
called discriminants for multiplicities. It is also known that discriminants
can be obtained by using repeated parametric gcd’s. The resulting
discriminants are usually nested determinants, that is, determinants of
matrices whose entries are determinants, and so on. In this paper, we give a
new type of discriminants which are not based on repeated gcd’s. The new
discriminants are simpler in the sense that they are non-nested determinants
and have smaller maximum degrees.

###### keywords:

Parametric polynomial, complex roots, discriminant, multiplicity, resultant

12D10, 68W30

##  1 Introduction

In this paper, we consider the problem of complex root classification, i.e.,
finding the conditions on the coefficients of a polynomial over the complex
field $\mathbb{C}$ for every potential multiplicity structure its complex
roots may have. For example, consider a quintic polynomial
$F=a_{5}x^{5}+a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}$ where $a_{i}$’s
take values over $\mathbb{C}$. We would like to find conditions
$C_{0},C_{1},\ldots,C_{6}$ on $a=(a_{0},\ldots,a_{5})$ such that

$\text{multiplicity structure of
}F=\left\\{\begin{array}[c]{lll}\left(1,1,1,1,1\right),&\text{if
}&C_{0}\left(a\right)\ \text{holds;}\\\
(2,1,1,1),&\text{if}&C_{1}\left(a\right)\ \text{holds};\\\
(2,2,1),&\text{if}&C_{2}\left(a\right)\ \text{holds};\\\
(3,1,1),&\text{if}&C_{3}\left(a\right)\ \text{holds};\\\
(3,2),&\text{if}&C_{4}\left(a\right)\ \text{holds};\\\
\left(4,1\right),&\text{if}&C_{5}\left(a\right)\ \text{holds};\\\
\left(5\right),&\text{if}&C_{6}\left(a\right)\
\text{holds}.\end{array}\right.$

In general, the problem is stated as follows:

Problem: _For every_ $\bm{\mu}=\left(\mu_{1},\ldots,\mu_{m}\right)$ _such
that_ $\mu_{1}\geq\ldots\geq\mu_{m}>0$ _and_ $\mu_{1}+\cdots+\mu_{m}=n$_, find
a condition on the coefficients of a polynomial_ $F$_over $\mathbb{C}$ of
degree _$n$ such that the multiplicity structure of $F$ is $\bm{\mu}$.

The problem is important because many tasks in mathematics, science and
engineering can be reduced to the problem. Due to its importance, the problem
and several related problems have been already carefully studied [5, 7, 8, 9,
10, 13].

The problem can be viewed as a generalization of the well known problem of
finding a condition on coefficients such that the polynomial has the given
number of distinct roots. This subproblem has been extensively studied. For
instance, the subdiscriminant theory provides a complete solution to the
subproblem: a univariate polynomial of degree $n$ has $m$ distinct roots if
and only if its $0$-th, $\ldots$, $(n-m-1)$-th psd’s (i.e., principal
subdiscriminant coefficient) vanish and the $(n-m)$-th psd does not. For
details, see standard textbooks on computational algebra (e.g., [1]).

In [13], Yang, Hou and Zeng gave an algorithm to generate conditions for
discriminating different multiplicity structures of a univariate polynomial
(referred as YHZ’s condition hereinafter) by making use of repeated gcd
computation for parametric polynomials [3, 4, 11]. It is based on a similar
idea adopted by Gonzalez-Vega et al. [5] for solving the real root
classification and quantifier elimination problems by using Sturm-Habicht
sequences. The conditions produced by these methods are conjunctions of
several polynomial equations and one inequation on the coefficients. Those
polynomials in the coefficients are called discriminants for multiplicities.
The maximum degree of the discriminants grows exponentially in the degree of
$F$. Furthermore, each discriminant is a “nested” determinant, that is, it is
a determinant of a matrix whose entries are again determinants and so on.

In [7], the authors developed a new type of multiplicity discriminants to
distinguish different multiplicities when the number of distinct roots is
fixed. The main idea is to convert the multiplicity condition expressed as a
permanent inequation in roots into a sum of determinants in coefficients. In
order to generate conditions for all the possible multiplicity structures of a
univariate polynomial, one may first use subdiscriminants in classical
resultant theory to decide the number of distinct complex roots and then add
one more inequation to discriminate different multiplicity structures with the
same number of distinct roots. In the new condition, the maximum degree of the
discriminants grows linearly in the degree of $F$, which makes the size of
discriminants significantly smaller. However, the form of resulting
discriminants is a sum of many determinants, which makes the further analysis
(reasoning) difficult.

The main contribution in this paper is to provide a new type of discriminants,
which are _non-nested_ determinants and whose maximum degrees are _smaller_
than those in the previous methods. The method is based on a significantly
different theory and techniques from the previous methods (which are
essentially based on repeated parametric gcd or subdiscriminant theory). The
new condition is given by a newly devised multiplicity discriminant in
coefficients for every potential multiplicity vector of a given degree, which
can be viewed as a generalization of subdiscriminant theory to higher order
derivatives. To build up the connection between the new discriminants and
multiple roots, we first convert it into the ratio of two determinants in
terms of generic roots (without considering the multiplicities). Then by
making use of the connection between divided difference with multiple nodes
and the derivatives of higher orders at the nodes, we integrate the
multiplicity information into the expression and convert it into an expression
in terms of multiple roots. After careful manipulation, it is shown that the
new discriminant can capture the multiplicity information.

The paper is structured as follows. In Section 2, we first present the problem
to be solved in a formal way. In Section 3, we give a precise statement of the
main result of the paper (Theorem 3.5). Then a proof of Theorem 3.5 is
provided in Section 4. The proof is long thus we divide the proof into three
subsections which are interesting on their own. In Section 5, we compare the
form and size of polynomials in the multiplicity-discriminant condition in
Theorem 3.5 and those given by previous works.

##  2 Problem

######  Definition 2.1 (Multiplicity vector).

Let $F\in\mathbb{C}\left[x\right]$ with $m$ distinct complex roots, say
$r_{1},\ldots,r_{m}$, with multiplicities $\mu_{1},\ldots,\mu_{m}$
respectively. Without losing generality, we assume that
$\mu_{1}\geq\cdots\geq\mu_{m}>0$. Then the _multiplicity_ vector of $F$,
written as $\operatorname*{mult}\left(F\right)$, is defined by

$\operatorname*{mult}\left(F\right)=\left(\mu_{1},\ldots,\mu_{m}\right).$

######  Example 2.2.

Let $F=x^{5}-5x^{4}+7x^{3}+x^{2}-8x+4$. Then
$\operatorname*{mult}\left(F\right)=\left(2,2,1\right)$, since it can be
verified that $F$
$=\left(x-1\right)^{2}\left(x+1\right)^{1}\left(x-2\right)^{2}$. Note that the
multiplicity vector is a partition of $5,$ which is the degree of $F$.

######  Definition 2.3 (Potential multiplicity vectors).

Let $n$ be a positive integer. Let $\mathcal{M}(n)$ stand for the set of all
the potential multiplicity vectors of polynomials of degree $n$, equivalently,
the set of all partitions of $n,$ that is,

$\mathcal{M}(n)=\left\\{(\mu_{1},\ldots,\mu_{m}):\,\mu_{1}+\cdots+\mu_{m}=n,\mu_{1}\geq\cdots\geq\mu_{m}>0\right\\}.$

######  Example 2.4.

$\mathcal{M}\left(5\right)=\left\\{\ \left(1,1,1,1,1\right),\ \
\left(2,1,1,1\right),\ \left(2,2,1\right),\ \left(3,1,1\right),\
\left(3,2\right),\ \left(4,1\right),\ \left(5\right)\ \right\\}$.

######  Problem 2.5 (Parametric multiplicity problem).

The parametric multiplicity problem is stated as:

  1. In :

$n$, a positive integer standing for the polynomial of degree $n$ with
parametric coefficients $a$, that is,

$F=\sum_{i=0}^{n}a_{i}x^{i}\ \ \text{where}\ \ a_{n}\neq 0.$

  2. Out:

For each $\bm{\mu}\in\mathcal{M}(n)$, find a condition $C_{\bm{\mu}}\ $on $a\
$such that $\operatorname*{mult}\left(F\right)=\bm{\mu}$.

##  3 Main Result

######  Definition 3.1 (Determinant polynomial).

Consider a vector of univariate polynomials

$P=\left[\begin{array}[c]{c}P_{0}\\\ \vdots\\\
P_{k}\end{array}\right]\in\mathbb{C}[x]^{k+1}$

where $\deg P_{i}\leq k\ \ \text{and}\ \ P_{i}=\sum_{0\leq j\leq
k}a_{ij}x^{j}.$ The _coefficient matrix_ of $P,$ written as $C\left(P\right),$
is defined by

$C\left(P\right)=\operatorname*{coef}\left(P\right)=\begin{bmatrix}\operatorname*{coef}\left(P_{0}\right)\\\
\vdots\\\
\operatorname*{coef}\left(P_{k}\right)\end{bmatrix}=\begin{bmatrix}a_{0k}&\cdots&a_{00}\\\
\vdots&&\vdots\\\ a_{kk}&\cdots&a_{k0}\end{bmatrix}.$

The _determinant polynomial_ of $P,$ written as
$\operatorname*{dp}\left(P\right),$ is defined by

$\operatorname*{dp}\left(P\right)=|C\left(P\right)|.$

######  Definition 3.2 (Multiplicity Discriminant).

Let $F=\sum_{i=0}^{n}a_{i}x^{i}\ \ $where$\ \ a_{n}\neq 0$. Let
$\bm{\gamma}=\left(\gamma_{1},\ldots,\gamma_{s}\right)\in\mathcal{M}\left(n\right)$.
The the $\bm{\gamma}$-_discriminant_ of $F$, written as
$D\left(\bm{\gamma}\right),\ $is defined by

$D\left(\bm{\gamma}\right)=\frac{1}{a_{n}}\operatorname*{dp}\left[\begin{array}[c]{l}F^{(0)}x^{\gamma_{0}-1}\\\
~{}~{}~{}~{}\ \vdots\\\ F^{(0)}x^{0}\\\ \hline\cr F^{(1)}x^{\gamma_{1}-1}\\\
~{}~{}~{}~{}\ \vdots\\\ F^{(1)}x^{0}\\\ \hline\cr~{}~{}~{}~{}\ \vdots\\\
\hline\cr F^{(s)}x^{\gamma_{s}-1}\\\ ~{}~{}~{}~{}\ \vdots\\\
F^{(s)}x^{0}\end{array}\right]$

where $\gamma_{0}$ is the smallest so that the above matrix is square and
$F^{(i)}$ is the $i$-th derivative of $F$ in terms of $x$. It is
straightforward to show that $\gamma_{0}=\gamma_{1}-1$.

######  Example 3.3.

Let $n=5$ and $F=\sum_{i=0}^{n}a_{i}x^{i}$ and $a_{n}\neq 0$. Then

$\begin{array}[c]{lll}D\left(5\right)&=&\operatorname*{dp}\left[\begin{array}[c]{l}F^{(0)}x^{3}\\\
F^{(0)}x^{2}\\\ F^{(0)}x^{1}\\\ F^{(0)}x^{0}\\\ \hline\cr F^{(1)}x^{4}\\\
F^{(1)}x^{3}\\\ F^{(1)}x^{2}\\\ F^{(1)}x^{1}\\\
F^{(1)}x^{0}\end{array}\right]=\dfrac{1}{a_{5}}\left|\begin{array}[c]{ccccccccc}a_{5}&a_{4}&a_{3}&a_{2}&a_{1}&a_{0}&&&\\\
&a_{5}&a_{4}&a_{3}&a_{2}&a_{1}&a_{0}&&\\\
&&a_{5}&a_{4}&a_{3}&a_{2}&a_{1}&a_{0}&\\\
&&&a_{5}&a_{4}&a_{3}&a_{2}&a_{1}&a_{0}\\\ \hline\cr
5a_{5}&4a_{4}&3a_{3}&2a_{2}&1a_{1}&&&&\\\
&5a_{5}&4a_{4}&3a_{3}&2a_{2}&1a_{1}&&&\\\
&&5a_{5}&4a_{4}&3a_{3}&2a_{2}&1a_{1}&&\\\
&&&5a_{5}&4a_{4}&3a_{3}&2a_{2}&1a_{1}&\\\
&&&&5a_{5}&4a_{4}&3a_{3}&2a_{2}&1a_{1}\end{array}\right|,\\\
D\left(4,1\right)&=&\operatorname*{dp}\left[\begin{array}[c]{l}F^{(0)}x^{2}\\\
F^{\left(0\right)}x^{1}\\\ F^{(0)}x^{0}\\\ \hline\cr F^{(1)}x^{3}\\\
F^{(1)}x^{2}\\\ F^{(1)}x^{1}\\\ F^{(1)}x^{0}\\\ \hline\cr
F^{(2)}x^{0}\end{array}\right]=\dfrac{1}{a_{5}}\left|\begin{array}[c]{cccccccc}a_{5}&a_{4}&a_{3}&a_{2}&a_{1}&a_{0}&&\\\
&a_{5}&a_{4}&a_{3}&a_{2}&a_{1}&a_{0}&\\\
&&a_{5}&a_{4}&a_{3}&a_{2}&a_{1}&a_{0}\\\ \hline\cr
5a_{5}&4a_{4}&3a_{3}&2a_{2}&1a_{1}&&&\\\
&5a_{5}&4a_{4}&3a_{3}&2a_{2}&1a_{1}&&\\\
&&5a_{5}&4a_{4}&3a_{3}&2a_{2}&1a_{1}&\\\
&&&5a_{5}&4a_{4}&3a_{3}&2a_{2}&1a_{1}\\\ \hline\cr&&&&5\cdot 4a_{5}&4\cdot
3a_{4}&3\cdot 2a_{3}&2\cdot 1a_{2}\end{array}\right|,\\\
D\left(3,2\right)&=&\operatorname*{dp}\left[\begin{array}[c]{l}F^{(0)}x^{1}\\\
F^{(0)}x^{0}\\\ \hline\cr F^{(1)}x^{2}\\\ F^{(1)}x^{1}\\\ F^{(1)}x^{0}\\\
\hline\cr F^{(2)}x^{1}\\\
F^{(2)}x^{0}\end{array}\right]=\dfrac{1}{a_{5}}\left|\begin{array}[c]{ccccccc}a_{5}&a_{4}&a_{3}&a_{2}&a_{1}&a_{0}&\\\
&a_{5}&a_{4}&a_{3}&a_{2}&a_{1}&a_{0}\\\ \hline\cr
5a_{5}&4a_{4}&3a_{3}&2a_{2}&1a_{1}&&\\\
&5a_{5}&4a_{4}&3a_{3}&2a_{2}&1a_{1}&\\\
&&5a_{5}&4a_{4}&3a_{3}&2a_{2}&1a_{1}\\\ \hline\cr&&5\cdot 4a_{5}&4\cdot
3a_{4}&3\cdot 2a_{3}&2\cdot 1a_{2}&\\\ &&&5\cdot 4a_{5}&4\cdot 3a_{4}&3\cdot
2a_{3}&2\cdot 1a_{2}\end{array}\right|,\end{array}$
$\begin{array}[c]{lll}D\left(3,1,1\right)&=&\operatorname*{dp}\left[\begin{array}[c]{l}F^{(0)}x^{1}\\\
F^{(0)}x^{0}\\\ \hline\cr F^{(1)}x^{2}\\\ F^{(1)}x^{1}\\\ F^{(1)}x^{0}\\\
\hline\cr F^{(2)}x^{0}\\\ \hline\cr
F^{(3)}x^{0}\end{array}\right]=\dfrac{1}{a_{5}}\left|\begin{array}[c]{ccccccc}a_{5}&a_{4}&a_{3}&a_{2}&a_{1}&a_{0}&\\\
&a_{5}&a_{4}&a_{3}&a_{2}&a_{1}&a_{0}\\\ \hline\cr
5a_{5}&4a_{4}&3a_{3}&2a_{2}&1a_{1}&&\\\
&5a_{5}&4a_{4}&3a_{3}&2a_{2}&1a_{1}&\\\
&&5a_{5}&4a_{4}&3a_{3}&2a_{2}&1a_{1}\\\ \hline\cr&&&5\cdot 4a_{5}&4\cdot
3a_{4}&3\cdot 2a_{3}&2\cdot 1a_{2}\\\ \hline\cr&&&&5\cdot 4\cdot 3a_{5}&4\cdot
3\cdot 2a_{4}&3\cdot 2\cdot 1a_{3}\end{array}\right|,\\\
D\left(2,2,1\right)&=&\operatorname*{dp}\left[\begin{array}[c]{l}F^{(0)}x^{0}\\\
\hline\cr F^{(1)}x^{1}\\\ F^{(1)}x^{0}\\\ \hline\cr F^{(2)}x^{1}\\\
F^{(2)}x^{0}\\\ \hline\cr
F^{(3)}x^{0}\end{array}\right]=\dfrac{1}{a_{5}}\left|\begin{array}[c]{cccccc}a_{5}&a_{4}&a_{3}&a_{2}&a_{1}&a_{0}\\\
\hline\cr 5a_{5}&4a_{4}&3a_{3}&2a_{2}&1a_{1}&\\\
&5a_{5}&4a_{4}&3a_{3}&2a_{2}&1a_{1}\\\ \hline\cr&5\cdot 4a_{5}&4\cdot
3a_{4}&3\cdot 2a_{3}&2\cdot 1a_{2}&\\\ &&5\cdot 4a_{5}&4\cdot 3a_{4}&3\cdot
2a_{3}&2\cdot 1a_{2}\\\ \hline\cr&&&5\cdot 4\cdot 3a_{5}&4\cdot 3\cdot
2a_{4}&3\cdot 2\cdot 1a_{3}\end{array}\right|,\\\
D\left(2,1,1,1\right)&=&\operatorname*{dp}\left[\begin{array}[c]{l}F^{(0)}x^{0}\\\
\hline\cr F^{(1)}x^{1}\\\ F^{(1)}x^{0}\\\ \hline\cr F^{(2)}x^{0}\\\ \hline\cr
F^{(3)}x^{0}\\\ \hline\cr
F^{(4)}x^{0}\end{array}\right]=\dfrac{1}{a_{5}}\left|\begin{array}[c]{cccccc}a_{5}&a_{4}&a_{3}&a_{2}&a_{1}&a_{0}\\\
\hline\cr 5a_{5}&4a_{4}&3a_{3}&2a_{2}&1a_{1}&\\\
&5a_{5}&4a_{4}&3a_{3}&2a_{2}&1a_{1}\\\ \hline\cr&&5\cdot 4a_{5}&4\cdot
3a_{4}&3\cdot 2a_{3}&2\cdot 1a_{2}\\\ \hline\cr&&&5\cdot 4\cdot 3a_{5}&4\cdot
3\cdot 2a_{4}&3\cdot 2\cdot 1a_{3}\\\ \hline\cr&&&&5\cdot 4\cdot 3\cdot
2a_{5}&4\cdot 3\cdot 2\cdot 1a_{4}\end{array}\right|,\\\
D\left(1,1,1,1,1\right)&=&\operatorname*{dp}\left[\begin{array}[c]{l}F^{(1)}x^{0}\\\
\hline\cr F^{(2)}x^{0}\\\ \hline\cr F^{(3)}x^{0}\\\ \hline\cr F^{(4)}x^{0}\\\
\hline\cr
F^{\left(5\right)}x^{0}\end{array}\right]=\dfrac{1}{a_{5}}\left|\begin{array}[c]{cccccc}{5a_{5}}&{4a_{4}}&{3a_{3}}&{2a_{2}}&{1a_{1}}\\\
\hline\cr&5\cdot 4a_{5}&4\cdot 3a_{4}&3\cdot 2a_{3}&2\cdot 1a_{2}\\\
\hline\cr&&5\cdot 4\cdot 3a_{5}&4\cdot 3\cdot 2a_{4}&3\cdot 2\cdot 1a_{3}\\\
\hline\cr&&&5\cdot 4\cdot 3\cdot 2a_{5}&4\cdot 3\cdot 2\cdot 1a_{4}\\\
\hline\cr&&&&5\cdot 4\cdot 3\cdot 2\cdot 1a_{5}\end{array}\right|.\end{array}$

Note that the last one
$D\left(1,1,1,1,1\right)=5^{5}4^{4}3^{3}2^{2}1^{1}a_{5}^{4}$. Since $a_{5}\neq
0$, we see that $D\left(1,1,1,1,1\right)\neq 0$.

To present the main theorem, we recall the following definition for the
conjugate of $\bm{\mu}\in\mathcal{M}(n)$.

######  Definition 3.4 (Conjugate).

Let $\bm{\mu}=\left(\mu_{1},\ldots,\mu_{m}\right)\in\mathcal{M}(n)$. Then the
_conjugate_
$\overline{\bm{\mu}}=\left(\overline{\mu}_{1},\ldots,\overline{\mu}_{s}\right)$
of $\bm{\mu}$ is defined by

$\displaystyle s$ $\displaystyle=\max_{1\leq i\leq m}\mu_{i}=\mu_{1},$
$\displaystyle\overline{\mu}_{i}$
$\displaystyle=\\#\left\\{\mu_{j}:\mu_{j}\geq i\right\\}\ \ \ \text{for
}i=1,\ldots,s.$

######  Theorem 3.5 (Main Result).

Let $F=\sum_{i=0}^{n}a_{i}x^{i}\ \ $where$\ \ a_{n}\neq 0$. Let
$\mathcal{M}(n)=\left\\{\bm{\mu}_{0},\bm{\mu}_{1},\ldots,\bm{\mu}_{p}\right\\}$
where the entries are ordered in the lexicographically decreasing order in
their conjugates $\overline{\bm{\mu}_{i}}$’s. Then we have the following
conditions for the multiplicity vectors.

$\operatorname*{mult}(F)=\left\\{\begin{array}[c]{llll}{\bm{\mu}_{0}},&\text{if
}&{D}\left({\overline{\bm{\mu}_{0}}}\right)&\neq 0;\\\ \ \vdots&\vdots&\
{\vdots}&\\\ {\bm{\mu}_{p-1},}&\text{else\
if}&{D}\left({\overline{\bm{\mu}_{p-1}}}\right)&\neq 0;\\\
{\bm{\mu}_{p},}&\text{else\ if}&{D}\left({\overline{\bm{\mu}_{p}}}\right)&\neq
0.\end{array}\right.$

Equivalently,

$\operatorname*{mult}(F)={\bm{\mu}_{i}}\ \ \ \Longleftrightarrow\ \ \
D\left({\overline{\bm{\mu}_{0}}}\right)=\cdots=D\left({\overline{\bm{\mu}_{i-1}}}\right)=0\wedge
D\left({\overline{\bm{\mu}_{i}}}\right)\neq 0.$

######  Example 3.6.

We have the following condition for each multiplicity vector for degree $5$.

$\operatorname*{mult}(F)=\left\\{\begin{array}[c]{llll}\left(1,1,1,1,1\right),&\text{if
}&D\left(5\right)&\neq 0;\\\ (2,1,1,1),&\text{else\
if}&{D}\left(4,1\right)&\neq 0;\\\ (2,2,1),&\text{else\
if}&{D}\left(3,2\right)&\neq 0;\\\ (3,1,1),&\text{else\
if}&{D}\left(3,1,1\right)&\neq 0;\\\ (3,2),&\text{else\
if}&{D}\left(2,2,1\right)&\neq 0;\\\ \left(4,1\right),&\text{else\
if}&{D}\left(2,1,1,1\right)&\neq 0;\\\ \left(5\right),&\text{else\
if}&{D}\left(1,1,1,1,1\right)&\neq 0.\end{array}\right.$

Equivalently, for instance,

$\operatorname*{mult}(F)=\left(2,2,1\right)\ \ \ \Longleftrightarrow\ \ \
D\left(5\right)={D}\left(4,1\right)=0\wedge D\left(3,2\right)\neq 0.$

######  Remark 3.7.

Note that ${\overline{\bm{\mu}_{p}}}=(1,\ldots,1)$ and

${D}\left({\overline{\bm{\mu}_{p}}}\right)=\frac{1}{a_{n}}\left|\begin{array}[c]{cccc}na_{n}&\cdots&\cdots&1a_{1}\\\
&n\left(n-1\right)a_{n}&\cdots&2\cdot 1a_{2}\\\ &&\ddots&\vdots{}{}{}\\\
&&&n\left(n-1\right)\cdots 1a_{n}\end{array}\right|=\prod_{i=1}^{n}i^{i}\cdot
a_{n}^{n-1}\neq 0.$

Hence the last condition is always satisfied and there is no need to check the
condition.

##  4 Proof of the Main Theorem

Here is a high level view of the proof. We start with converting
${D}\left(\bm{\mu}\right)$ into the equivalent symmetric polynomials in
generic roots (though displayed as a ratio of two determinants) which is
easier to embed the multiplicity information. Then by making use of the
connection between divided difference with multiple nodes and the derivatives
of higher orders at the nodes, we convert the expression in generic roots to
that in distinct roots with multiplicity information integrated. The theorem
will be proved by eliminating the entries in the determinantal expression
obtained from the second stage which may vanish under the given multiplicity
structure.

###  4.1 Multiplicity discriminant in terms of roots

We first understand what the multiplicity discriminants look like in terms of
roots. .

######  Notation 4.1.

$V(\alpha_{1},\ldots,\alpha_{n}):=\left|\begin{array}[c]{ccc}\alpha_{1}^{n-1}&\cdots&\alpha_{n}^{n-1}\\\
\vdots&&\vdots\\\ \alpha_{1}^{0}&\cdots&\alpha_{n}^{0}\end{array}\right|$.

######  Lemma 4.2 (Multiplicity discriminant in generic roots).

Let $F=a_{n}(x-\alpha_{1})\cdots(x-\alpha_{n})$ and
$\bm{\gamma}=(\gamma_{1},\ldots,\gamma_{s})\in\mathcal{M}(n)$. Then

$D(\bm{\gamma})=\dfrac{a_{n}^{\gamma_{1}-2}\cdot\left|\begin{array}[c]{lcl}F^{(1)}(\alpha_{1})\alpha_{1}^{\gamma_{1}-1}&\cdots&F^{(1)}(\alpha_{n})\alpha_{n}^{\gamma_{1}-1}\\\
~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}\vdots\\\
F^{(1)}(\alpha_{1})\alpha_{1}^{0}&\cdots&F^{(1)}(\alpha_{n})\alpha_{n}^{0}\\\
\hline\cr~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}\vdots\\\ \hline\cr
F^{(s)}(\alpha_{1})\alpha_{1}^{\gamma_{s}-1}&\cdots&F^{(s)}(\alpha_{n})\alpha_{n}^{\gamma_{s}-1}\\\
~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}\vdots\\\
F^{(s)}(\alpha_{1})\alpha_{1}^{0}&\cdots&F^{(s)}(\alpha_{n})\alpha_{n}^{0}\end{array}\right|}{V(\alpha_{1},\ldots,\alpha_{n})}.$
(1)

######  Proof 4.3.

  1. 1.

Since $\gamma_{1}\geq\cdots\geq\gamma_{s}$ and $\gamma_{0}=\gamma_{1}-1$, we
have

$\deg(F^{(0)}x^{n-2})>\cdots>\deg(F^{(0)}x^{\gamma_{1}-1})>\max({\deg(F^{(0)}x^{\gamma_{0}-1}),\deg(F^{(1)}x^{\gamma_{1}-1}),\ldots,\deg(F^{(s)}x^{\gamma_{s}-1})}).$

Thus

$D(\bm{\gamma})=\frac{1}{a_{n}}\operatorname*{dp}\left[\begin{array}[c]{l}F^{(0)}x^{\gamma_{1}-2}\\\
~{}~{}~{}~{}\vdots\\\ F^{(0)}x^{0}\\\ \hline\cr F^{(1)}x^{\gamma_{1}-1}\\\
~{}~{}~{}~{}\vdots\\\ F^{(1)}x^{0}\\\ \hline\cr~{}~{}~{}~{}\vdots\\\ \hline\cr
F^{(s)}x^{\gamma_{s}-1}\\\ ~{}~{}~{}~{}\vdots\\\
F^{(s)}x^{0}\end{array}\right]=\frac{1}{a_{n}}\cdot
a_{n}^{\gamma_{1}-n}\operatorname*{dp}\left[\begin{array}[c]{l}F^{(0)}x^{n-2}\\\
~{}~{}~{}~{}\vdots\\\ F^{(0)}x^{\gamma_{1}-1}\\\ F^{(0)}x^{\gamma_{1}-2}\\\
~{}~{}~{}~{}\vdots\\\ F^{(0)}x^{0}\\\ \hline\cr F^{(1)}x^{\gamma_{1}-1}\\\
~{}~{}~{}~{}\vdots\\\ F^{(1)}x^{0}\\\ \hline\cr~{}~{}~{}~{}\vdots\\\ \hline\cr
F^{(s)}x^{\gamma_{s}-1}\\\ ~{}~{}~{}~{}\vdots\\\
F^{(s)}x^{0}\end{array}\right]=a_{n}^{\gamma_{1}-n-1}\operatorname*{dp}\left[\begin{array}[c]{l}F^{(0)}x^{n-2}\\\
~{}~{}~{}~{}\vdots\\\ F^{(0)}x^{0}\\\ \hline\cr F^{(1)}x^{\gamma_{1}-1}\\\
~{}~{}~{}~{}\vdots\\\ F^{(1)}x^{0}\\\ \hline\cr~{}~{}~{}~{}\vdots\\\ \hline\cr
F^{(s)}x^{\gamma_{s}-1}\\\ ~{}~{}~{}~{}\vdots\\\
F^{(s)}x^{0}\end{array}\right].$

  2. 2.

Now we recall the following result from [7] which is the key for proving the
lemma. Let $G_{1},\ldots,G_{n}\in\mathbb{C}\left[x\right]_{2n-2}$ where
$\mathbb{C}\left[x\right]_{2n-2}$ consists of all the polynomials in $x$ with
degree no greater than $2n-2$. Then

$\operatorname*{dp}\left[\begin{array}[c]{l}F^{(0)}x^{n-2}\\\
~{}~{}~{}~{}\vdots\\\ F^{(0)}x^{0}\\\ ~{}~{}~{}~{}G_{1}\\\
~{}~{}~{}~{}\vdots\\\
~{}~{}~{}~{}G_{n}\end{array}\right]=\dfrac{a_{n}^{n-1}\cdot\left|\begin{array}[c]{ccc}G_{1}(\alpha_{1})&\cdots&G_{1}(\alpha_{n})\\\
\vdots&&\vdots\\\
G_{n}(\alpha_{1})&\cdots&G_{n}\left(\alpha_{n}\right)\end{array}\right|}{V(\alpha_{1},\ldots,\alpha_{n})}.$
(2)

  3. 3.

After specializing $G_{1},\ldots,G_{n}$ in (2) with
$F^{(1)}x^{\gamma_{1}-1},\ldots,F^{(1)}x^{0},\ldots,F^{(s)}x^{\gamma_{s}-1},\ldots,F^{(s)}x^{0}$,
respectively, we have

$D(\bm{\gamma})=a_{n}^{\gamma_{1}-n-1}\cdot\dfrac{a_{n}^{n-1}\cdot\left|\begin{array}[c]{lcl}\left(F^{(1)}x^{\gamma_{1}-1}\right)(\alpha_{1})&\cdots&\left(F^{(1)}x^{\gamma_{1}-1}\right)(\alpha_{n})\\\
~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}\vdots\\\
\left(F^{(1)}x^{0}\right)(\alpha_{1})&\cdots&\left(F^{(1)}x^{0}\right)(\alpha_{n})\\\
\hline\cr~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}\vdots\\\
\hline\cr\left(F^{(s)}x^{\gamma_{s}-1}\right)(\alpha_{1})&\cdots&\left(F^{(s)}x^{\gamma_{s}-1}\right)(\alpha_{n})\\\
~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}\vdots\\\
\left(F^{(s)}x^{0}\right)(\alpha_{1})&\cdots&\left(F^{(s)}x^{0}\right)(\alpha_{n})\end{array}\right|}{V(\alpha_{1},\ldots,\alpha_{n})}$

which can be easily simplified into (1).

######  Remark 4.4.

It is very important to note that the right hand side is a polynomial function
in $\alpha_{1},\ldots,\alpha_{n}$, even though written as a rational function,
since the numerator is exactly divisible by the denominator. Hence the above
definition should be read as follows:

  1. 1.

Treating $\alpha_{1},\ldots,\alpha_{n}$ as distinct indeterminates, carry out
the exact division obtaining a polynomial.

  2. 2.

Treating $\alpha_{1},\ldots,\alpha_{n}$ as numbers, evaluate the resulting
polynomial.

######  Lemma 4.5 (Multiplicity discriminant in multiple roots).

Let $F$ be of degree $n$ with $m$ distinct roots $r_{1},\ldots,r_{m}$, of
multiplicities $\mu_{1},\ldots,\mu_{m}$, that is $\mu_{1}+\cdots+\mu_{m}=n$.
Let $\bm{\gamma}=\left(\gamma_{1},\ldots,\gamma_{s}\right)\in\Gamma(n)$. Then
we have

$D(\bm{\gamma})=\frac{c\cdot\left|\begin{array}[c]{lcl|c|lcl}({F^{(1)}x^{\gamma_{1}-1}})^{(0)}(r_{1})&\cdots&({F^{(1)}x^{\gamma_{1}-1}})^{(\mu_{1}-1)}(r_{1})&\cdots\cdots&({F^{(1)}x^{\gamma_{1}-1}})^{(0)}(r_{m})&\cdots&({F^{(1)}x^{\gamma_{1}-1}})^{(\mu_{m}-1)}(r_{m})\\\
~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots\\\
({F^{(1)}x^{0}})^{(0)}(r_{1})&\cdots&({F^{(1)}x^{0}})^{(\mu_{1}-1)}(r_{1})&\cdots\cdots&({F^{(1)}x^{0}})^{(0)}(r_{m})&\cdots&({F^{(1)}x^{0}})^{(\mu_{m}-1)}(r_{m})\\\
\hline\cr~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots\\\
\hline\cr({F^{(s)}}x^{\gamma_{s}-1})^{(0)}(r_{1})&\cdots&({F}^{(s)}x^{\gamma_{s}-1})^{(\mu_{1}-1)}(r_{1})&\cdots\cdots&({F^{(s)}}x^{\gamma_{s}-1})^{(0)}(r_{m})&\cdots&({F}^{(s)}x^{\gamma_{s}-1})^{(\mu_{m}-1)}(r_{m})\\\
~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots\\\
({F^{(s)}}x^{0})^{(0)}(r_{1})&\cdots&({F^{(s)}}x^{0})^{(\mu_{1}-1)}(r_{1})&\cdots\cdots&({F^{(s)}}x^{0})^{(0)}(r_{m})&\cdots&({F^{(s)}}x^{0})^{(\mu_{m}-1)}(r_{m})\end{array}\right|}{\prod\limits_{1\leq
i<j\leq m}(r_{i}-r_{j})^{\mu_{i}\mu_{j}}}$ (3)

where $c=\pm
1\Big{/}\left(\prod_{i=1}^{m}\prod_{j=0}^{\mu_{i}-1}j!\right)\cdot
a_{n}^{\gamma_{1}-2}$.

######  Proof 4.6.

  1. 1.

Let $F=a_{n}(x-\alpha_{1})\cdots(x-\alpha_{n})$. When
$\alpha_{1},\ldots,\alpha_{n}$ are treated as numbers, without loss of
generality, we may assume that $\alpha_{1},\ldots,\alpha_{n}$ are grouped into
$m$ sets as follows:

$\begin{array}[c]{llccccc}\mathcal{S}_{1}:=&\\{\alpha_{1}&\cdots&\cdots&\cdots&\cdots&\alpha_{\mu_{1}}\\},\\\
\mathcal{S}_{2}:=&\\{\alpha_{\mu_{1}+1}&\cdots&\cdots&\cdots&\alpha_{\mu_{1}+\mu_{2}}\\},&\\\
\qquad\vdots&&&&&&\\\
\mathcal{S}_{m}:=&\\{\alpha_{\mu_{1}+\cdots+\mu_{m-1}+1}&\cdots&\alpha_{\mu_{1}+\cdots+\mu_{m-1}+\mu_{m}}\\}.&&&\end{array}$

where elements in $\mathcal{S}_{i}$ are all equal to $r_{i}$.

  2. 2.

Recall that

${D}(\bm{\gamma})=a_{n}^{\gamma_{1}-2}\cdot{\left|\begin{array}[c]{lcl}({F}^{(1)}x^{\gamma_{1}-1})(\alpha_{1})&\cdots&({F}^{(1)}x^{\gamma_{1}-1})(\alpha_{n})\\\
~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}\vdots\\\
({F}^{(1)}x^{0})(\alpha_{1})&\cdots&({F}^{(1)}x^{0})(\alpha_{n})\\\
\hline\cr~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}\vdots\\\
~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}\vdots\\\
\hline\cr({F}^{(s)}x^{\gamma_{s}-1})(\alpha_{1})&\cdots&({F}^{(s)}x^{\gamma_{s}-1})(\alpha_{n})\\\
~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}\vdots\\\
({F}^{(s)}x^{0})(\alpha_{1})&\cdots&({F}^{(s)}x^{0})(\alpha_{n})\end{array}\right|\big{/}V(\alpha_{1},\ldots,\alpha_{n}).}$

Next we will treat $\alpha_{1},\ldots,\alpha_{n}$ as indeterminates and carry
out the exact division so that difference between the collapsed $\alpha_{i}$’s
do not appear in the denominator.

  3. 3.

For the sake of simplicity, we use the follow shorthand notion:

$\bm{F}:=\left[F^{(1)}x^{\gamma_{1}-1},\ldots,F^{(1)}x^{0},\ldots,F^{(s)}x^{\gamma_{s}-1},\ldots,F^{(s)}x^{0}\right]^{T}.$

  4. 4.

Let $P[x_{1},\ldots,x_{i}]$ denote the $(i-1)$th divided difference of
$P\in\mathbb{C}[x]$ at $x_{1},\ldots,x_{i}$ defined recursively as follows:

$P[x_{1},\ldots,x_{i}]=\left\\{\begin{array}[]{ll}P(x_{1}),&\text{if}\
i=1;\\\\[3.0pt]
\dfrac{P[x_{1},\ldots,x_{i-2},x_{i}]-P[x_{1},\ldots,x_{i-2},x_{i-1}]}{x_{i}-x_{i-1}},&\text{if}\
i>1.\end{array}\right.$

Let

$\displaystyle\bm{F}[\alpha_{1},\ldots,\alpha_{i}]:=$
$\displaystyle\left[(F^{(1)}x^{\gamma_{1}-1})[\alpha_{1},\ldots,\alpha_{i}],\ldots,(F^{(1)}x^{0})[\alpha_{1},\ldots,\alpha_{i}],\right.$
$\displaystyle\left.\quad\ldots\ldots,(F^{(s)}x^{\gamma_{s}-1})[\alpha_{1},\ldots,\alpha_{i}],\ldots,(F^{(s)}x^{0})[\alpha_{1},\ldots,\alpha_{i}]\right]^{T}.$

  5. 5.

It follows that

$\displaystyle D(\bm{\gamma})$
$\displaystyle=a_{n}^{\gamma_{1}-2}\cdot\frac{\left|\begin{array}[c]{ccc}\bm{F}(\alpha_{1})&\cdots&\bm{F}(\alpha_{n})\end{array}\right|}{V(\alpha_{1},\ldots,\alpha_{n})}$
$\displaystyle=a_{n}^{\gamma_{1}-2}\cdot\frac{\left|\begin{array}[c]{cccccc}\bm{F}[\alpha_{1}]&\cdots&\bm{F}[\alpha_{\mu_{1}}]&\bm{F}[\alpha_{\mu_{1}+1}]&\cdots&\bm{F}[\alpha_{n}]\end{array}\right|}{\prod\limits_{\begin{subarray}{c}\alpha_{i},\alpha_{j}\in\mathcal{S}_{1}\\\
j-i>0\end{subarray}}(\alpha_{i}-\alpha_{j})\prod\limits_{\begin{subarray}{c}\alpha_{i},\alpha_{j}\notin\mathcal{S}_{1}\\\
j-i>0\end{subarray}}(\alpha_{i}-\alpha_{j})\prod\limits_{\begin{subarray}{c}\alpha_{i}\in\mathcal{S}_{1}\\\
\alpha_{j}\notin\mathcal{S}_{1}\end{subarray}}(\alpha_{i}-\alpha_{j})}$
$\displaystyle=\pm
a_{n}^{\gamma_{1}-2}\cdot\frac{\left|\begin{array}[c]{ccccccc}\bm{F}[\alpha_{1}]&\bm{F}[\alpha_{1},\alpha_{2}]&\cdots&\bm{F}[\alpha_{\mu_{1}-1},\alpha_{\mu_{1}}]&\bm{F}[\alpha_{\mu_{1}+1}]&\cdots&\bm{F}[\alpha_{n}]\end{array}\right|}{\prod\limits_{\begin{subarray}{c}\alpha_{i},\alpha_{j}\in\mathcal{S}_{1}\\\
j-i>1\end{subarray}}(\alpha_{i}-\alpha_{j})\prod\limits_{\begin{subarray}{c}\alpha_{i},\alpha_{j}\notin\mathcal{S}_{1}\\\
j-i>0\end{subarray}}(\alpha_{i}-\alpha_{j})\prod\limits_{\begin{subarray}{c}\alpha_{i}\in\mathcal{S}_{1}\\\
\alpha_{j}\notin\mathcal{S}_{1}\end{subarray}}(\alpha_{i}-\alpha_{j})}$
$\displaystyle=\pm
a_{n}^{\gamma_{1}-2}\cdot\frac{\left|\begin{array}[c]{cccccccc}\bm{F}[\alpha_{1}]&\bm{F}[\alpha_{1},\alpha_{2}]&\bm{F}[\alpha_{1},\alpha_{2},\alpha_{3}]&\cdots&\bm{F}[\alpha_{\mu_{1}-2},\alpha_{\mu_{1}-1},\alpha_{\mu_{1}}]&\bm{F}[\alpha_{\mu_{1}+1}]&\cdots&\bm{F}[\alpha_{n}]\end{array}\right|}{\prod\limits_{\begin{subarray}{c}\alpha_{i},\alpha_{j}\in\mathcal{S}_{1}\\\
j-i>2\end{subarray}}(\alpha_{i}-\alpha_{j})\prod\limits_{\begin{subarray}{c}\alpha_{i},\alpha_{j}\notin\mathcal{S}_{1}\\\
j-i>0\end{subarray}}(\alpha_{i}-\alpha_{j})\prod\limits_{\begin{subarray}{c}\alpha_{i}\in\mathcal{S}_{1}\\\
\alpha_{j}\notin\mathcal{S}_{1}\end{subarray}}(\alpha_{i}-\alpha_{j})}$
$\displaystyle~{}\vdots$ $\displaystyle=\pm
a_{n}^{\gamma_{1}-2}\cdot\frac{\left|\begin{array}[c]{ccccccc}\bm{F}[\alpha_{1}]&\bm{F}[\alpha_{1},\alpha_{2}]&\cdots&\bm{F}[\alpha_{1},\ldots,\alpha_{\mu_{1}}]&\bm{F}(\alpha_{\mu_{1}+1})&\cdots&\bm{F}(\alpha_{n})\end{array}\right|}{\prod\limits_{\begin{subarray}{c}\alpha_{i},\alpha_{j}\notin\mathcal{S}_{1}\\\
j-i>0\end{subarray}}(\alpha_{i}-\alpha_{j})\prod\limits_{\begin{subarray}{c}\alpha_{i}\in\mathcal{S}_{1}\\\
\alpha_{j}\notin\mathcal{S}_{1}\end{subarray}}(\alpha_{i}-\alpha_{j})}.$

  6. 6.

Repeating the procedure for $\alpha_{j}$’s in each $\mathcal{S}_{i}$ for
$i=2,\ldots,m$ successively, we get

$D(\bm{\gamma})=\pm
a_{n}^{\gamma_{1}-2}\cdot\frac{\left|\begin{array}[c]{ccc|cc|ccc}\bm{F}[\alpha_{1}]&\cdots&\bm{F}[\alpha_{1},\ldots,\alpha_{\mu_{1}}]&\cdots&\cdots&\bm{F}[\alpha_{\mu_{1}+\cdots+\mu_{m-1}+1}]&\cdots&\bm{F}[\alpha_{\mu_{1}+\cdots+\mu_{m-1}+1},\ldots,\alpha_{n}]\end{array}\right|}{\prod\limits_{1\leq
i<j\leq m}\prod\limits_{\begin{subarray}{c}\alpha_{p}\in\mathcal{S}_{i}\\\
\alpha_{q}\in\mathcal{S}_{j}\end{subarray}}(\alpha_{p}-\alpha_{q})}.$

  7. 7.

Now we substitute $\alpha_{1}=\cdots=\alpha_{\mu_{1}}=r_{1}$, …,
$\alpha_{\mu_{1}+\cdots+\mu_{m-1}+1}=\cdots=\alpha_{n}=r_{m}$ into
$D(\bm{\gamma})$ and obtain

$D(\bm{\gamma})=\pm
a_{n}^{\gamma_{1}-2}\cdot\frac{\left|\begin{array}[c]{ccc|cc|ccc}\bm{F}[r_{1}]&\cdots&\bm{F}[r_{1},\ldots,r_{1}]&\cdots&\cdots&\bm{F}[r_{m}]&\cdots&\bm{F}[r_{m},\ldots,r_{m}]\end{array}\right|}{\prod\limits_{1\leq
i<j\leq m}(r_{i}-r_{j})^{\mu_{i}\mu_{j}}}.$ (4)

  8. 8.

By [12, Equation (2.1.5.7a)], for any given polynomial $P\in\mathbb{C}[x]$,

$P[\underbrace{r_{i},\ldots,r_{i}}_{k\
r_{i}\text{'s}}]=\frac{P^{(k-1)}(r_{i})}{(k-1)!}.$

Hence

$\bm{F}[\underbrace{r_{i},\ldots,r_{i}}_{k\
r_{i}\text{'s}}]=\left[\frac{({F^{(1)}x^{\gamma_{1}-1}})^{(k-1)}(r_{i})}{(k-1)!},\ldots,\frac{({F^{(1)}x^{0}})^{(k-1)}(r_{i})}{(k-1)!},\ldots,\frac{({F}^{(s)}x^{\gamma_{s}-1})^{(k-1)}(r_{i})}{(k-1)!},\ldots,\frac{({F}^{(s)}x^{0})^{(k-1)}(r_{i})}{(k-1)!}\right]^{T}.$
(5)

  9. 9.

Substituting (5) into (4) , we have

$\displaystyle D(\bm{\gamma})$ $\displaystyle=\pm
a_{n}^{\gamma_{1}-2}\cdot\frac{\left|\begin{array}[c]{lcl|c|lcl}\frac{({F^{(1)}x^{\gamma_{1}-1}})^{(0)}(r_{1})}{0!}&\cdots&\frac{({F^{(1)}x^{\gamma_{1}-1}})^{(\mu_{1}-1)}(r_{1})}{(\mu_{1}-1)!}&\cdots\cdots&\frac{({F^{(1)}x^{\gamma_{1}-1}})^{(0)}(r_{m})}{0!}&\cdots&\frac{({F^{(1)}x^{\gamma_{1}-1}})^{(\mu_{m}-1)}(r_{m})}{(\mu_{m}-1)!}\\\
~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots\\\
\frac{({F^{(1)}x^{0}})^{(0)}(r_{1})}{0!}&\cdots&\frac{({F^{(1)}x^{0}})^{(\mu_{1}-1)}(r_{1})}{(\mu_{1}-1)!}&\cdots\cdots&\frac{({F^{(1)}x^{0}})^{(0)}(r_{m})}{0!}&\cdots&\frac{({F^{(1)}x^{0}})^{(\mu_{m}-1)}(r_{m})}{(\mu_{m}-1)!}\\\
\hline\cr~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots\\\
\hline\cr\frac{({F}^{(s)}x^{\gamma_{s}-1})^{(0)}(r_{1})}{0!}&\cdots&\frac{({F}^{(s)}x^{\gamma_{s}-1})^{(\mu_{1}-1)}(r_{1})}{(\mu_{1}-1)!}&\cdots\cdots&\frac{({F}^{(s)}x^{\gamma_{s}-1})^{(0)}(r_{m})}{0!}&\cdots&\frac{({F^{(s)}}x^{\gamma_{s}-1})^{(\mu_{m}-1)}(r_{m})}{(\mu_{m}-1)!}\\\
~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots\\\
\frac{({F}^{(s)}x^{0})^{(0)}(r_{1})}{0!}&\cdots&\frac{({F^{(s)}}x^{0})^{(\mu_{1}-1)}(r_{1})}{(\mu_{1}-1)!}&\cdots\cdots&\frac{({F^{(s)}}x^{0})^{(0)}(r_{m})}{0!}&\cdots&\frac{({F^{(s)}}x^{0})^{(\mu_{m}-1)}(r_{m})}{(\mu_{m}-1)!}\end{array}\right|}{\prod\limits_{1\leq
i<j\leq m}(r_{i}-r_{j})^{\mu_{i}\mu_{j}}}$
$\displaystyle=\frac{c\cdot\left|\begin{array}[c]{lcl|c|lcl}({F^{(1)}x^{\gamma_{1}-1}})^{(0)}(r_{1})&\cdots&({F^{(1)}x^{\gamma_{1}-1}})^{(\mu_{1}-1)}(r_{1})&\cdots\cdots&({F^{(1)}x^{\gamma_{1}-1}})^{(0)}(r_{m})&\cdots&({F^{(1)}x^{\gamma_{1}-1}})^{(\mu_{m}-1)}(r_{m})\\\
~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots\\\
({F^{(1)}x^{0}})^{(0)}(r_{1})&\cdots&({F^{(1)}x^{0}})^{(\mu_{1}-1)}(r_{1})&\cdots\cdots&({F^{(1)}x^{0}})^{(0)}(r_{m})&\cdots&({F^{(1)}x^{0}})^{(\mu_{m}-1)}(r_{m})\\\
\hline\cr~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots\\\
\hline\cr({F^{(s)}}x^{\gamma_{s}-1})^{(0)}(r_{1})&\cdots&({F}^{(s)}x^{\gamma_{s}-1})^{(\mu_{1}-1)}(r_{1})&\cdots\cdots&({F^{(s)}}x^{\gamma_{s}-1})^{(0)}(r_{m})&\cdots&({F}^{(s)}x^{\gamma_{s}-1})^{(\mu_{m}-1)}(r_{m})\\\
~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots\\\
({F^{(s)}}x^{0})^{(0)}(r_{1})&\cdots&({F^{(s)}}x^{0})^{(\mu_{1}-1)}(r_{1})&\cdots\cdots&({F^{(s)}}x^{0})^{(0)}(r_{m})&\cdots&({F^{(s)}}x^{0})^{(\mu_{m}-1)}(r_{m})\end{array}\right|}{\prod\limits_{1\leq
i<j\leq m}(r_{i}-r_{j})^{\mu_{i}\mu_{j}}}$

where $c=\pm
1\Big{/}\left(\prod_{i=1}^{m}\prod_{j=0}^{\mu_{i}-1}j!\right)\cdot
a_{n}^{\gamma_{1}-2}$.

###  4.2 Connection between the multiplicity discriminants and multiplicity
vectors

By decompiling Theorem 3.5, we identify the two essential ingredients therein,
which are re-stated as Lemmas 4.8 and 4.10 below. From now on, we will use
$\overline{\bm{\gamma}}$ to denote the conjugate of
$\bm{\gamma}\in\mathcal{M}(n)$. To prove the lemmas, we recall the following
well known fact [2] which depicts the connection between $\gamma$ and its
conjugate.

######  Lemma 4.7.

Let $\gamma\in\mathcal{M}(n)$. Then $\overline{\overline{\gamma}}=\gamma$.
Moreover, if $\gamma$ and $\mu$ are conjugates to each other, then
$\\#\\{\mu_{j}:\mu_{j}\geq i\\}=\gamma_{i}$ and
$\\#\\{\lambda_{j}:\lambda_{j}\geq i\\}=\mu_{i}$.

######  Lemma 4.8.

Let $\operatorname*{mult}(F)=\bm{\mu}$. Then $D(\bar{\bm{\mu}})\neq 0$.

######  Proof 4.9.

In order to convey the main underlying ideas effectively, we will show the
proof for a particular case first. After that, we will generalize the ideas to
arbitrary cases.

Particular case: Consider the case $n=5$ and
$\operatorname*{mult}(F)=\bm{\mu}=\left(3,2\right)$.

  1. 1.

Assume that $r_{1}$ and $r_{2}$ are the two distinct roots with multiplicities
$3$ and $2,$ respectively. In other words,
$F=a_{5}(x-r_{1})^{3}(x-r_{2})^{2}$.

  2. 2.

Let $\bm{\gamma}=\bar{\bm{\mu}}$. Then

$\gamma_{1}=\\#\\{\mu_{j}:\,\mu_{j}\geq
1\\}=2,\quad\gamma_{2}=\\#\\{\mu_{j}:\,\mu_{j}\geq
2\\}=2,\quad\gamma_{3}=\\#\\{\mu_{j}:\,\mu_{j}\geq 3\\}=1.$

Thus $\bm{\gamma}=(2,2,1)$.

  3. 3.

By Lemma 4.5,

$\displaystyle D(\bm{\gamma})$
$\displaystyle=\frac{c\cdot\left|\begin{array}[c]{lcl|cc}({F^{(1)}x^{1}})^{(0)}(r_{1})&({F^{(1)}x^{1}})^{(1)}(r_{1})&({F^{(1)}x^{1}})^{(2)}(r_{1})&({F^{(1)}x^{1}})^{(0)}(r_{2})&({F^{(1)}x^{1}})^{(1)}(r_{2})\\\
({F^{(1)}x^{0}})^{(0)}(r_{1})&({F^{(1)}x^{0}})^{(1)}(r_{1})&({F^{(1)}x^{0}})^{(2)}(r_{1})&({F^{(1)}x^{0}})^{(0)}(r_{2})&({F^{(1)}x^{0}})^{(1)}(r_{2})\\\
\hline\cr({F^{(2)}x^{1}})^{(0)}(r_{1})&({F^{(2)}x^{1}})^{(1)}(r_{1})&({F^{(2)}x^{1}})^{(2)}(r_{1})&({F^{(2)}x^{1}})^{(0)}(r_{2})&({F^{(2)}x^{1}})^{(1)}(r_{2})\\\
({F^{(2)}x^{0}})^{(0)}(r_{1})&({F^{(2)}x^{0}})^{(1)}(r_{1})&({F^{(2)}x^{0}})^{(2)}(r_{1})&({F^{(2)}x^{0}})^{(0)}(r_{2})&({F^{(2)}x^{0}})^{(1)}(r_{2})\\\
\hline\cr({F^{(3)}x^{0}})^{(0)}(r_{1})&({F^{(3)}x^{0}})^{(1)}(r_{1})&({F^{(3)}x^{0}})^{(2)}(r_{1})&({F^{(3)}x^{0}})^{(0)}(r_{2})&({F^{(3)}x^{0}})^{(1)}(r_{2})\end{array}\right|}{(r_{1}-r_{2})^{6}}$

where

$c=\pm 1\big{/}\left[(0!\cdot 1!\cdot 2!)\cdot(0!\cdot 1!)\right]\cdot
a_{5}^{0}=\pm 1/2.$

  4. 4.

Since

$F^{(i)}(r_{1})\left\\{\begin{array}[c]{ll}=0,&\text{for}\ i=0,1,2;\\\ \neq
0,&\text{for}\ i=3;\end{array}\right.\qquad
F^{(i)}(r_{2})\left\\{\begin{array}[c]{ll}=0,&\text{for}\ i=0,1;\\\ \neq
0,&\text{for}\ i=2,\end{array}\right.$ (6)

by the Leibniz’s rule for derivatives, we immediately know

$\begin{array}[c]{l}({F^{(1)}x^{1}})^{(0)}(r_{1})=0,\
({F^{(1)}x^{1}})^{(1)}(r_{1})=0,\
({F^{(1)}x^{1}})^{(2)}(r_{1})=F^{(3)}(r_{1})r_{1}^{1},\
({F^{(1)}x^{1}})^{(0)}(r_{2})=0,\
({F^{(1)}x^{1}})^{(1)}(r_{2})=F^{(2)}(r_{2})r_{2}^{1},\\\\[2.0pt]
({F^{(1)}x^{0}})^{(0)}(r_{1})=0,\ ({F^{(1)}x^{0}})^{(1)}(r_{1})=0,\
({F^{(1)}x^{0}})^{(2)}(r_{1})=F^{(3)}(r_{1})r_{1}^{0},\
({F^{(1)}x^{0}})^{(0)}(r_{2})=0,\
({F^{(1)}x^{0}})^{(1)}(r_{2})=F^{(2)}(r_{2})r_{2}^{0},\\\\[2.0pt]
({F^{(2)}x^{1}})^{(0)}(r_{1})=0,\
({F^{(2)}x^{1}})^{(1)}(r_{1})=F^{(3)}(r_{1})r_{1}^{1},\hskip
85.50014pt({F^{(2)}x^{1}})^{(0)}(r_{2})={F^{(2)}(r_{2})r_{2}^{1}},\\\\[2.0pt]
({F^{(2)}x^{0}})^{(0)}(r_{1})=0,\
({F^{(2)}x^{0}})^{(1)}(r_{1})=F^{(3)}(r_{1})r_{1}^{0},\hskip
85.50014pt({F^{(2)}x^{0}})^{(0)}(r_{2})={F^{(2)}(r_{2})r_{2}^{0}},\\\\[2.0pt]
({F^{(3)}x^{0}})^{(0)}(r_{1})=F^{(3)}(r_{1})r_{1}^{0}.\end{array}$

  5. 5.

Therefore,

$D(\bm{\gamma})=\frac{c\cdot\left|\begin{array}[c]{ccc|cc}0&0&F^{(3)}(r_{1})r_{1}^{1}&0&F^{(2)}(r_{2})r_{2}^{1}\\\
0&0&F^{(3)}(r_{1})r_{1}^{0}&0&F^{(2)}(r_{2})r_{2}^{0}\\\ \hline\cr
0&F^{(3)}(r_{1})r_{1}^{1}&\cdot&F^{(2)}(r_{2})r_{2}^{1}&\cdot\\\
0&F^{(3)}(r_{1})r_{1}^{0}&\cdot&F^{(2)}(r_{2})r_{2}^{0}&\cdot\\\ \hline\cr
F^{(3)}(r_{1})r_{1}^{0}&\cdot&\cdot&\cdot&\cdot\end{array}\right|}{(r_{1}-r_{2})^{6}}.$

  6. 6.

By rearranging the columns of the determinant in the numerator, we have

$\displaystyle D(\bm{\gamma})$
$\displaystyle=\pm\frac{c\cdot\left|\begin{array}[c]{c|cc|cc}&&&F^{(3)}(r_{1})r_{1}^{1}&F^{(2)}(r_{2})r_{2}^{1}\\\
&&&F^{(3)}(r_{1})r_{1}^{0}&F^{(2)}(r_{2})r_{2}^{0}\\\
\hline\cr&F^{(3)}(r_{1})r_{1}^{1}&F^{(2)}(r_{2})r_{2}^{1}&\cdot&\cdot\\\
&F^{(3)}(r_{1})r_{1}^{0}&F^{(2)}(r_{2})r_{2}^{0}&\cdot&\cdot\\\ \hline\cr
F^{(3)}(r_{1})r_{1}^{0}&\cdot&\cdot&\cdot&\cdot\end{array}\right|}{(r_{1}-r_{2})^{6}}$
$\displaystyle=\pm\frac{c\cdot\left|\begin{array}[c]{ccc}&&M_{1}\\\
&M_{2}&\cdot\\\ M_{3}&\cdot&\cdot\end{array}\right|}{(r_{1}-r_{2})^{6}}$

where

$M_{1}=\left[\begin{array}[c]{cc}F^{(3)}(r_{1})r_{1}^{1}&F^{(2)}(r_{2})r_{2}^{1}\\\
F^{(3)}(r_{1})r_{1}^{0}&F^{(2)}(r_{2})r_{2}^{0}\end{array}\right],\qquad
M_{2}=\left[\begin{array}[c]{cc}F^{(3)}(r_{1})r_{1}^{1}&F^{(2)}(r_{2})r_{2}^{1}\\\
F^{(3)}(r_{1})r_{1}^{0}&F^{(2)}(r_{2})r_{2}^{0}\end{array}\right],\qquad
M_{3}=\left[F^{(3)}(r_{1})r_{1}^{0}\right].$

  7. 7.

Obviously,

$D(\bm{\gamma})=\pm\frac{c\cdot\left|M_{1}\right|\cdot\left|M_{2}\right|\cdot\left|M_{3}\right|}{(r_{1}-r_{2})^{6}}.$

We only need to show that $M_{i}\neq 0$ for $i=1,2,3$. The claim follows from
the following observations:

$\displaystyle\left|M_{1}\right|$
$\displaystyle=F^{(3)}(r_{1})F^{(2)}(r_{2})V(r_{1},r_{2})\neq 0,$
$\displaystyle\left|M_{2}\right|$
$\displaystyle=F^{(3)}(r_{1})F^{(2)}(r_{2})V(r_{1},r_{2})\neq 0,$
$\displaystyle\left|M_{3}\right|$ $\displaystyle=F^{(3)}(r_{1})V(r_{1})\neq
0.$

The proof is completed.

Arbitrary case. Now we generalize the above ideas to arbitrary cases.

  1. 1.

Let $\bm{\mu}=(\mu_{1},\ldots,\mu_{m})$. Assume that $r_{1},\ldots,r_{m}$ are
the $m$ distinct roots with multiplicities $\mu_{1},\ldots,\mu_{m}$
respectively. In other words,
$F=a_{n}(x-r_{1})^{\mu_{1}}\cdots(x-r_{m})^{\mu_{m}}$.

  2. 2.

Let $\bm{\gamma}=\bar{\bm{\mu}}=(\gamma_{1},\ldots,\gamma_{s})$, i.e.,
$\gamma_{i}=\\#\\{\mu_{j}:\,\mu_{j}\geq i\\}$. Note that $\gamma_{1}=m$ and
$s=\mu_{1}$ since $\mu_{1}\geq\cdots\geq\mu_{m}\geq 1$.

  3. 3.

Recall that

$D(\bm{\gamma})=\frac{c\cdot\left|\begin{array}[c]{lcl|c|lcl}({F^{(1)}x^{\gamma_{1}-1}})^{(0)}(r_{1})&\cdots&({F^{(1)}x^{\gamma_{1}-1}})^{(\mu_{1}-1)}(r_{1})&\cdots\cdots&({F^{(1)}x^{\gamma_{1}-1}})^{(0)}(r_{m})&\cdots&({F^{(1)}x^{\gamma_{1}-1}})^{(\mu_{m}-1)}(r_{m})\\\
~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots\\\
({F^{(1)}x^{0}})^{(0)}(r_{1})&\cdots&({F^{(1)}x^{0}})^{(\mu_{1}-1)}(r_{1})&\cdots\cdots&({F^{(1)}x^{0}})^{(0)}(r_{m})&\cdots&({F^{(1)}x^{0}})^{(\mu_{m}-1)}(r_{m})\\\
\hline\cr~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots\\\
\hline\cr({F^{(\mu_{1})}}x^{\gamma_{\mu_{1}}-1})^{(0)}(r_{1})&\cdots&({F}^{(\mu_{1})}x^{\gamma_{\mu_{1}}-1})^{(\mu_{1}-1)}(r_{1})&\cdots\cdots&({F^{(\mu_{1})}}x^{\gamma_{\mu_{1}}-1})^{(0)}(r_{m})&\cdots&({F}^{(\mu_{1})}x^{\gamma_{\mu_{1}}-1})^{(\mu_{m}-1)}(r_{m})\\\
~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots\\\
({F^{(\mu_{1})}}x^{0})^{(0)}(r_{1})&\cdots&({F^{(\mu_{1})}}x^{0})^{(\mu_{1}-1)}(r_{1})&\cdots\cdots&({F^{(\mu_{1})}}x^{0})^{(0)}(r_{m})&\cdots&({F^{(\mu_{1})}}x^{0})^{(\mu_{m}-1)}(r_{m})\end{array}\right|}{\prod\limits_{1\leq
i<j\leq m}(r_{i}-r_{j})^{\mu_{i}\mu_{j}}}$ (7)

where $c=\pm
1\Big{/}\left(\prod_{i=1}^{m}\prod_{j=0}^{\mu_{i}-1}j!\right)\cdot
a_{n}^{\gamma_{1}-2}$.

  4. 4.

Since $F$ and its first $\mu_{j}-1$ derivatives are equal to zero at
$x=r_{j}$, by the Leibniz’s rule for derivatives, we immediately know that for
$i=1,\ldots,s$ and $j$ satisfying $\mu_{j}\geq i$:

$(F^{(i)}x^{k})^{(\ell)}(r_{j})=\left\\{\begin{array}[]{ll}0,&\text{if}\
\ell<\mu_{j}-i;\\\ F^{(\mu_{j})}(r_{j})\cdot r_{j}^{k},&\text{if}\
\ell=\mu_{j}-i;\\\ ,&\text{if}\ \ell>\mu_{j}-i.\end{array}\right.$ (8)

  5. 5.

Plugging (8) into (7), we have

$D(\bm{\gamma})=\frac{c\cdot{\footnotesize\left|\begin{array}[c]{lcll|c|lcll|l}~{}~{}~{}~{}~{}~{}~{}~{}0&\cdots&~{}~{}~{}~{}~{}~{}~{}~{}0&F^{(\mu_{1})}(r_{1})r_{1}^{\gamma_{1}-1}&\cdots&~{}~{}~{}~{}~{}~{}~{}~{}0&\cdots&~{}~{}~{}~{}~{}~{}~{}~{}0&\quad
F^{(\mu_{i})}(r_{i})r_{i}^{\gamma_{1}-1}&\cdots\\\
~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&\\\
~{}~{}~{}~{}~{}~{}~{}~{}0&\cdots&~{}~{}~{}~{}~{}~{}~{}~{}0&F^{(\mu_{1})}(r_{1})r_{1}^{0}&\cdots&~{}~{}~{}~{}~{}~{}~{}~{}0&\cdots&~{}~{}~{}~{}~{}~{}~{}~{}0&\quad
F^{(\mu_{i})}(r_{i})r_{i}^{0}&\cdots\\\
\hline\cr~{}~{}~{}~{}~{}~{}~{}~{}0&\cdots&F^{(\mu_{1})}(r_{1})r_{1}^{\gamma_{2}-1}&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&\cdots&~{}~{}~{}~{}~{}~{}~{}~{}0&\cdots&F^{(\mu_{i})}(r_{i})r_{i}^{\gamma_{2}-1}&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&\cdots\\\
~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&\\\
~{}~{}~{}~{}~{}~{}~{}~{}0&\cdots&F^{(\mu_{1})}(r_{1})r_{1}^{0}&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&\cdots&~{}~{}~{}~{}~{}~{}~{}~{}0&\cdots&F^{(\mu_{i})}(r_{i})r_{i}^{0}&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&\cdots\\\
\hline\cr~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&\\\
\hline\cr~{}~{}~{}~{}~{}~{}~{}~{}0&&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&~{}~{}~{}~{}~{}~{}~{}~{}\cdot&\cdots&F^{(\mu_{i})}(r_{i})r_{i}^{\gamma_{\mu_{i}}-1}&&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&\cdots\\\
~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&\\\
~{}~{}~{}~{}~{}~{}~{}~{}0&&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&~{}~{}~{}~{}~{}~{}~{}~{}\cdot&\cdots&F^{(\mu_{i})}(r_{i})r_{i}^{0}&&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&\cdots\\\
\hline\cr~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&\\\
\hline\cr
F^{(\mu_{1})}(r_{1})r_{1}^{\gamma_{\mu_{1}}-1}&\cdots&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&\cdots&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&\cdots&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&\cdots\\\
~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&\\\
F^{(\mu_{1})}(r_{1})r_{1}^{0}&&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&\cdots&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&\cdots&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&\cdots\end{array}\right|}}{\prod\limits_{1\leq
i<j\leq m}(r_{i}-r_{j})^{\mu_{i}\mu_{j}}}$

where the block $D_{ij}$ at the $i$-th row and the $j$-th column is a matrix
$\begin{pmatrix}0&\cdots&0&F^{(\mu_{j})}(r_{j})r_{j}^{\gamma_{i}-1}&*&\cdots&*\\\
\vdots&\ddots&\vdots&\vdots&\vdots&\ddots&\vdots&\\\
0&\cdots&0&F^{(\mu_{j})}(r_{j})&*&\cdots&*\end{pmatrix}$ of size
$\gamma_{i}\times\mu_{j}$.

  6. 6.

By rearranging the columns of the determinant in the numerator, we have

$\displaystyle D(\bm{\gamma})$
$\displaystyle=\pm\frac{c\cdot{\scriptsize\left|\begin{array}[c]{lcl|ccc|lcl|lcl}&&&&&&&&&F^{(\mu_{1})}(r_{1})r_{1}^{\gamma_{1}-1}&\cdots&F^{(\mu_{m})}(r_{m})r_{m}^{\gamma_{1}-1}\\\
&&&&&&&&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots\\\
&&&&&&&&&F^{(\mu_{1})}(r_{1})r_{1}^{0}&\cdots&F^{(\mu_{m})}(r_{m})r_{m}^{0}\\\
\hline\cr&&&&&&F^{(\mu_{1})}(r_{1})r_{1}^{\gamma_{2}-1}&\cdots&F^{(\mu_{\gamma_{2}})}(r_{\gamma_{2}})r_{\gamma_{2}}^{\gamma_{2}-1}&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&\cdots&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot\\\
&&&&&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots\\\
&&&&&&F^{(\mu_{1})}(r_{1})r_{1}^{0}&\cdots&F^{(\mu_{\gamma_{2}})}(r_{\gamma_{2}})r_{\gamma_{2}}^{0}&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&\cdots&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot\\\
\hline\cr&&&\cdot&\cdots&\cdot&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&\cdots&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&\cdots&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot\\\
&&&\vdots&&\vdots&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots\\\
&&&\cdot&\cdots&\cdot&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&\cdots&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&\cdots&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot\\\
\hline\cr
F^{(\mu_{1})}(r_{1})r_{1}^{\gamma_{\mu_{1}}-1}&\cdots&F^{(\mu_{\gamma_{\mu_{1}}})}(r_{\gamma_{\mu_{1}}})r_{\gamma_{\mu_{1}}}^{\gamma_{\mu_{1}}-1}&\cdot&\cdots&\cdot&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&\cdots&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&\cdots&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot\\\
~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&\vdots&&\vdots&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots\\\
F^{(\mu_{1})}(r_{1})r_{1}^{0}&\cdots&F^{(\mu_{\gamma_{\mu_{1}}})}(r_{\gamma_{\mu_{1}}})r_{\gamma_{\mu_{1}}}^{0}&\cdot&\cdots&\cdot&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&\cdots&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&\cdots&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot\end{array}\right|}}{\prod\limits_{1\leq
i<j\leq m}(r_{i}-r_{j})^{\mu_{i}\mu_{j}}}$
$\displaystyle=\pm\frac{c\cdot\left|\begin{array}[c]{cccc}&&&M_{1}\\\
&&M_{2}&\cdot\\\\[3.0pt]
&\\!\\!\\!\\!\\!\\!\\!\begin{sideways}$\ddots$\end{sideways}&&\cdot\\\
M_{\mu_{1}}&\cdots&\cdot&\cdot\end{array}\right|}{\prod\limits_{1\leq i<j\leq
m}(r_{i}-r_{j})^{\mu_{i}\mu_{j}}}$

where

$M_{i}=\left[\begin{array}[c]{lcl}F^{(\mu_{1})}(r_{1})r_{1}^{\gamma_{i}-1}&\cdots&F^{(\mu_{\gamma_{i}})}(r_{\gamma_{i}})r_{\gamma_{i}}^{\gamma_{i}-1}\\\
~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots\\\
F^{(\mu_{1})}(r_{1})r_{1}^{0}&\cdots&F^{(\mu_{\gamma_{i}})}(r_{\gamma_{i}})r_{\gamma_{i}}^{0}\end{array}\right]$

for $i=1,\ldots,\mu_{1}$. Then

$D(\bm{\gamma})=\pm\frac{c\cdot\left|M_{1}\right|\cdots\left|M_{\mu_{1}}\right|}{\prod\limits_{1\leq
i<j\leq m}(r_{i}-r_{j})^{\mu_{i}\mu_{j}}}.$

  7. 7.

It only remains to show that $\left|M_{i}\right|\neq 0$. The claim follows
from the following observations:

$\left|M_{i}\right|=\left(\prod_{j=1}^{\gamma_{i}}F^{(\mu_{j})}(r_{j})\right)V(r_{1},\ldots,r_{\gamma_{i}})\neq
0\ \ \text{for}\ \ i=1,\ldots,\mu_{1}.$

The proof is completed.

######  Lemma 4.10.

Let $\operatorname*{mult}(F)=\bm{\mu}$. Then $D(\bm{\lambda})=0$ for any
$\bm{\lambda}$ such that
$\bm{\bar{\mu}}\prec_{\operatorname*{lex}}\bm{\lambda}$.

######  Proof 4.11.

In order to convey the main underlying ideas, we will show the proof for a
particular case first. After that, we will generalize the ideas to arbitrary
cases.

Particular case: Consider the case $n=5$ and
$\operatorname*{mult}(F)=\bm{\mu}=\left(3,2\right)$. Let
$\bm{\gamma}=\bar{\bm{\mu}}=(2,2,1)$ and $\bm{\lambda}=(3,1,1)$. Obviously,
$\bar{\bm{\mu}}\prec_{\operatorname*{lex}}\bm{\lambda}$. We will show that
$D(\bm{\lambda})=0$.

  1. 1.

Assume that $r_{1}$ and $r_{2}$ are the two distinct roots with multiplicities
$3$ and $2,$ respectively. In other words,
$F=a_{5}(x-r_{1})^{3}(x-r_{2})^{2}$.

  2. 2.

By Lemma 4.5,

$\displaystyle D(\bm{\lambda})$
$\displaystyle=\frac{c\cdot\left|\begin{array}[c]{lcl|cc}({F^{(1)}x^{2}})^{(0)}(r_{1})&({F^{(1)}x^{2}})^{(1)}(r_{1})&({F^{(1)}x^{1}})^{(2)}(r_{1})&({F^{(1)}x^{2}})^{(0)}(r_{2})&({F^{(1)}x^{2}})^{(1)}(r_{2})\\\
({F^{(1)}x^{1}})^{(0)}(r_{1})&({F^{(1)}x^{1}})^{(1)}(r_{1})&({F^{(1)}x^{1}})^{(2)}(r_{1})&({F^{(1)}x^{1}})^{(0)}(r_{2})&({F^{(1)}x^{1}})^{(1)}(r_{2})\\\
({F^{(1)}x^{0}})^{(0)}(r_{1})&({F^{(1)}x^{0}})^{(1)}(r_{1})&({F^{(1)}x^{0}})^{(2)}(r_{1})&({F^{(1)}x^{0}})^{(0)}(r_{2})&({F^{(1)}x^{0}})^{(1)}(r_{2})\\\
\hline\cr({F^{(2)}x^{0}})^{(0)}(r_{1})&({F^{(2)}x^{0}})^{(1)}(r_{1})&({F^{(2)}x^{0}})^{(2)}(r_{1})&({F^{(2)}x^{0}})^{(0)}(r_{2})&({F^{(2)}x^{0}})^{(1)}(r_{2})\\\
\hline\cr({F^{(3)}x^{0}})^{(0)}(r_{1})&({F^{(3)}x^{0}})^{(1)}(r_{1})&({F^{(3)}x^{0}})^{(2)}(r_{1})&({F^{(3)}x^{0}})^{(0)}(r_{2})&({F^{(3)}x^{0}})^{(1)}(r_{2})\end{array}\right|}{(r_{1}-r_{2})^{6}}$

where

$c=\pm 1/\big{[}(0!\cdot 1!\cdot 2!)\cdot(0!\cdot 1!)]\cdot a_{5}^{1}=\pm
a_{5}/2.$

  3. 3.

Recall (6). Then we immediately have

$\begin{array}[c]{l}({F^{(1)}x^{2}})^{(0)}(r_{1})=0,\
({F^{(1)}x^{2}})^{(1)}(r_{1})=0,\
({F^{(1)}x^{2}})^{(2)}(r_{1})=F^{(3)}(r_{1})r_{1}^{2},\
({F^{(1)}x^{2}})^{(0)}(r_{2})=0,\
({F^{(1)}x^{2}})^{(1)}(r_{2})=F^{(2)}(r_{2})r_{2}^{2},\\\\[2.0pt]
({F^{(1)}x^{1}})^{(0)}(r_{1})=0,\ ({F^{(1)}x^{1}})^{(1)}(r_{1})=0,\
({F^{(1)}x^{1}})^{(2)}(r_{1})=F^{(3)}(r_{1})r_{1}^{1},\
({F^{(1)}x^{1}})^{(0)}(r_{2})=0,\
({F^{(1)}x^{1}})^{(1)}(r_{2})=F^{(2)}(r_{2})r_{2}^{1},\\\\[2.0pt]
({F^{(1)}x^{0}})^{(0)}(r_{1})=0,\ ({F^{(1)}x^{0}})^{(1)}(r_{1})=0,\
({F^{(1)}x^{0}})^{(2)}(r_{1})=F^{(3)}(r_{1})r_{1}^{0},\
({F^{(1)}x^{0}})^{(0)}(r_{2})=0,\
({F^{(1)}x^{0}})^{(1)}(r_{2})=F^{(2)}(r_{2})r_{2}^{0},\\\\[2.0pt]
({F^{(2)}x^{0}})^{(0)}(r_{1})=0,\
({F^{(2)}x^{0}})^{(1)}(r_{1})=F^{(3)}(r_{1})r_{1}^{0},\hskip
85.50014pt({F^{(2)}x^{0}})^{(0)}(r_{2})={F^{(2)}(r_{2})r_{2}^{0}},\\\\[2.0pt]
({F^{(3)}x^{0}})^{(0)}(r_{1})=F^{(3)}(r_{1})r_{1}^{0}.\end{array}$

Therefore,

$D(\bm{\lambda})=\frac{c\cdot\left|\begin{array}[c]{ccc|cc}0&0&F^{(3)}(r_{1})r_{1}^{2}&0&F^{(2)}(r_{2})r_{2}^{2}\\\
0&0&F^{(3)}(r_{1})r_{1}^{1}&0&F^{(2)}(r_{2})r_{2}^{1}\\\
0&0&F^{(3)}(r_{1})r_{1}^{0}&0&F^{(2)}(r_{2})r_{2}^{0}\\\ \hline\cr
0&F^{(3)}(r_{1})r_{1}^{0}&\cdot&F^{(2)}(r_{2})r_{2}^{0}&\cdot\\\ \hline\cr
F^{(3)}(r_{1})r_{1}^{0}&\cdot&\cdot&\cdot&\cdot\end{array}\right|}{(r_{1}-r_{2})^{6}}.$

  4. 4.

By rearranging the columns of the determinant in the numerator, we have

$\displaystyle D(\bm{\lambda})$
$\displaystyle=\pm\frac{c\cdot\left|\begin{array}[c]{c|cc|cc}&&&F^{(3)}(r_{1})r_{1}^{2}&F^{(2)}(r_{2})r_{2}^{2}\\\
&&&F^{(3)}(r_{1})r_{1}^{1}&F^{(2)}(r_{2})r_{2}^{1}\\\
&&&F^{(3)}(r_{1})r_{1}^{0}&F^{(2)}(r_{2})r_{2}^{0}\\\
\hline\cr&F^{(3)}(r_{1})r_{1}^{0}&F^{(2)}(r_{2})r_{2}^{0}&\cdot&\cdot\\\
\hline\cr
F^{(3)}(r_{1})r_{1}^{0}&\cdot&\cdot&\cdot&\cdot\end{array}\right|}{(r_{1}-r_{2})^{6}}$
$\displaystyle=\pm\frac{c\cdot\left|\begin{array}[c]{ccc}&&M_{1}\\\
&M_{2}&\cdot\\\ M_{3}&\cdot&\cdot\end{array}\right|}{(r_{1}-r_{2})^{6}}$

where

$M_{1}=\left[\begin{array}[c]{ll}F^{(3)}(r_{1})r_{1}^{2}&F^{(2)}(r_{2})r_{2}^{2}\\\
F^{(3)}(r_{1})r_{1}^{1}&F^{(2)}(r_{2})r_{2}^{1}\\\
F^{(3)}(r_{1})r_{1}^{0}&F^{(2)}(r_{2})r_{2}^{0}\end{array}\right],\quad
M_{2}=\left[\begin{array}[c]{ll}F^{(3)}(r_{1})r_{1}^{0}&F^{(2)}(r_{2})r_{2}^{0}\end{array}\right],\quad
M_{3}=\left[\begin{array}[c]{l}F^{(3)}(r_{1})r_{1}^{0}\end{array}\right].$

  5. 5.

We repartition the columns so that the reverse diagonal consists of two square
matrices and obtain the following:

$\displaystyle D(\bm{\lambda})$
$\displaystyle=\pm\frac{c\cdot\left|\begin{array}[c]{cc|ccc}&&&F^{(3)}(r_{1})r_{1}^{2}&F^{(2)}(r_{2})r_{2}^{2}\\\
&&&F^{(3)}(r_{1})r_{1}^{1}&F^{(2)}(r_{2})r_{2}^{1}\\\
&&&F^{(3)}(r_{1})r_{1}^{0}&F^{(2)}(r_{2})r_{2}^{0}\\\
\hline\cr&F^{(3)}(r_{1})r_{1}^{0}&F^{(2)}(r_{2})r_{2}^{0}&\cdot&\cdot\\\
F^{(3)}(r_{1})r_{1}^{0}&\cdot&\cdot&\cdot&\cdot\end{array}\right|}{(r_{1}-r_{2})^{6}}$
$\displaystyle=\pm\frac{c\cdot\left|\begin{array}[c]{cc}&T\\\
B&\cdot\end{array}\right|}{(r_{1}-r_{2})^{6}}$

where the size of the square matrix $T$ is $\lambda_{1}=3$, namely,

$T=\left[\begin{array}[c]{cc}0&M_{1}\end{array}\right],$

where $0$ is the $\lambda_{1}\times(\lambda_{1}-\gamma_{1})$ matrix.

  6. 6.

Since $\lambda_{1}-\gamma_{1}=3-2>0$, the first column of $T$ is all zeros.
Hence $|T|=0$ and in turn $D(\bm{\lambda})=0$.

Arbitrary case. Now we generalize the above ideas to arbitrary cases.

  1. 1.

Let $\bm{\mu}=(\mu_{1},\ldots,\mu_{m})$. Assume that $r_{1},\ldots,r_{m}$ are
the $m$ distinct roots of $F$ with multiplicities $\mu_{1},\ldots,\mu_{m}$
respectively. In other words,
$F=a_{n}(x-r_{1})^{\mu_{1}}\cdots(x-r_{m})^{\mu_{m}}$.

  2. 2.

Let $\bm{\gamma}=\bar{\bm{\mu}}=(\gamma_{1},\ldots,\gamma_{s})$. By the
definition of conjugate, $\gamma_{i}=\\#\\{\mu_{j}:\,\mu_{j}\geq i\\}$. Note
that $\gamma_{1}=m$ and $s=\mu_{1}$ since $\mu_{1}\geq\cdots\geq\mu_{m}\geq
1$.

  3. 3.

Consider $\bm{\lambda}=(\lambda_{1},\ldots,\lambda_{t})\in\mathcal{M}(n)$ such
that $\bm{\gamma}\prec_{\text{lex}}\bm{\lambda}$. By Lemma 4.5, we have

$D(\bm{\lambda})=\frac{c\cdot{\footnotesize\left|\begin{array}[c]{lcl|c|lcl}({F^{(1)}x^{\lambda_{1}-1}})^{(0)}(r_{1})&\cdots&({F^{(1)}x^{\lambda_{1}-1}})^{(\mu_{1}-1)}(r_{1})&\cdots\cdots&({F^{(1)}x^{\lambda_{1}-1}})^{(0)}(r_{m})&\cdots&({F^{(1)}x^{\lambda_{1}-1}})^{(\mu_{m}-1)}(r_{m})\\\\[-2.0pt]
~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots\\\\[-2.0pt]
({F^{(1)}x^{0}})^{(0)}(r_{1})&\cdots&({F^{(1)}x^{0}})^{(\mu_{1}-1)}(r_{1})&\cdots\cdots&({F^{(1)}x^{0}})^{(0)}(r_{m})&\cdots&({F^{(1)}x^{0}})^{(\mu_{m}-1)}(r_{m})\\\\[-2.0pt]
\hline\cr~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots\\\\[-2.0pt]
\hline\cr({F^{(t)}}x^{\lambda_{t}-1})^{(0)}(r_{1})&\cdots&({F}^{(t)}x^{\lambda_{t}-1})^{(\mu_{1}-1)}(r_{1})&\cdots\cdots&({F^{(t)}}x^{\lambda_{t}-1})^{(0)}(r_{m})&\cdots&({F}^{(t)}x^{\lambda_{t}-1})^{(\mu_{m}-1)}(r_{m})\\\\[-2.0pt]
~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots\\\\[-2.0pt]
({F^{(t)}}x^{0})^{(0)}(r_{1})&\cdots&({F^{(t)}}x^{0})^{(\mu_{1}-1)}(r_{1})&\cdots\cdots&({F^{(t)}}x^{0})^{(0)}(r_{m})&\cdots&({F^{(t)}}x^{0})^{(\mu_{m}-1)}(r_{m})\end{array}\right|}}{\prod_{i<j}(r_{i}-r_{j})^{\mu_{i}\mu_{j}}}$
(9)

where $c=\pm
1\Big{/}\left(\prod_{i=1}^{m}\prod_{j=0}^{\mu_{i}-1}j!\right)\cdot
a_{n}^{\lambda_{1}-2}$.

  4. 4.

Recall (8) and plug (8) into (9). Then we get

$D(\bm{\gamma})=\frac{c\cdot{\scriptsize\left|\begin{array}[c]{lcll|c|lcll|l}~{}~{}~{}~{}~{}~{}~{}~{}0&\cdots&~{}~{}~{}~{}~{}~{}~{}~{}0&F^{(\mu_{1})}(r_{1})r_{1}^{\lambda_{1}-1}&\cdots&~{}~{}~{}~{}~{}~{}~{}~{}0&\cdots&~{}~{}~{}~{}~{}~{}~{}~{}0&\quad
F^{(\mu_{i})}(r_{i})r_{i}^{\lambda_{1}-1}&\cdots\\\\[-2.0pt]
~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&\\\\[-2.0pt]
~{}~{}~{}~{}~{}~{}~{}~{}0&\cdots&~{}~{}~{}~{}~{}~{}~{}~{}0&F^{(\mu_{1})}(r_{1})r_{1}^{0}&\cdots&~{}~{}~{}~{}~{}~{}~{}~{}0&\cdots&~{}~{}~{}~{}~{}~{}~{}~{}0&\quad
F^{(\mu_{i})}(r_{i})r_{i}^{0}&\cdots\\\\[-2.0pt]
\hline\cr~{}~{}~{}~{}~{}~{}~{}~{}0&\cdots&F^{(\mu_{1})}(r_{1})r_{1}^{\lambda_{2}-1}&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&\cdots&~{}~{}~{}~{}~{}~{}~{}~{}0&\cdots&F^{(\mu_{i})}(r_{i})r_{i}^{\lambda_{2}-1}&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&\cdots\\\\[-2.0pt]
~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&\\\\[-2.0pt]
~{}~{}~{}~{}~{}~{}~{}~{}0&\cdots&F^{(\mu_{1})}(r_{1})r_{1}^{0}&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&\cdots&~{}~{}~{}~{}~{}~{}~{}~{}0&\cdots&F^{(\mu_{i})}(r_{i})r_{i}^{0}&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&\cdots\\\\[-2.0pt]
\hline\cr~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&\\\\[-2.0pt]
\hline\cr~{}~{}~{}~{}~{}~{}~{}~{}0&&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&~{}~{}~{}~{}~{}~{}~{}~{}\cdot&\cdots&F^{(\mu_{i})}(r_{i})r_{i}^{\lambda_{\mu_{i}}-1}&&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&\cdots\\\\[-2.0pt]
~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&\\\\[-2.0pt]
~{}~{}~{}~{}~{}~{}~{}~{}0&&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&~{}~{}~{}~{}~{}~{}~{}~{}\cdot&\cdots&F^{(\mu_{i})}(r_{i})r_{i}^{0}&&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&\cdots\\\
\hline\cr~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&\\\\[-2.0pt]
\hline\cr
F^{(\mu_{1})}(r_{1})r_{1}^{\lambda_{\mu_{1}}-1}&\cdots&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&\cdots&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&\cdots&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&\cdots\\\\[-2.0pt]
~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&\\\\[-2.0pt]
F^{(\mu_{1})}(r_{1})r_{1}^{0}&&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&\cdots&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&\cdots&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&\cdots\end{array}\right|}}{\prod\limits_{1\leq
i<j\leq m}(r_{i}-r_{j})^{\mu_{i}\mu_{j}}}.$

  5. 5.

By rearranging the columns of the determinant in the numerator, we have

$\displaystyle D(\bm{\lambda})$
$\displaystyle=\pm\frac{c\cdot{\scriptsize\left|\begin{array}[c]{lcl|ccc|lcl|lcl}&&&&&&&&&F^{(\mu_{1})}(r_{1})r_{1}^{\lambda_{1}-1}&\cdots&F^{(\mu_{\gamma_{1}})}(r_{\gamma_{1}})r_{\gamma_{1}}^{\lambda_{1}-1}\\\
&&&&&&&&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots\\\
&&&&&&&&&F^{(\mu_{1})}(r_{1})r_{1}^{0}&\cdots&F^{(\mu_{\gamma_{1}})}(r_{\gamma_{1}})r_{\gamma_{1}}^{0}\\\
\hline\cr&&&&&&F^{(\mu_{1})}(r_{1})r_{1}^{\lambda_{2}-1}&\cdots&F^{(\mu_{\gamma_{2}})}(r_{\gamma_{2}})r_{\gamma_{2}}^{\lambda_{2}-1}&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&\cdots&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot\\\
&&&&&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots\\\
&&&&&&F^{(\mu_{1})}(r_{1})r_{1}^{0}&\cdots&F^{(\mu_{\gamma_{2}})}(r_{\gamma_{2}})r_{\gamma_{2}}^{0}&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&\cdots&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot\\\
\hline\cr&&&\cdot&\cdots&\cdot&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&\cdots&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&\cdots&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot\\\
&&&\vdots&&\vdots&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots\\\
&&&\cdot&\cdots&\cdot&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&\cdots&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&\cdots&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot\\\
\hline\cr
F^{(\mu_{1})}(r_{1})r_{1}^{\lambda_{t}-1}&\cdots&F^{(\mu_{\gamma_{s}})}(r_{\gamma_{s}})r_{\gamma_{s}}^{\lambda_{t}-1}&\cdot&\cdots&\cdot&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&\cdots&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&\cdots&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot\\\
~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&\vdots&&\vdots&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}\vdots\\\
F^{(\mu_{1})}(r_{1})r_{1}^{0}&\cdots&F^{(\mu_{\gamma_{s}})}(r_{\gamma_{s}})r_{\gamma_{s}}^{0}&\cdot&\cdots&\cdot&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&\cdots&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot&\cdots&~{}~{}~{}~{}~{}~{}~{}~{}\,\cdot\end{array}\right|}}{\prod\limits_{1\leq
i<j\leq m}(r_{i}-r_{j})^{\mu_{i}\mu_{j}}}$
$\displaystyle=\pm\frac{c\cdot\left|\begin{array}[c]{cccc}&&&M_{1}\\\
&&M_{2}&\cdot\\\\[3.0pt]
&\\!\\!\\!\\!\\!\\!\\!\begin{sideways}$\ddots$\end{sideways}&&\cdot\\\
M_{\mu_{1}}&\cdots&\cdot&\cdot\end{array}\right|}{\prod\limits_{1\leq i<j\leq
m}(r_{i}-r_{j})^{\mu_{i}\mu_{j}}}$

where $M_{i}$ is $\lambda_{i}$ by $\gamma_{i}$.

  6. 6.

Since $\bm{\gamma}\prec_{\text{lex}}\bm{\lambda}$, there exists $\ell$ such
that $\gamma_{j}=\lambda_{j}$ for $j<\ell$ and $\gamma_{\ell}<\lambda_{\ell}$.
Thus

$\gamma_{1}+\cdots+\gamma_{\ell}<\lambda_{1}+\cdots+\lambda_{\ell}.$

  7. 7.

We repartition the numerator matrix so that the reverse diagonal consists of
two square matrices $T$ and $B$ as follows:

$D(\bm{\lambda})=\pm\frac{c\cdot\left|\begin{array}[c]{cc}&T\\\
B&{\cdot}\end{array}\right|}{\prod\limits_{1\leq i<j\leq
m}(r_{i}-r_{j})^{\mu_{i}\mu_{j}}}$

where the size of the square matrix $T$ is
$\lambda_{1}+\cdots+\lambda_{\ell}$, namely,

$T=\left[\begin{array}[c]{c|ccc}&&&M_{1}\\\
0{}&&\begin{sideways}$\ddots$\end{sideways}&\vdots\\\
&M_{\ell}&\cdots&\cdot\end{array}\right],$

where again $0$ is the $\gamma_{\ell}\times p$ and
$p=(\lambda_{1}+\cdots+\lambda_{\ell})-(\gamma_{1}+\cdots+\gamma_{\ell})$.

  8. 8.

Obviously,

$D(\bm{\lambda})=\pm\frac{c\cdot\left|T\right|\cdot\left|B\right|}{\prod\limits_{1\leq
i<j\leq m}(r_{i}-r_{j})^{\mu_{i}\mu_{j}}}.$

  9. 9.

Since $p>0$, the first column of $T$ is all zeros. Hence $\left|T\right|=0$,
which implies that $D(\bm{\lambda})=0$.

###  4.3 Proof of Theorem 3.5

Now we are ready to prove Theorem 3.5.

######  Proof 4.12 (Proof of Theorem 3.5).

Theorem 3.5 is equivalent to the following claim: let

$\bm{\delta}=\max\limits_{\begin{subarray}{c}\bm{\gamma}\in\mathcal{M}(n)\\\
D(\bm{\gamma})\neq 0\end{subarray}}\bm{\gamma}$

where $\max$ is with respect to the lexicographic ordering
$\prec_{\operatorname*{lex}}$. Then
$\operatorname*{mult}(F)=\overline{\bm{\delta}}$.

Next we will show the correctness of the claim.

  1. 1.

Assume that $\text{mult}(F)=\bm{\mu}$. We will show
$\bm{\mu}=\overline{\bm{\delta}}$ by disproving
$\bm{\delta}\prec_{\text{lex}}\overline{\bm{\mu}}$ and
$\overline{\bm{\mu}}~{}\prec_{\text{lex}}\bm{\delta}$.

  2. 2.

If $\bm{\delta}\prec_{\text{lex}}\overline{\bm{\mu}}$, then by the condition
for determining $\bm{\delta}$, we immediately have $D(\overline{\bm{\mu}})=0$,
leading to a contradiction with Lemma 4.8.

  3. 3.

If $\overline{\bm{\mu}}\prec_{\text{lex}}\bm{\delta}$, then by Lemma 4.10,
$D(\bm{\delta})=0.$ However, it contradicts the condition for determining
$\bm{\delta}$.

  4. 4.

Therefore, the only possibility is $\bm{\mu}=\overline{\bm{\delta}}$.

##  5 Comparison

In this section, we compare the multiplicity discriminant condition given by
Theorem 3.5 (mentioned as HY22 hereinafter) and that given by a complex root
version of YHZ’s condition [13] as well as the one given by the authors in [7,
Theorem 6] (mentioned as HY21 hereinafter). In particular, we will make
comparison on the forms and the maximum degrees of discriminants appearing in
the conditions.

###  5.1 Form of discriminants

We will illustrate the forms of conditions generated by the three methods for
a fixed $\bm{\mu}$. For example, we consider the polynomial
$F=a_{5}x^{5}+a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}$ and
$\bm{\mu}=(2,2,1)$. The condition for $F$ having the multiplicity structure
$\bm{\mu}$ is given as follows:

  1. 1.

YHZ’s condition: $P_{1}=0\wedge P_{2}=0\wedge P_{3}\neq 0$ where

$\displaystyle P_{1}=$
$\displaystyle\left|\begin{array}[c]{ccccccccc}a_{5}&a_{4}&a_{3}&a_{2}&a_{1}&a_{0}&&&\\\\[-2.0pt]
&a_{5}&a_{4}&a_{3}&a_{2}&a_{1}&a_{0}&&\\\\[-2.0pt]
&&a_{5}&a_{4}&a_{3}&a_{2}&a_{1}&a_{0}&\\\\[-2.0pt]
&&&a_{5}&a_{4}&a_{3}&a_{2}&a_{1}&a_{0}\\\\[-2.0pt]
5a_{5}&4a_{4}&3a_{3}&2a_{2}&a_{1}&&&&\\\\[-2.0pt]
&5a_{5}&4a_{4}&3a_{3}&2a_{2}&a_{1}&&&\\\\[-2.0pt]
&&5a_{5}&4a_{4}&3a_{3}&2a_{2}&a_{1}&&\\\\[-2.0pt]
&&&5a_{5}&4a_{4}&3a_{3}&2a_{2}&a_{1}&\\\\[-2.0pt]
&&&&5a_{5}&4a_{4}&3a_{3}&2a_{2}&a_{1}\end{array}\right|,$ $\displaystyle
P_{2}=$
$\displaystyle\left|\begin{array}[c]{ccccccc}a_{5}&a_{4}&a_{3}&a_{2}&a_{1}&a_{0}&\\\\[-2.0pt]
&a_{5}&a_{4}&a_{3}&a_{2}&a_{1}&a_{0}\\\\[-2.0pt]
&&a_{5}&a_{4}&a_{3}&a_{2}&a_{1}\\\\[-2.0pt]
5a_{5}&4a_{4}&3a_{3}&2a_{2}&a_{1}&&\\\\[-2.0pt]
&5a_{5}&4a_{4}&3a_{3}&2a_{2}&a_{1}&\\\\[-2.0pt]
&&5a_{5}&4a_{4}&3a_{3}&2a_{2}&a_{1}\\\\[-2.0pt]
&&&5a_{5}&4a_{4}&3a_{3}&2a_{2}\end{array}\right|,$ $\displaystyle P_{3}=$
$\displaystyle\left|\begin{array}[c]{rrr}\left|\begin{array}[c]{ccccc}a_{5}&a_{4}&a_{3}&a_{2}&a_{1}\\\\[-2.0pt]
&a_{5}&a_{4}&a_{3}&a_{2}\\\\[-2.0pt]
5a_{5}&4a_{4}&3a_{3}&2a_{2}&a_{1}\\\\[-2.0pt]
&5a_{5}&4a_{4}&3a_{3}&2a_{2}\\\\[-2.0pt]
&&5a_{5}&4a_{4}&3a_{3}\end{array}\right|&\left|\begin{array}[c]{ccccc}a_{5}&a_{4}&a_{3}&a_{2}&a_{0}\\\\[-2.0pt]
&a_{5}&a_{4}&a_{3}&a_{1}\\\\[-2.0pt] 5a_{5}&4a_{4}&3a_{3}&2a_{2}&\\\\[-2.0pt]
&5a_{5}&4a_{4}&3a_{3}&a_{1}\\\\[-2.0pt]
&&5a_{5}&4a_{4}&2a_{2}\end{array}\right|&\left|\begin{array}[c]{ccccc}a_{5}&a_{4}&a_{3}&a_{2}&\\\\[-2.0pt]
&a_{5}&a_{4}&a_{3}&a_{0}\\\\[-2.0pt] 5a_{5}&4a_{4}&3a_{3}&2a_{2}&\\\\[-2.0pt]
&5a_{5}&4a_{4}&3a_{3}&\\\\[-2.0pt]
&&5a_{5}&4a_{4}&a_{1}\end{array}\right|\\\\[35.0pt]
2\left|\begin{array}[c]{ccccc}a_{5}&a_{4}&a_{3}&a_{2}&a_{1}\\\\[-2.0pt]
&a_{5}&a_{4}&a_{3}&a_{2}\\\\[-2.0pt]
5a_{5}&4a_{4}&3a_{3}&2a_{2}&a_{1}\\\\[-2.0pt]
&5a_{5}&4a_{4}&3a_{3}&2a_{2}\\\\[-2.0pt]
&&5a_{5}&4a_{4}&3a_{3}\end{array}\right|&\left|\begin{array}[c]{ccccc}a_{5}&a_{4}&a_{3}&a_{2}&a_{0}\\\\[-2.0pt]
&a_{5}&a_{4}&a_{3}&a_{1}\\\\[-2.0pt] 5a_{5}&4a_{4}&3a_{3}&2a_{2}&\\\\[-2.0pt]
&5a_{5}&4a_{4}&3a_{3}&a_{1}\\\\[-2.0pt]
&&5a_{5}&4a_{4}&2a_{2}\end{array}\right|&\\\\[35.0pt]
&2\left|\begin{array}[c]{ccccc}a_{5}&a_{4}&a_{3}&a_{2}&a_{1}\\\\[-2.0pt]
&a_{5}&a_{4}&a_{3}&a_{2}\\\\[-2.0pt]
5a_{5}&4a_{4}&3a_{3}&2a_{2}&a_{1}\\\\[-2.0pt]
&5a_{5}&4a_{4}&3a_{3}&2a_{2}\\\\[-2.0pt]
&&5a_{5}&4a_{4}&3a_{3}\end{array}\right|&\left|\begin{array}[c]{ccccc}a_{5}&a_{4}&a_{3}&a_{2}&a_{0}\\\\[-2.0pt]
&a_{5}&a_{4}&a_{3}&a_{1}\\\\[-2.0pt] 5a_{5}&4a_{4}&3a_{3}&2a_{2}&\\\\[-2.0pt]
&5a_{5}&4a_{4}&3a_{3}&a_{1}\\\\[-2.0pt]
&&5a_{5}&4a_{4}&2a_{2}\end{array}\right|\end{array}\right|.$

  2. 2.

HY21’s condition: ${Q_{1}=0\wedge Q_{2}=0}\wedge Q_{3}\neq 0\wedge Q_{4}\neq
0$ where $Q_{1}=P_{1}$, $Q_{2}=P_{2}$ and

$\displaystyle\hskip 32.00002ptQ_{3}=$
$\displaystyle\left|\begin{array}[c]{ccccc}a_{5}&a_{4}&a_{3}&a_{2}&a_{1}\\\\[-2.0pt]
&a_{5}&a_{4}&a_{3}&a_{2}\\\\[-2.0pt]
5a_{5}&4a_{4}&3a_{3}&2a_{2}&a_{1}\\\\[-2.0pt]
&5a_{5}&4a_{4}&3a_{3}&2a_{2}\\\\[-2.0pt]
&&5a_{5}&4a_{4}&3a_{3}\end{array}\right|,$ $\displaystyle Q_{4}=$
$\displaystyle\left|\begin{array}[c]{ccccccccc}a_{5}&a_{4}&a_{3}&a_{2}&a_{1}&a_{0}&&&\\\\[-2.0pt]
&a_{5}&a_{4}&a_{3}&a_{2}&a_{1}&a_{0}&&\\\\[-2.0pt]
&&a_{5}&a_{4}&a_{3}&a_{2}&a_{1}&a_{0}&\\\\[-2.0pt]
&&&a_{5}&a_{4}&a_{3}&a_{2}&a_{1}&a_{0}\\\\[-2.0pt]
&10a_{5}&6a_{4}&3a_{3}&a_{2}&&&&\\\\[-2.0pt]
&&10a_{5}&6a_{4}&3a_{3}&a_{2}&&&\\\\[-2.0pt]
&&&10a_{5}&6a_{4}&3a_{3}&a_{2}&&\\\\[-2.0pt]
&&&&10a_{5}&6a_{4}&3a_{3}&a_{2}&\\\\[-2.0pt]
&&&&5a_{5}&4a_{4}&3a_{3}&2a_{2}&a_{1}\end{array}\right|+\left|\begin{array}[c]{ccccccccc}a_{5}&a_{4}&a_{3}&a_{2}&a_{1}&a_{0}&&&\\\\[-2.0pt]
&a_{5}&a_{4}&a_{3}&a_{2}&a_{1}&a_{0}&&\\\\[-2.0pt]
&&a_{5}&a_{4}&a_{3}&a_{2}&a_{1}&a_{0}&\\\\[-2.0pt]
&&&a_{5}&a_{4}&a_{3}&a_{2}&a_{1}&a_{0}\\\\[-2.0pt]
&10a_{5}&6a_{4}&3a_{3}&a_{2}&&&&\\\\[-2.0pt]
&&10a_{5}&6a_{4}&3a_{3}&a_{2}&&&\\\\[-2.0pt]
&&&10a_{5}&6a_{4}&3a_{3}&a_{2}&&\\\\[-2.0pt]
&&&5a_{5}&4a_{4}&3a_{3}&2a_{2}&a_{1}&\\\\[-2.0pt]
&&&&&10a_{5}&6a_{4}&3a_{3}&a_{2}\end{array}\right|$ $\displaystyle+$
$\displaystyle\left|\begin{array}[c]{ccccccccc}a_{5}&a_{4}&a_{3}&a_{2}&a_{1}&a_{0}&&&\\\\[-2.0pt]
&a_{5}&a_{4}&a_{3}&a_{2}&a_{1}&a_{0}&&\\\\[-2.0pt]
&&a_{5}&a_{4}&a_{3}&a_{2}&a_{1}&a_{0}&\\\\[-2.0pt]
&&&a_{5}&a_{4}&a_{3}&a_{2}&a_{1}&a_{0}\\\\[-2.0pt]
&10a_{5}&6a_{4}&3a_{3}&a_{2}&&&&\\\\[-2.0pt]
&&10a_{5}&6a_{4}&3a_{3}&a_{2}&&&\\\\[-2.0pt]
&&5a_{5}&4a_{4}&3a_{3}&2a_{2}&a_{1}&&\\\\[-2.0pt]
&&&&10a_{5}&6a_{4}&3a_{3}&a_{2}&\\\\[-2.0pt]
&&&&&10a_{5}&6a_{4}&3a_{3}&a_{2}\end{array}\right|+\left|\begin{array}[c]{ccccccccc}a_{5}&a_{4}&a_{3}&a_{2}&a_{1}&a_{0}&&&\\\\[-2.0pt]
&a_{5}&a_{4}&a_{3}&a_{2}&a_{1}&a_{0}&&\\\\[-2.0pt]
&&a_{5}&a_{4}&a_{3}&a_{2}&a_{1}&a_{0}&\\\\[-2.0pt]
&&&a_{5}&a_{4}&a_{3}&a_{2}&a_{1}&a_{0}\\\\[-2.0pt]
&10a_{5}&6a_{4}&3a_{3}&a_{2}&&&&\\\\[-2.0pt]
&5a_{5}&4a_{4}&3a_{3}&2a_{2}&a_{1}&&&\\\\[-2.0pt]
&&&10a_{5}&6a_{4}&3a_{3}&a_{2}&&\\\\[-2.0pt]
&&&&10a_{5}&6a_{4}&3a_{3}&a_{2}&\\\\[-2.0pt]
&&&&&10a_{5}&6a_{4}&3a_{3}&a_{2}\end{array}\right|$ $\displaystyle+$
$\displaystyle\left|\begin{array}[c]{ccccccccc}a_{5}&a_{4}&a_{3}&a_{2}&a_{1}&a_{0}&&&\\\\[-2.0pt]
&a_{5}&a_{4}&a_{3}&a_{2}&a_{1}&a_{0}&&\\\\[-2.0pt]
&&a_{5}&a_{4}&a_{3}&a_{2}&a_{1}&a_{0}&\\\\[-2.0pt]
&&&a_{5}&a_{4}&a_{3}&a_{2}&a_{1}&a_{0}\\\\[-2.0pt]
5a_{5}&4a_{4}&3a_{3}&2a_{2}&a_{1}&&&&\\\\[-2.0pt]
&&10a_{5}&6a_{4}&3a_{3}&a_{2}&&&\\\\[-2.0pt]
&&&10a_{5}&6a_{4}&3a_{3}&a_{2}&&\\\\[-2.0pt]
&&&&10a_{5}&6a_{4}&3a_{3}&a_{2}&\\\\[-2.0pt]
&&&&&10a_{5}&6a_{4}&3a_{3}&a_{2}\end{array}\right|.$

  3. 3.

HY22’s condition: ${R_{1}}=0\wedge R_{2}=0\wedge R_{3}\neq 0$ where
$R_{1}=P_{1}$ and

$\displaystyle R_{2}=$
$\displaystyle\frac{1}{a_{5}}\left|\begin{array}[c]{cccccccc}a_{5}&a_{4}&a_{3}&a_{2}&a_{1}&a_{0}&&\\\\[-2.0pt]
&a_{5}&a_{4}&a_{3}&a_{2}&a_{1}&a_{0}&\\\\[-2.0pt]
&&a_{5}&a_{4}&a_{3}&a_{2}&a_{1}&a_{0}\\\\[-2.0pt]
5a_{5}&4a_{4}&3a_{3}&2a_{2}&a_{1}&&&\\\\[-2.0pt]
&5a_{5}&4a_{4}&3a_{3}&2a_{2}&a_{1}&&\\\\[-2.0pt]
&&5a_{5}&4a_{4}&3a_{3}&2a_{2}&a_{1}&\\\\[-2.0pt]
&&&5a_{5}&4a_{4}&3a_{3}&2a_{2}&a_{1}\\\\[-2.0pt]
&&&&20a_{5}&12a_{4}&6a_{3}&2a_{2}\end{array}\right|$ $\displaystyle R_{3}=$
$\displaystyle\frac{1}{a_{5}}\left|\begin{array}[c]{ccccccc}a_{5}&a_{4}&a_{3}&a_{2}&a_{1}&a_{0}&\\\
&a_{5}&a_{4}&a_{3}&a_{2}&a_{1}&a_{0}\\\ 5a_{5}&4a_{4}&3a_{3}&2a_{2}&a_{1}&&\\\
&5a_{5}&4a_{4}&3a_{3}&2a_{2}&a_{1}&\\\ &&5a_{5}&4a_{4}&3a_{3}&2a_{2}&a_{1}\\\
&&20a_{5}&12a_{4}&6a_{3}&2a_{2}&\\\
&&&20a_{5}&12a_{4}&6a_{3}&2a_{2}\end{array}\right|$

From the above conditions, we make the following observations which are also
true in general.

  1. 1.

YHZ’s discriminant involves one nested determinant;

  2. 2.

HY21’s discriminant involves a sum of several non-nested determinants;

  3. 3.

HY22’s discriminant involves one non-nested determinant.

###  5.2 Maximum degree of discriminants

For the sake of simplicity, we use the following short-hands:

  * •

$d_{\mathrm{YHZ}}$ : the maximum of the degrees of the polynomials appearing
in YHZ’s conditions ([13]);

  * •

$d_{\mathrm{HY21}}$ : the maximum of the degrees of the polynomials appearing
in HY21’s conditions ([7]);

  * •

$d_{\mathrm{HY22}}$ : the maximum of the degrees of the polynomials appearing
in the new conditions (Theorem 3.5).

######  Lemma 5.1.

Let $d_{\mathrm{YHZ}}(\bm{\mu})$,$d_{\mathrm{HY21}}(\bm{\mu})$ and
$d_{\mathrm{HY22}}(\bm{\mu})$ denote the maximum degrees of the polynomials
appearing in YHZ’s condition, HY21’s condition and HY22’s condition for a
given $\bm{\mu}=(\mu_{1},\ldots,\mu_{m})\in\mathcal{M}(n)$, respectively. Then
we have:

  1. 1.

Under some minor and reasonable assumption (see [6, Assumption 2]),

$d_{\mathrm{YHZ}}(\bm{\mu})=\left\\{\begin{array}[c]{ll}\prod\limits_{j=1}^{\mu_{2}}(2\,m_{j}-1)\left\\{\begin{array}[c]{lll}1&\text{if}&\mu_{1}=\mu_{2}\\\
1+\frac{2}{2m_{\mu_{2}}-1}&\text{if}&\mu_{1}=\mu_{2}+1\\\
\left(2\left(\mu_{1}-\mu_{2}\right)-1\right)&\text{if}&\mu_{1}>\mu_{2}+1\end{array}\right.\;\;\;\;\;\;\geq\;\;2n+3^{\mu_{2}}-4\mu_{2},&\text{for}\
m>1;\\\ 2n-1,&\text{for}\ m=1,\end{array}\right.$

where $m_{i}=\\#\\{\mu_{k}:\,\mu_{k}\geq i\\}$;

  2. 2.

$d_{\mathrm{HY21}}(\bm{\mu})=2n-1;$

  3. 3.

$d_{\mathrm{HY22}}(\bm{\mu})=2n-2.$

######  Proof 5.2.

  1. 1.

When $m=1$, $\bm{\mu}=(n)$. In this case, the condition for the polynomial
having multiplicity structure $\bm{\mu}$ is given by the $0$-th,…,$(n-1)$-th
subdiscriminants. Thus the maximum degree $d_{\mathrm{YHZ}}(\bm{\mu})$ is
$2n-1$, achieved at the $0$-th subdiscriminant.

When $m>1$, see [6, Appendix] for a detailed proof.

  2. 2.

Recall that HY21’s condition consists of two parts: (i) the
$0$-th,…,$(n-m)$-th subdiscriminants whose highest degree is $2n-1$; (ii) the
multiplicity discriminant given by

$\sum_{{\sigma}\in
S_{p}}\operatorname*{dp}\left[\begin{array}[c]{c}x^{n-\mu_{m}-1}F\\\ \vdots\\\
x^{0}F\\\ x^{n-1}F^{\left(\sigma_{{}_{1}}\right)}/\sigma_{1}!\\\ \vdots\\\
x^{0}F^{\left(\sigma_{{}_{n}}\right)}/\sigma_{n}!\end{array}\right]$

where $p=(\underset{\mu_{1}}{\underbrace{\mu_{1},\ldots,\mu_{1}}},\ldots,$
$\underset{\mu_{m}}{\underbrace{\mu_{m},\ldots,\mu_{m}}})$ and $S_{p}$ is the
set of all permutations of $p$. It is easy to see that the degree of the
multiplicity discriminant is $2n-\mu_{m}$. Hence the maximum degree of the
above discriminants is $2n-1$.

  3. 3.

HY22’s condition only consists of the multiplicity discriminants given by

$D\left(\bm{\gamma}\right)=\frac{1}{a_{n}}\operatorname*{dp}\left[\begin{array}[c]{l}F^{(0)}x^{\gamma_{0}-1}\\\
~{}~{}~{}~{}\ \vdots\\\ F^{(0)}x^{0}\\\ \hline\cr F^{(1)}x^{\gamma_{1}-1}\\\
~{}~{}~{}~{}\ \vdots\\\ F^{(1)}x^{0}\\\ \hline\cr~{}~{}~{}~{}\ \vdots\\\
\hline\cr F^{(s)}x^{\gamma_{s}-1}\\\ ~{}~{}~{}~{}\ \vdots\\\
F^{(s)}x^{0}\end{array}\right]$

where $\bm{\gamma}=(\gamma_{1},\ldots,\gamma_{s})$ ranges over
$\overline{\bm{\mu}}\prec_{\operatorname*{lex}}\cdots\prec_{\operatorname*{lex}}(n)$.
Note that the highest degree is achieved when $\bm{\gamma}=(n)$. In this case,
the degree of $D\left(\bm{\gamma}\right)$ is $2n-2$.

######  Remark 5.3.

It is noted that in HY21’s condition, the multiplicity discriminant is always
divisible by the leading coefficient $a_{n}$ and thus with this division
carried out, the degree can be made smaller by $1$.

By Lemma 5.1, the maximum degree in YHZ’s condition grows exponentially with
respect to $n$ while the maximum degrees in HY21 and HY22’s conditions grow
linearly. Below we show a comparison with examples where $n<10$.

$n$ | $d_{\mathrm{YHZ}}$ | $d_{\mathrm{HY21}}$ | $d_{\mathrm{HY22}}$
---|---|---|---
3 | 5 | 5 | 4
4 | 9 | 7 | 6
5 | 15 | 9 | 8
6 | 27 | 11 | 10
7 | 45 | 13 | 12
8 | 81 | 15 | 14
9 | 135 | 17 | 16
Table 1: Comparison on the maximal degrees
of polynomials in the conditions generated with
the three methods Figure 1: An illustration on the changes of maximal degrees
of polynomials in the conditions generated with the three methods along with
the degree $n$

Hoon Hong’s work was supported by National Science Foundations of USA (Grant
Nos: 2212461 and 1813340). Jing Yang’s work was supported by National Natural
Science Foundation of China (Grant Nos.: 12261010 and 11801101).

## References

  * [1] Basu S, Pollack R, Roy M-F. Algorithms in real algebraic geometry. Springer-Verlag, Berlin-Heidelberg, 2006
  * [2] Bóna M. A Walk Through Combinatorics: An Introduction to Enumeration and Graph Theory (4th edition). World Scientific Publishing, 2016
  * [3] Brown W, Traub J. On Euclid’s algorithm and the theory of subresultants. Journal of the Association for Computing Machinery, 1971, 18:505–514
  * [4] Collins G. Subresultants and reduced polynomial remainder sequences. Journal of the Association for Computing Machinery, 1967, 14:128–142
  * [5] González-Vega L, Recio T, Lombardi H, et al.  Sturm-Habicht Sequences, Determinants and real roots of univariate polynomials. In Quantifier Elimination and Cylindrical Algebraic Decomposition. Texts and Monographs in Symbolic Computation (A Series of the Research Institute for Symbolic Computation, Johannes-Kepler-University, Linz, Austria). Springer, 1998, 300–316
  * [6] Hong H, Yang J. A condition for multiplicity structure of univariate polynomials. arXiv:2001.02388, 2020
  * [7] Hong H, Yang J. A condition for multiplicity structure of univariate polynomials. Journal of Symbolic Computation, 2021, 104:523–538
  * [8] Liang S, Jeffrey D J.  An algorithm for computing the complete root classification of a parametric polynomial. In Calmet J, Ida T, Wang D, eds. Proceedings of the Artificial Intelligence and Symbolic Computation (AISC 2006). Lecture Notes in Computer Science, vol 4120. Springer Berlin Heidelberg, 2006, 116–130
  * [9] Liang S, Jeffrey D J, Maza M M. The complete root classification of a parametric polynomial on an interval. In Proceedings of the Twenty-first International Symposium on Symbolic and Algebraic Computation (ISSAC’08). New York: ACM, 2008, 189–196
  * [10] Liang S, Zhang J.  A complete discrimination system for polynomials with complex coefficients and its automatic generation. Science in China Series E: Technological Sciences, 1999, 42:113–128
  * [11] Loos R. Generalized polynomial remainder sequences. In Computer Algebra. Computing Supplementa (Computing), vol 4. Springer Vienna, 1983, 115–137
  * [12] Stoer J, Bulirsch R. Interpolation. In: Introduction to Numerical Analysis. Texts in Applied Mathematics, vol 12. Springer, New York, 2002, 37–144
  * [13] Yang L, Hou X, Zeng Z. A complete discrimination system for polynomials. Science in China (Series E), 1996, 39(6):628–646