%28.01.2005
\documentclass[a4paper,11pt]{article}
\usepackage{amsmath,amscd,amsfonts,amssymb}

\def\A{{\mathbb A}}
\def\B{{\mathbb B}}
\def\N{{\mathbb N}}
\def\C{{\mathbb C}}
\def\D{{\mathbb D}}
\def\S{{\mathbb S}}
\def\Z{{\mathbb Z}}
\def\E{{\mathbb E}}
\def\F{{\mathbb F}}
\def\G{{\mathbb G}}
\def\H{{\mathbb H}}
\def\K{{\mathbb K}}
\def\L{{\mathbb L}}
\def\AB{ {\mathbb A}\moins {\mathbb B}}
\def\xA{x\moins {\mathbb A}}
\def\xB{x\moins {\mathbb B}}
\def\cR{{\mathcal R}}
\def\cQ{{\mathcal Q}}
\def\cD{{\mathcal D}}
\def\cN{{\mathcal N}}

\def\Bez{\mathop{\rm Bez}}
\def\card{\mathop{\rm card}}

\def\moins{\raise 1pt\hbox{{$\scriptstyle -$}}}
\def\plus{\raise 1pt\hbox{{$\scriptstyle +$}} }

\def\phi{\varphi}

\newtheorem{theorem}{Theorem}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}

\def\proof{\noindent{\bf Proof.\ }}
\def\qed{~\hbox{$\Box$}}

\begin{document}
\title{\bf Bezoutians, Euclidean algorithm,
and orthogonal polynomials}

\author{Alain Lascoux\\
\small CNRS, Institut Gaspard Monge, Universit\'e de Marne-la-Vall\'ee\\
\small 77454 Marne-la-Vall\'ee Cedex, France\\
\small Center for Combinatorics, Nankai University\\
\small Tianjin 300071, P.R. China\\
\small  Alain.Lascoux@univ-mlv.fr
\and
Piotr Pragacz\\
\small Institute of Mathematics, Polish Academy of Sciences\\
\small \'Sniadeckich 8, 00-956 Warszawa, Poland\\
\small  P.Pragacz@impan.gov.pl\\}
\date{}

\author{Alain Lascoux\thanks{Supported by
the EC's IHRP Program, within the Research Training Network
``Algebraic Combinatorics in Europe", grant HPRN-CT-2001-00272.}\\
\small CNRS, Institut Gaspard Monge, Universit\'e de Marne-la-Vall\'ee\\
\small 77454 Marne-la-Vall\'ee Cedex, France\\
\small Center for Combinatorics, Nankai University\\
\small Tianjin 300071, P.R. China\\
\small  Alain.Lascoux@univ-mlv.fr
\and
Piotr Pragacz\thanks{Research supported by KBN
grant 2P03A 024 23.}\\
\small Institute of Mathematics of Polish Academy of Sciences\\
\small \'Sniadeckich 8, 00-956 Warszawa, Poland\\
\small P.Pragacz@impan.gov.pl}
\date{}

\maketitle

\begin{abstract}
We prove a quadratic expression for the Bezoutian of two
univariate polynomials in terms of the remainders
for the Euclidean algorithm.
In case of two polynomials of the same degree,
or of consecutive degrees, this allows us to interpret
their Bezoutian as the Christoffel-Darboux kernel for a finite
family of orthogonal polynomials, arising from the
Euclidean algorithm. We give orthogonality
properties of remainders, and reproducing properties of Bezoutians.
\end{abstract}

%\tableofcontents

\section{Introduction}\label{2intro}

In the classical theory of elimination, or of Sturm sequences,
we encounter relations which are better understood when translated
in terms of orthogonal polynomials.
The main goal of the present paper is to study the Bezoutians
associated with a pair of univariate polynomials, and discuss
their relations to the Euclidean algorithm and orthogonal
polynomials.
By linking Bezoutians of general pairs of polynomials,
of the same degree or of consecutive degrees, to the Christoffel-Darboux
kernels of the families of orthogonal polynomials coming from
the Euclidean algorithm, we establish some reproducing properties
of Bezoutians.\footnote
{As a matter of fact, we may safely associate to the
Christoffel-Darboux kernel the names of B\'ezout \cite{Bezout},
Cayley \cite{Cayley}, Sylvester \cite{Sylvester53},
Hermite \cite{Hermite}, Brioschi \cite{Brioschi1},
\cite{Brioschi2}, and recognize that these mathematicians
treated the case of a finite discrete measure, with the help
of the Euclidean algorithm.}

Neither attempting to give a full survey on Bezoutians
(or Bezoutian matrices) in mathematics nor quote all relevant bibliography,
we note that there has recently been a revival of interest
in Bezoutians because of their importance in many diverse fields
of numerical and symbolical computing as well as in control theory.
Bezoutians themselves are classical algebraic objects
associated with a pair of monic polynomials
(nowadays, also ``Bezoutian matrices'' associated
with a pair of polynomial matrices are studied -- cf., e.g.,
\cite{Wimmer0} -- but we shall not treat this more general
notion here).
This concept was invented by B\'ezout \cite{Bezout0}, \cite{Bezout},
and developed by Sylvester \cite{Sylvester53}, Jacobi \cite{Jacobi}, 
Cayley \cite{Cayley}, and others for the purposes of elimination theory:
the non-vanishing of the determinant of the Bezoutian matrix
is equivalent to the fact that the polynomials are relatively prime.
(As pointed out in \cite{Wimmer},
the Bezoutian matrix evolved, in fact, from Euler's work \cite{Euler}
in elimination theory.)
Also, Bezoutian matrices turned out to be useful tools to study
the real root counting problem cf., e.g., \cite{Gonzalez}, and
Routh-Hurwtiz (resp. Schur-Cohn) problems of counting the roots
of a given polynomial in a given half-plane (resp. in a circle),
cf., e.g., \cite{Heinig}.
This is closely related to the study of stability criteria for
polynomials or linear discrete-time systems (cf., e.g., \cite{Helmke},
\cite{Jury}). Bezoutian matrices show up as the inverse matrices
to Hankel matrices (cf., e.g., \cite{Heinig0.1}, \cite{Heinig1})
and can be used for extremal problems in set theory (cf., e.g.,
\cite{Yu}).

As pointed out above, the present paper discusses {\it other}
properties of Bezoutians.
We describe now in more detail the content of the paper. The theorems,
propositions (apart from \ref{P7.1} and \ref{P10}), and corollaries
mentioned in the following, appear to be new -- we have not seen
them in the literature.

In Section 2, we recall the notions associated
with the Euclidean algorithm for polynomials in one variable,
and the corresponding continued fraction. Besides
remainders and quotients, we define successive convergents,
and associated numerators and denominators and discuss relations
between them.

In Section 3, we prove our basic formula expressing the
Bezoutian of two polynomials as a quadratic form in their
remainders (Theorem \ref{T3}).
This formula leads to a short proof of classical
formulas of Sylvester \cite{Sylvester53} and Brioschi \cite{Brioschi1},
\cite{Brioschi2} related to the theory of Sturm; this is discussed
in Proposition \ref{P7.1}.
In a certain limit case, Theorem \ref{T3} gives also the Sylvester formula
\cite{Sylvester53} for the Jacobian; this is the content of 
Proposition \ref{P10}. In Section 6, for general pairs of polynomials 
of degrees $(n\plus 1,n)$ and $(n,n)$, we derive from
Theorem \ref{T3} expressions presenting the Bezoutians solely
in terms of the roots of the polynomials (Propositions
\ref{P11}, \ref{P12}, and Theorems \ref{T14} and \ref{T15}).

In Section 7, we interpret, in terms of orthogonal polynomials,
some identities of Brioschi \cite{Brioschi1} and show more
general relations for denominators and remainders
(Proposition \ref{P17} and Corollary \ref{C19}).
Moreover, we use the Schur function expression of orthogonal
polynomials given in Proposition \ref{P16} to relate
Bezoutians and Christoffel-Darboux kernels of the orthogonal
polynomials coming from the Euclidean algorithm.
We infer some ``reproducing"
congruences for Bezoutians (Propositions \ref{P21} and \ref{P24}).


The appendix contains a proof of two useful properties
of orthogonal polynomials using the techniques of multi-Schur
functions.

\section{Euclidean algorithm and continued fractions}\label{2eu}

Let $f=f(x)$ and $\phi=\phi(x)$ be two monic polynomials in $\C[x]$
\footnote{The polynomials in this paper,
unless otherwise explicitly stated, have complex coefficients.},
of degrees $m\ge n$ respectively. Performing the
Euclidean algorithm for division (note that we use here
{\it nonstandard} signs\footnote{This variant of the Euclidean algorithm
is called sometimes the {\it extended Euclidean algorithm}.}), we get
$$
f=\cQ_0\phi-\cR_1, \ \ \phi=\cQ_1\cR_1-\cR_2, \ \ \cR_1=\cQ_2\cR_2-\cR_3 \ ,...
$$
$$
..., \cR_{n-2}=\cQ_{n-1}\cR_{n-1}-\cR_n, \ \ \cR_{n-1}=\cQ_n\cR_n\,.
$$

\begin{definition}\label{D0.1}
The polynomial $\cR_i$ is called the $i$th remainder and
$\cQ_j$ is called the $j$th quotient.
\end{definition}

Obviously, $\deg \cQ_i=\deg\cR_{i-1}\moins \deg\cR_i$.
In the generic case, $\deg \cQ_i=1$
and $\deg \cR_i=n-i$ \ for $i=1,...,n$.

\begin{definition}\label{D0.2}
We say that $(f,\phi)$ is a general pair if
$\deg \cQ_i=1$ for $i=1,...,n$.
\end{definition}

It is convenient to set $\cR_{-1}:=f$,
$\cR_{0}:=\phi$, and $\cR_{n\plus 1}:=0$. Then the above sequence
of equations is given compactly by the equations
\begin{equation}\label{1}
\cR_{i-1} = \cQ_i\cR_i - \cR_{i\plus 1} \ , \ \  i=0,\ldots, n.
\end{equation}

We may rewrite the equation for $i=0$ as follows:
$$
\frac{\varphi}{f}=\frac{1}{\cQ_0-\frac{\cR_1}{\varphi}}\,.
$$
Iterating, we get a continued fraction:
$$  \frac{\varphi}{f} =  \cfrac{1}{\cQ_0 -
      \cfrac{1}{\cQ_1 -
       \cfrac{1}{\ddots}
        \cfrac{}{
        \cfrac{1}{\cQ_n}}}} \ \ .  $$

This is a continued fraction with $n+1$ levels numbered
from top to bottom with $k=1,...,n+1$. We recall the following definition.

\begin{definition}\label{D0.21}
The intermediate fractions (stopping at the level $k=1,...,n$
instead of the last one) are rational functions
$$
        \cN_k/\cD_k =\cN_k(\cQ_0,...,\cQ_{n-1})/\cD_k(\cQ_0,...,\cQ_{n-1})\,,
$$
where we set inductively
\begin{equation}\label{1.1}
\cN_k(\cQ_0,...,\cQ_{n-1}):=\cD_{k-1}(\cQ_1,...,\cQ_{n-1})
\end{equation}
and
\begin{equation}\label{1.2}
\cD_k(\cQ_0,...,\cQ_{n-1}):=\cQ_0 \cD_{k-1}(\cQ_1,...,\cQ_{n-1})
- \cN_{k-1}(\cQ_1,...,\cQ_{n-1})\,.
\end{equation}
The rational function $\cN_k/\cD_k$ is called the $k$th convergent.
The polynomial $\cN_k$ is called the numerator and $\cD_k$ is called
the denominator of the $k$th convergent.

\end{definition}

\noindent
(Note that Equations (\ref{1.1}) and (\ref{1.2}) match with the
equation:
$$
\frac{\cN_k(\cQ_0,\ldots,\cQ_{n-1})}{\cD_k(\cQ_0,\ldots,\cQ_{n-1})}
=\frac{1}{\cQ_0-\frac{\cN_{k-1}(\cQ_1,\ldots,\cQ_{n-1})}
{\cD_{k-1}(\cQ_1,\ldots,\cQ_{n-1})}}\,. \ )
$$

We have the following expressions for the first three convergents:
$$
\frac{\cN_1}{\cD_1}=\frac{1}{\cQ_0}\,, \ \ \
\frac{\cN_2}{\cD_2}=\frac{\cQ_1}{\cQ_0\cQ_1-1}\,, \ \ \
\frac{\cN_3}{\cD_3}=\frac{\cQ_1\cQ_2-1}{\cQ_0\cQ_1\cQ_2-\cQ_0-\cQ_2}\,.
$$

\medskip

The denominators $\cD_k$ (resp. numerators $\cN_k$),
are in the generic case of degrees $m-n+k-1$ and $k-1$, respectively.

\medskip

We now recall the following result.

\begin{lemma}\label{L0} For $i=1,\ldots,n$\,,
\begin{equation}\label{3}
\cR_i=\phi \cD_i - f \cN_i\,.
\end{equation}
\end{lemma}
\proof
We proceed by induction on the number of the $\cQ_j$'s. Consider the
continued fraction associated
with division of $\phi$ by $\cR_1$. This continued fraction depends
on $\cQ_1,\ldots,\cQ_n$, and by the induction assumption we have
\begin{equation}\label{3.1}
\cR_i=\cR_1 \cD_{i-1}(\cQ_1,\ldots,\cQ_{n-1})-
\phi \cN_{i-1}(\cQ_1,\ldots,\cQ_{n-1})\,.
\end{equation}
By Equations (\ref{1.1}) and (\ref{1.2}), the right-hand side of
Equation (\ref{3.1}) is equal to
\begin{equation}\label{3.2}
(-f+\cQ_0 \phi) \cN_i(\cQ_0,\ldots,\cQ_{n-1})
-\phi \bigl(\cD_i(\cQ_0,\ldots,\cQ_{n-1})-\cQ_0 \cD_{k-1}(\cQ_1,\ldots,
\cQ_{n-1})\bigr)\,.
\end{equation}
Using Equation (\ref{1.1}) once again, we see that the polynomial (\ref{3.2})
equals
$$
\phi \cD_i(\cQ_0,\ldots,\cQ_{n-1}) - f \cN_i(\cQ_0,\ldots,\cQ_{n-1})\,,
$$
which is the wanted expression for $\cR_i$. The lemma is proved.\qed


\begin{corollary}\label{C0} For $i=1,\ldots, n$, we have the following
congruence:
\begin{equation}\label{4} \cR_i \equiv \phi \cD_i \ \mod \ f \,.
\end{equation}
\end{corollary}

\begin{remark}\label{R0}
Alternatively, we recover this fact by observing that the denominators
satisfy the recurrent equations:
\begin{equation}\label{5} \cD_{i-1} = \cQ_i \cD_i - \cD_{i\plus 1} \ ,
 \ \ i=0,\ldots, n \,,
\end{equation}
where we set $\cD_{-1}:=0$ and $\cD_0:=1$.
Indeed, Equations (\ref{5}) and (\ref{1}) give the
same recurrent relations for denominators and remainders,
and $\cR_0=\varphi \cD_0$.
\end{remark}

We refer the reader to \cite{Brioschi1} and \cite{LascouxE} (see also 
\cite{Lascoux}) for a more systematic account of the (extended) Euclidean 
algorithm and related objects.

\section{A quadratic expression for the Bezoutian}\label{2bez}

In this section, we prove a formula
expressing the Bezoutian of two monic
polynomials as a quadratic form in their remainders.

\begin{definition}\label{D0.5}
If $g=g(x,y)$ is a polynomial in $x$,$y$, then we set
\begin{equation}\label{6}
\partial_{xy}(g):=\frac{g(x,y)-g(y,x)}{x-y}\,.
\end{equation}
\end{definition}

Given two arbitrary monic polynomials $f$ and $\phi$,
B\'ezout \cite{Bezout0}, \cite{Bezout} (cf. also \cite{Muir},
I, p.~41--52), considered quadratic expressions in the
coefficients of $f,\phi$, that he called
\emph{\'Equations d\'eriv\'ees}.
Following Jacobi \cite{Jacobi} and Cayley \cite{Cayley}
(cf. also \cite{Muir}, II, p.~138),
we can reinterpret B\'ezout's system of equations as the following
polynomial.

\begin{definition}\label{D0.6}
We set
\begin{equation}\label{7}
\Bez(f,\phi) := \partial_{xy}\bigl(f(x) \phi(y) \bigr)\,,
\end{equation}
and call this bivariate polynomial (in $x$ and $y$) the {\it Bezoutian}
(of $f$ and $\phi$).
\end{definition}
We refer to \cite{Lascoux}, p. 53, for some examples of Bezoutians.

\medskip

We have the following two relations:

\begin{equation}\label{8}
\Bez(f,\phi)=-\Bez(\phi,f)\,,
\end{equation}
\begin{equation}\label{9}
\Bez(\alpha f+\beta g,\phi)=\alpha \Bez(f,\phi)+\beta \Bez(g,\phi)\,,
\end{equation}
where $\alpha$ and $\beta$ do not depend on $x, y$.

Also, the coefficients of the bivariate polynomial $\Bez(f,\phi)$
are symmetric, but we shall not need this property.
Basic properties of Bezoutians are collected, e.g., in \cite {Fuhrmann},
Chap.~8. For the purposes of the present paper, it is recommended
to consult also Chapter 3 of \cite{Lascoux}.

Assume $m=\deg f \ge n=\deg \phi$ and use the notation of Section 2.
We record first the following lemma.
\begin{lemma}\label{L1} For $i=0,1,\ldots,n$,
\begin{equation}\label{10}
\Bez(\cR_{i-1},\cR_i)=p_i \cR_i(x)\cR_i(y) +
\Bez(\cR_i,\cR_{i\plus 1})\,,
\end{equation}
where $p_i=p_i(x,y)=\partial_{xy}(\cQ_i(x))$.
\end{lemma}
\proof We have
\begin{equation}\label{11}
\frac{\cR_{i-1}(x) \cR_i(y) -\cR_{i-1}(y) \cR_i(x)}{\cR_i(x)\cR_i(y)}
=\frac{\cR_{i-1}(x)}{\cR_i(x)} - \frac{\cR_{i-1}(y)}{\cR_i(y)} \,.
\end{equation}
Since
$$
\cR_{i-1}(x)/{\cR_i(x)}=\cQ_i(x)-\cR_{i\plus 1}(x)/\cR_i(x)\,,
$$
we can rewrite the right-hand side of Equation (\ref{11}) as
\begin{eqnarray}\label{12}
\lefteqn
{\cQ_i(x)-\cQ_i(y) - \frac{\cR_{i\plus 1}(x)}{\cR_i(x)}
+\frac{\cR_{i\plus 1}(y)}{\cR_i(y)} = {}}
                               \nonumber\\
& & = (x - y) p_i + \frac{\cR_i(x)\cR_{i\plus 1}(y)
-\cR_i(y)\cR_{i\plus 1}(x)}{\cR_i(x)\cR_i(y)} \,.
\end{eqnarray}
Equating the left-hand side of Equation (\ref{11}) with
the right-hand side of
Equation (\ref{12}), multiplying by
$\cR_i(x)\cR_i(y)$, and dividing by $x - y$, we obtain
Equation (\ref{10}).
The lemma has been proved.\qed

\begin{remark}\label{R2} Note that the derivative of $\cQ_i$ is the
limit case:
\begin{equation}\label{13}
\lim_{y\to x}p_i(x,y)=\cQ_i'(x)\,.
\end{equation}
This will be used in Section 5.
\end{remark}

The following theorem is basic for the present paper.

\begin{theorem}\label{T3} With the above notation,
\begin{equation}\label{14}
\Bez(f,\phi)= p_0 \phi(x)\phi(y)
+\sum_{i=1}^n p_i \cR_i(x)\cR_i(y)\,.
\end{equation}
\end{theorem}
\proof Indeed, iterating Equation (\ref{10}), we arrive at
$$
\Bez(f,\phi)= p_0\cR_0(x)\cR_0(y) + \Bez(\cR_0,\cR_1) = \cdots =
\sum_{i=0}^n p_i \cR_i(x)\cR_i(y)\,.
$$
This proves the theorem.\qed

We now discuss two special cases of this formula: the case of a general
pair of polynomials $(f,\phi)$ of degrees $(n\plus 1,n)$ and the case
of a general pair of polynomials of the same degree $n$.
The remainders for these two cases will be interpreted in the last
two sections in terms of orthogonal polynomials, and their Bezoutian
will be linked with the corresponding Christoffel-Darboux kernel.

\begin{lemma}\label{L4} If $(f,\phi)$ is a general pair
of polynomials of degrees $(n\plus 1,n)$, then for $i=0,1,\ldots,n$,
\begin{equation}\label{16}
\Bez(\cR_{i-1},\cR_i)=\alpha_i\cR_i(x)\cR_i(y)
+ \Bez(\cR_i,\cR_{i\plus 1})\,,
\end{equation}
where $\alpha_i$ is the coefficient of $x$ in $\cQ_i(x)$.
\end{lemma}
\proof
This is just a special case of Theorem \ref{T3} because
we have
$$
\partial_{xy}(\cQ_i(x))=\alpha_i \ \ \ \ \hbox{for} \ \
i=0,1,\ldots,n\,. \qed
$$

Iterating Equation (\ref{16}), we get the following result
(note that $\alpha_0=1$).

\begin{proposition}\label{P5} If $(f,\phi)$ is a general pair
of polynomials of degrees $(n\plus 1,n)$, then
\begin{equation}\label{17}
\Bez(f,\phi)= \phi(x)\phi(y) +\sum_{i=1}^n \alpha_i \cR_i(x)\cR_i(y)\,.
\end{equation}
\end{proposition}

\medskip
For example take $n=2$, $f(x)=x^3+x^2+2x+1$, and $\phi(x)=x^2+x+1$.
Then
$$
\Bez(f,\phi)=1+x^2+y^2+x^2y+xy^2+x^2y^2.
$$
We have $\cR_1=-x-1$, $\cR_2=-1$, $\cQ_1=-x$, $\cQ_2=x+1$, so that
$\alpha_1=-1$, $\alpha_2=1$, and Proposition \ref{P5} says:
$$
\Bez(f,\phi)=(x^2+x+1)(y^2+y+1)-(x+1)(y+1)+1\,.
$$

\bigskip
We record the following consequence of Equations (\ref{4}) and (\ref{17}):

\begin{corollary}\label{C6} If $(f,\phi)$ is a general pair
of polynomials of degrees $(n\plus 1,n)$, then the following congruence
holds modulo the ideal $(f(x),f(y))$:
\begin{equation}\label{18}
\Bez(f,\phi) \equiv \phi(x)\phi(y)\bigl(1 +\sum_{i=1}^n \alpha_i
\cD_i(x)\cD_i(y)\bigr)\,.
\end{equation}
\end{corollary}

If we specialize  $x$ and $y$ to roots of $f$, then this congruence
is replaced by an equality.

Similarly, we infer the following result.

\begin{proposition}\label{P7} If $(f,\phi)$ is a general pair
of polynomials of the same degree $n$, then
\begin{equation}\label{19}
\Bez(f,\phi)=\sum_{i=1}^{n} \alpha_i\cR_i(x)\cR_i(y)\,,
\end{equation}
where $\alpha_i$ is the coefficient of $x$ in $\cQ_i(x)$.
\end{proposition}


\section{Nullity of Sylvester's and Brioschi's determinants}\label{2app1}

In this section, we give a new simple proof of two
classical identities of Sylvester and Brioschi, based on
the quadratic expression for the Bezoutian from Theorem \ref{T3}.

Let $f$ be a monic polynomial of degree $n\plus 1$.
Suppose that $(n\plus 1) \phi(x)$ is the derivative of $f(x)$.
In his study of Sturm sequences, Sylvester \cite {Sylvester53}, vol.~I,
p.~502 stated (without proof) the vanishing of the following
determinant:
\begin{equation}\label{20}
\Bigl|   \cD_i(a_j)^2  \Bigr|_{0\leq i,j\leq n}  \ ,
\end{equation}
where $a_0,\ldots, a_n$ are the roots\footnote{Since we are interested
in the nullity of the determinant (\ref{20}), we can order these roots
arbitrarily.} of $f$, $\cD_i(x)$, $i=1,\ldots,n$, are the denominators 
of convergents from Section 2, and $\cD_0=1$.

Brioschi \cite {Brioschi1} generalized Sylvester's relation to
any polynomial $\phi$ of degree $n$ such that $f$ and $\phi$ are
relatively prime and the pair $(f,\phi)$ is general.

\begin{proposition}\label{P7.1} With these assumptions, if we set
\begin{equation}\label{21}
\cD_0(x)^2: = 1 - \frac{ f'(x)}{\phi(x)}\,,
\end{equation}
then the same determinant (\ref{20}) vanishes.
\end{proposition}
\proof
It suffices to show that
there exist complex numbers $c_i$, $i=0,\ldots,n$, such that
for any root $a$ of $f$ the following equation holds:
\begin{equation}\label{22}
c_0\cD_0(a)^2 + c_1\cD_1(a)^2+ \cdots +c_n\cD_n(a)^2  = 0 \ .
\end{equation}
This follows from the expression of the specializations
$x=y=a$ of the Bezoutian, divided by $\phi(a)^2$:
\begin{equation}\label{23}
\phi(a)^{-2}\,\Bez(f,\phi)\Big|_{x=y=a} =  \frac{f'(a)\phi(a)}{\phi(a)^2}
        = 1 +\alpha_1 \cD_1(a)^2 + \cdots + \alpha_n \cD_n(a)^2 \,,
\end{equation}
where Corollary \ref{C6} has been used.\footnote{For another proof of
Sylvester's relation, see \cite{Lascoux}, Ex.2.34.}\qed

\section{Sylvester's formula for the Jacobian}
\label{2app2}

Sylvester \cite{Sylvester53} p.~506 stated a formula
relating the Jacobian of two forms in two variables (of the same degree)
with the remainders and quotients.
The purpose of this section is to give a simple proof of this
formula, based on the quadratic expression for the Bezoutian
from Theorem \ref{T3}.

Let $f$ and $\phi$ be two monic polynomials with
$\deg f \ge \deg \phi$.

\begin{definition}\label{D7.2}
We set
\begin{equation}\label{24}
W(f,\phi):= f'\phi - f \phi'\,.
\end{equation}
\end{definition}

This is the simplest Wronskian. Note that
\begin{equation}\label{25}
W(f,\phi)=\lim_{y \to x} \Bez(f,\phi)\,.
\end{equation}

By virtue of Equation (\ref{13}), we get, as the ``limit $x=y$ case"
of Equation (\ref{10}) the following result.

\begin{lemma}\label{L8} With the notation of Section 2,
for $i=0,1,\ldots,n$,
\begin{equation}\label{26}
W(\cR_{i-1},\cR_i)=\cQ_i'\cR_i^2 + W(\cR_i,\cR_{i\plus 1})\,.
\end{equation}
\end{lemma}

Iterating Equation (\ref{26}), we arrive at the following formula.

\begin{proposition}\label{P9} We have the following quadratic
expression for the Wronskian:
\begin{equation}\label{27}
W(f,\phi)=\sum_{i=0}^n \cQ_i' \cR_i^2\,.
\end{equation}
\end{proposition}

Suppose now that two (homogeneous) forms $F=F(x,y)$, $\Phi=\Phi(x,y)$
in two variables $x$ and $y$ are given with $N=\deg F=\deg \Phi$.
Set $f=F(x,1)$, $\phi=\Phi(x,1)$ and suppose $\deg f \ge \deg \phi =n$.
Following Sylvester \cite{Sylvester53}, we state
the following quadratic expression of the specialized Jacobian
$J(F,\Phi)$ in terms of the remainders (for the division
of $f$ by $\phi$). Its proof consists into recognizing that
the Jacobian is a Wronskian.

\begin{proposition}\label{P10} We have the following formula for the
Jacobian:
\begin{equation}\label{28}
  \frac{1}{N} J(F,\Phi)\Bigr|_{y=1}=\sum_{i=0}^n \cQ_i' \cR_i^2\,.
\end{equation}
\end{proposition}
\proof By the well-known identity
\begin{equation}\label{29}
NF=x\frac{\partial F}{\partial x} + y\frac{\partial F}{\partial y}\,,
\end{equation}
and Equation (\ref{27}), we have
$$
\sum_{i=0}^n \cQ_i' \cR_i^2=f'\phi-f\phi'
$$
$$
=\frac{\partial F}{\partial x} \cdot \frac{1}{N}
\bigl(x\frac{\partial \Phi}{\partial x}+y\frac{\partial \Phi}{\partial y}
\bigr) \Bigr|_{y=1}
 - \frac{1}{N}\bigl(x\frac{\partial F}{\partial x}
+y\frac{\partial F}{\partial y}\bigr) \frac{\partial \Phi}
{\partial x} \Bigr|_{y=1}
$$
$$
=\frac{1}{N}\bigl(\frac{\partial F}{\partial x}\frac{\partial \Phi}
{\partial y}-\frac{\partial F}{\partial y}\frac{\partial \Phi}{\partial x}
\bigr)\Bigr|_{y=1} = \frac{1}{N} J(F,\Phi)\Bigr|_{y=1}\,.
$$

\medskip

The proposition has been proved.\qed

\section{Bezoutians in terms of the roots of
polynomials}\label{2roo}

In this section, we give some expressions for the Bezoutians of
general pairs of polynomials of degrees $(n\plus 1,n)$ and $(n,n)$,
solely in terms of the roots of the two polynomials.

First we need to introduce {\it multi-Schur functions}. Here are
some basic definitions (for a more detailed account, we refer
the reader to \cite{Lascoux}; see also \cite{LascouxE} and
\cite{LasPrag}).

\begin{definition}\label{D10.01}
By an {\it alphabet} $\A$, we understand a (finite)
multi-set of elements in a commutative ring.
\end{definition}

We shall often identify an alphabet $\A$ with the sum \
$\sum_{a\in \A} a$ \ , and perform the usual algebraic operations on
such elements. We shall give priority to the algebraic notation over
the set-theoretic one.

\begin{definition}\label{D10.1}
Given two alphabets $\A$, $\B$, the {\it complete functions} $S_i(\AB)$
of the difference of the alphabets
are defined by the generating series (with $z$ an extra variable):

\begin{equation}\label{30}
\sum S_i(\AB) z^i =\prod_{b\in \B} (1\moins bz)/\prod_{a\in \A}
(1\moins az)\,.
\end{equation}
\end{definition}

For example, if $\A$ is of cardinality $m$, then
$$
S_m(x\moins\A)=\prod_{a\in \A}(x-a)
$$
is a monic polynomial with the multi-set of roots $\A$.

\begin{definition}\label{D10.2}
Given  $I=(i_1,i_2, \ldots, i_r) \in  \N^r $, and alphabets
$\A_1,\ldots ,\A_r$, $\B_1,\ldots ,\B_r$, the {\it multi-Schur function}
$S_I(\A_1\moins \B_1,\ldots,\A_r\moins \B_r)$ is
\begin{equation}\label{31}
S_I(\A_1\moins\B_1,\ldots, \A_r\moins \B_r) := \Bigl|
     S_{i_k+k-h}(\A_k\moins \B_k) \Bigr|_{1\leq h,k \leq r}  \ .
\end{equation}
\end{definition}

For example,
$$
S_{5,7,2,0,3}(\A-\B,\C-\D,\E-\F,\G-\H,\K-\L)=
$$

\smallskip

$$
=\left|\,\CD
S_5(\A-\B) \ & \ S_8(\C-\D) \ & \ S_4(\E-\F) \ & \ S_3(\G-\H) \ & \ S_7(\K-\L) \\
S_4(\A-\B) & S_7(\C-\D) & S_3(\E-\F) & S_2(\G-\H) & S_6(\K-\L)\\
S_3(\A-\B) & S_6(\C-\D) & S_2(\E-\F) & S_1(\G-\H) & S_5(\K-\L)\\
S_2(\A-\B) & S_5(\C-\D) & S_1(\E-\F) & 1 & S_4(\K-\L)\\
S_1(\A-\B) & S_4(\C-\D) & 1 & 0 & S_3(\K-\L)\\
\endCD\,\right|\,.
$$

\medskip

In case of a multi-Schur function such that
$$
\A_1-\B_1=\cdots=\A_r-\B_r=\AB \,,
$$
we write more compactly $S_I(\AB)$ and call it a {\it Schur function}.

\medskip

In case of a multi-Schur function such that for some $p$,
$$i_1=\cdots =i_p =i\,, \ \  i_{p\plus 1}=\cdots=i_r=j\,,$$
$$\A_1\moins\B_1=\cdots = \A_p\moins\B_p = \A\moins\B\,, \ \
\A_{p\plus 1}\moins\B_{p\plus 1}=\cdots=\A_r\moins\B_r=\C\moins\D\,,$$
we write more compactly
$$
S_{i^p\, ;\, j^{r-p}}(\A\moins\B\, ;\,\C\moins\D)\,.
$$

For example, the multi-Schur function
$$S_{1,1,1,1,5,5,5}(\AB,\AB,\AB,\AB,\C\moins \D,\C\moins \D,\C\moins \D) $$
is written
$$
S_{1^4;5^3}(\AB;\C\moins \D)\,.
$$

We record the following well known property ({\it loc.cit.}):
for a partition $I\in \N^r$ (i.e., \ $I$ is weakly increasing),
\begin{equation}\label{30.9}
S_I(\AB)= (-1)^{|I|}S_J(\B \moins \A)\,,
\end{equation}
where $J$ is the conjugate partition of $I$ (i.e. the consecutive
rows of $J$ are equal to the corresponding columns of $I$).

\begin{definition}\label{D10.3}
Given two alphabets $\A$, $\B$, we set
\begin{equation}\label{31.1}
R(\A,\B):=\prod_{a\in \A,\, b\in \B}(a\moins b)\,.
\end{equation}
\end{definition}

We have (cf., e.g. \cite{Lascoux} (1.4))
\begin{equation}\label{32}
R(\A,\B)= S_{n^m}(\AB)\,,
\end{equation}
where $m=\card(\A)$ and $n=\card(\B)$.

In particular, the polynomial $S_n(x\moins\B)$ equals $R(x,\B)$,
and $R(\A,\B)$ is the resultant of the polynomials $R(x,\A)$
and $R(x,\B)$.

From now on, the alphabets will consist of complex numbers,
and for two alphabets $\A,\B$, $f(x)$ will denote the polynomial
$R(x,\A)$ and $\phi(x)$ will denote $R(x,\B)$.\footnote{We can
assume as well that $\A$ and $\B$ consist of independent variables,
and work over the base field of rational functions in $\A$ and $\B$.
All the results that we shall obtain, remain true in this
situation (and, obviously, the pair $(f,\varphi)$ is general).}

\begin{definition}\label{D10.4}
We say that a pair of alphabets $(\A,\B)$ is {\it general}
if $(f,\varphi)$ is a general pair of polynomials.
\end{definition}

For a pair $(\A,\B)$ of alphabets, we set
$$
\Bez(\A,\B):=\Bez(f,\varphi)\,.
$$

Suppose first that $\A$ and $\B$ form a general pair of alphabets
of cardinalities $n\plus 1$ and $n$.

\begin{proposition}\label{P11} With the above notation,
\begin{eqnarray}\label{33}
\lefteqn
{\Bez(\A,\B) = R(x\plus y,\B){} }
                                          \nonumber\\
& & {}
+\sum_{i=1}^n (-1)^i \frac{S_{1^{n-i};(i\plus 1)^i}(\B\moins x;\B\moins \A)
S_{1^{n-i};(i\plus 1)^i}(\B\moins y;\B\moins \A)}
{S_{i^{i-1}}(\B\moins \A)S_{(i\plus 1)^i}(\B\moins \A)}\,.
\end{eqnarray}
\end{proposition}
\proof
We use the formula for $\Bez(\A,\B)$ from Proposition \ref{P5}.

Second, we invoke the following expressions for successive remainders
as multi-Schur functions:
\begin{equation}\label{33.2}
\cR_i(x)=S_{1^{n-i};(i\plus 1)^i}(\B\moins x;\B\moins \A)
\end{equation}
(cf. \cite{LascouxE}, Theorem 2.6.1 and \cite{Lascoux}, Theorem 3.2.1.)

Third, the coefficients $\alpha_i$ admit the following expressions
in terms of Schur functions in $\A\moins \B$ coming from
\cite{LascouxE} \ (2.7) p.~23
(cf. also \cite{Lascoux} (3.8), pp.~57--58):
\begin{equation}\label {33.1}
\alpha_i=\frac{(-1)^i}{S_{i^{i-1}}(\B\moins \A)S_{(i\plus 1)^i}
(\B\moins \A)}\,.
\end{equation}
(In these references, this expression is given in a slightly different
but equivalent form.)

By combining these three facts, Equation (\ref{33}) follows.
\qed

\smallskip

The polynomial $S_{1^{n-i};(i\plus 1)^i}(\B\moins x;\B\moins \A)$
is, up to sign, a multi-Schur presentation of the $i$th {\it subresultant}
(cf. \cite{LasPrag}, Proposition 2.2). For a vast discussion
of subresultants, see, e.g., \cite{Collins}, \cite{Loos}, \cite{Diaz},
\cite{LasPrag}, and the references there.

\medskip

In a similar way, for a general pair $(\A,\B)$ of alphabets
with the same cardinality $n$, invoking Proposition \ref{P7},
we get the following identity.

\begin{proposition}\label{P12} We have
\begin{equation}\label{34}
\Bez(\A,\B)= \sum_{i=1}^n (-1)^i \frac{S_{1^{n-i};i^i}
(\B\moins x;\B\moins \A)
S_{1^{n-i};i^i}(\B\moins y;\B\moins \A)}
{S_{(i-1)^{i-1}}(\B\moins \A)S_{i^i}(\B\moins \A)}\,.
\end{equation}
\end{proposition}

We now record the following result.

\begin{lemma}\label{L13} Suppose that $\A$ and $\B$ are alphabets
of cardinalities
$m$ and $n$ with $m\ge n$. For $i=1,\ldots,n$, we have the following
congruence:
\begin{equation}\label{35}
\cR_i(x) \equiv S_{i^{m-n+i-1}}(\AB\moins x)\, \phi(x)  \  \mod\ f(x)  \ .
\end{equation}
\end{lemma}
\proof It is a consequence of the explicit expression of
 $\cR_i(x)$ as an element of the ideal generated by
$f(x)$ and $\phi(x)$:
\begin{equation}\label{36}
\cR_i(x) = (-1)^{n-i+1}S_{(m-n+i)^{i-1}}(\B\moins \A\moins x)f(x)
+ S_{i^{m-n+i-1}}(\AB\moins x)\phi(x) \,.
\end{equation}
To prove Equation (\ref{36}), we proceed in the same way as
in \cite{Lascoux} (3.1.5), p.~49. We only leave to the reader
to check that the signs are as stated.
\qed

\medskip

We shall now establish two expressions (\ref{38}) and (\ref{39}),
solely in terms of roots, that are congruent to the above Bezoutians 
modulo the ideal $(f(x),f(y))$.
In Section 8, these expressions will be compared to some 
Christoffel-Darboux kernels and will imply some ``reproducing
congruences" of Bezoutians given in Propositions \ref{P21} and \ref{P24}.

We start with a general pair of alphabets $(\A,\B)$ of cardinalities
$n\plus 1$ and $n$.
It follows from Proposition \ref{P5} and Lemma \ref{L13} that, modulo the
ideal $(f(x), f(y))$,
\begin{equation}\label{37}
\Bez(\A,\B)
\equiv R(x\plus y,\B)
\Bigl(1 + \sum_{i=1}^n \alpha_i S_{i^i}(\AB\moins x)
S_{i^i}(\AB\moins y)\Bigr)\,.
\end{equation}

We now record the following result.

\begin{theorem}\label{T14} For a general pair of alphabets $(\A,\B)$
of cardinalities
$n+1$ and $n$, we have the following congruence modulo the ideal
$(f(x), f(y))$:
\begin{equation}\label{38}
\Bez(\A,\B)
\equiv R(x\plus y,\B)
\Bigl(1 + \sum_{i=1}^n (-1)^i \frac {S_{i^i}(\AB\moins x)
S_{i^i}(\AB\moins y)}{S_{(i-1)^i}(\AB)S_{i^{i+1}}(\AB)} \Bigr)\,.
\end{equation}
\end{theorem}
\proof
We combine Equations (\ref{37}), (\ref{33.1}), and (\ref{30.9}).
\qed

\smallskip

We now pass to a pair of alphabets of the same cardinality.

\begin{theorem}\label{T15} For a general pair of alphabets
$(\A, \B)$ of the same
cardinality $n$, we have the following congruence modulo
the ideal $(f(x), f(y))$:
\begin{equation}\label{39}
\Bez(\A,\B) \equiv R(x\plus y,\B) \Bigl(\frac{1}{S_1(\AB)}
+ \sum_{i=1}^{n-1}(-1)^i \frac{S_{(i\plus 1)^i}(\AB\moins x)S_{(i\plus 1)^i}
(\AB\moins y)}
{S_{i^i}(\AB) S_{(i\plus 1)^{i+1}}(\AB)} \Bigr)\,.
\end{equation}
\end{theorem}
\proof
We shall deduce the present case from the one of alphabets
of cardinalities
$(n,n\moins 1)$.
For this purpose, suppose that
$$
\psi(x):=S_{n-1}(x\moins\C)
$$
is the monic remainder of $\phi(x)$ modulo $f(x)$, i.e.
\begin{equation}\label{40}
\phi(x)=f(x) + S_1(\AB) \psi(x) \,.
\end{equation}
Then the pair $(\A,\C)$ is general.
We have, by combining Equations (\ref{40}) and (\ref{9})
(note that $S_1(\AB)$ is a scalar),
\begin{equation}\label{41}
\Bez(\A,\B)= \Bez(\A,\C) S_1(\AB)\,.
\end{equation}
Thanks to Equation ({\ref{40}), the remainders of $f(x)$ by
$\psi(x)$ coincide
with those of $f(x)$ by $\phi(x)$, up to powers of $u:=S_1(\AB)$.
We get from Equation (\ref{36}), equalizing degrees by an appropriate
power of $u$,
$$
S_{(i\plus 1)^i}(\AB\moins x) \phi(x)  \equiv
  u^{i\plus 1}\,  S_{(i\plus 1)^i}(\A\moins\C\moins x) \psi(x)
\  \mod\ f(x)  \ .
$$

Similarly, comparing the top coefficients of
$S_{1^{n-i};\, i^i}(\A\moins x;\, \AB)$ and\\
$S_{1^{n-i};\, (i-1)^i}(\A\moins x;\, \A\moins \C)$,
we infer
$$
S_{i^i}(\A\moins \B) = u^i S_{ (i-1)^i}(\A\moins \C) \ .
$$
Thus, using Equation (\ref{41}), the congruence (\ref{38}):
\begin{multline*}
\Bez(\A,\C)\equiv \psi(x)\psi(y) \Bigl(1-\frac{S_1(\A\moins\C\moins x)
S_1(\A\moins\C\moins y)}{S_{11}(\A\moins\C)}     \\
    + \  \frac{S_{22}(\A\moins\C\moins x)S_{22}(\A\moins\C\moins y)}
{S_{11}(\A\moins\C)S_{222}(\A \moins\C)} -\cdots  \Bigr)
\end{multline*}
becomes
\begin{multline*}
\Bez(\A,\B)\equiv R(x\plus y,\B)\Bigl(\frac{1}{S_1(\AB)} -
\frac{S_2(\AB\moins x)S_2(\AB\moins y)}{S_1(\AB)S_{22}(\AB)}  \\
 +\  \frac{S_{33}(\AB\moins x)S_{33}(\AB\moins y)}{S_{22}(\AB)S_{333}(\AB)}
- \cdots \Bigr)\,.
\end{multline*}

This proves the theorem.\qed

Note that the congruences (\ref{38}) and (\ref{39}) become equalities,
when specializing $x$ and $y$ to roots of $f$.

\section{Euclidean remainders versus orthogonal polynomials}\label{2ort}

In this section, we investigate two families of orthogonal
polynomials (we refer to \cite{Brezinski1}, \cite{Brezinski2}, and
\cite{Szego} for basic information on orthogonal polynomials,
needed in the present section).
The first family comes from the Euclidean
algorithm for division of a pair of polynomials
of degrees $(n\plus 1,n)$, and the second one comes from division
of a pair of polynomials of degrees $(n,n)$.

Assume first that $(\A,\B)$ is a pair of alphabets
of cardinalities $n+1$ and $n$ (and follow the convention of the
previous section). One finds in the book of Brioschi \cite{Brioschi1},
p.~167, the following identities (we invoke the notation of Section 2
and write $\A \moins a$ for $\A \setminus \{a \}$).
For a fixed $i$,

\begin{equation}\label{42}
\sum_{a\in\A}\cD_i(a) \cN_i(a) \frac{\phi(a)}{ R(a,\A \moins a)} =0\,,
\end{equation}
\begin{equation}\label{43}
\sum_{a\in\A}\cD_i(a) \cD_{i-1}(a) \frac{\phi(a)}{ R(a,\A \moins a)} =0\,,
\end{equation}
\begin{equation}\label{44}
\sum_{a\in\A} \cD_i^2(a) \cQ_{i}(a) \frac{\phi(a)}{ R(a,\A \moins a)} =0\,.
\end{equation}
These three identities make use of the following functional on $\C[x]$:
\begin{equation}\label{45}
\mu: g(x) \mapsto
\mu\bigl(g(x)  \bigr) :=
 \sum_{a\in\A}  g(a) \frac{\phi(a)}{R(a,\A\moins a)} \ .
\end{equation}
The functional $\mu$ is an avatar of the Lagrange interpolation
in the points $a\in\A$ (cf., e.g., \cite{Brezinski1} and \cite{Lascoux}).
It is characterized by the fact that it sends each
$x^i$, $i\in \N$, onto the complete function $S_i(\AB)$.

Given a linear functional, it is known how to write an orthogonal
basis in terms of \emph{moments} (cf. \cite{Szego} and
\cite{Brezinski1}).
These expressions are still valid in the case of
our functional $\mu$,
restricted to the space of polynomials of degree $\leq n$, and the
following proposition is a rewriting, in terms of Schur functions,
of the classical expressions of orthogonal polynomials
in terms of Hankel determinants involving moments.

\begin{proposition}\label{P16}
The Schur polynomials
\begin{equation}\label{46}
P_k=S_{k^k}(\AB\moins x)\,,
\end{equation}
$k=0,1,\ldots,n$, form a unique (up to normalization) orthogonal basis
with respect to the functional $\mu$,
of polynomials of respective degrees $0,1,\ldots,n$.
\end{proposition}
(For the reader's convenience, we recall a proof in Appendix \ref{A1}.)

By combining the congruence (\ref{4}) with Lemma \ref{L13}, 
we thus get the following basic result.

\begin{proposition}\label{P17}
For any fixed $0\le i < j \le n$, the following relations hold:
\begin{equation}\label{47}
\sum_{a\in\A}  a^i \cD_j(a) \frac{\phi(a)}{R(a,\A\moins a)} =0  \,,
\end{equation}
\begin{equation}\label{48}
  \sum_{a\in\A}   \cD_i(a) \cD_j(a) \frac{\phi(a)}{R(a,\A\moins a)} =0 \,.
\end{equation}
\end{proposition}

Brioschi's relation (\ref{43}) is just a particular case
of Equation (\ref{48}) with consecutive $i$ and $j$.

We can replace in Equation (\ref{42}) the polynomial $\cN_i$,
which is of degree $i-1$,
by any polynomial of degree strictly less than $i$, because
the set $\{\cD_i \}$ forms an orthogonal basis.
Since by Equation (\ref{5}) we have
$$
\cQ_{i} \cD_i = \cD_{i-1} + \cD_{i\plus 1}\,,
$$
Equation ({\ref{44}) results from the orthogonality of
$\cD_i$ with $\cD_{i\plus 1}$, and with $\cD_{i-1}$.

\begin{remark}\label{R18}
Let $f$ be a monic polynomial of degree $n\plus 1$.
Suppose that $(n\plus 1) \phi(x)$ is the derivative of $f(x)$
In \cite{Brioschi2},
Brioschi gives the following relations:
for fixed $0\le i<j \le n$,
\begin{equation}\label{49}
    \sum_{a\in\A}   a^i \cD_j(a)  =0\,.
\end{equation}
The sum looks different from Equation (\ref{47}) because the term
\begin{equation}\label{50}
\frac{\phi(a)}{R(a,\A\moins a)}= \frac{\phi(a)}{f'(a)}=\frac{1}{n\plus 1}
\end{equation}
can be erased.
\end{remark}

We infer from Proposition \ref{P17} and Equation (\ref{4}):

\begin{corollary}\label{C19}
For any fixed $0\le i < j \le n$, the following relations hold:
\begin{equation}\label{51}
\sum_{a\in\A}  a^i \cR_j(a) \frac{1}{\phi(a) R(a,\A\moins a)} =0  \,,
\end{equation}
\begin{equation}\label{52}
\sum_{a\in\A}   \cR_i(a) \cR_j(a) \frac{1}{\phi(a)
R(a,\A\moins a)} =0 \,.
\end{equation}
\end{corollary}

\begin{remark}\label{R20}
Note that
\begin{equation}\label{53}
\frac{f'(x)}{f(x)} =  \sum_{a\in\A}  \frac{1}{x-a} \ ,
\end{equation}
so that the case treated by Sylvester and Brioschi is the case of
orthogonal
polynomials for a discrete uniform measure, in relation with Lagrange
interpolation. As a matter of fact, it was also the starting point of
Chebyshev (cf. \cite{Chebyshev}, and \cite{Brezinski2} for an account
to Chebyshev's work on orthogonal polynomials). Taking a more general
$\phi(x)$ of degree $n$, we have similarly
\begin{equation}\label{54}
 \frac{\phi(x)}{f(x)}=   \sum_{a\in\A}  \phi(a)\,
            \frac{R(x,\A\moins a)}{R(a,\A\moins a)}  \ ,
\end{equation}
thanks to Lagrange interpolation, and this time the measure
is no more uniform, but still concentrated in the points of $\A$.
\end{remark}

We shall now investigate the Christoffel-Darboux kernels associated
with the just studied family of orthogonal polynomials and another one
associated with a pair of monic polynomials of the same degree.
We shall derive ``reproducing'' congruences for the related Bezoutians.

We first recall the following definition (cf. \cite{Brezinski1},
\cite{Szego}).

\begin{definition}\label{D20.1}
Given a family of orthogonal polynomials $P_0(x),\ldots, P_n(x)$
associated with a linear functional $\mu$, then the
{\it Christoffel-Darboux kernel} is:
\begin{equation}\label{55}
K(x,y):=  \sum_{i=0}^n P_i(x) P_i(y) / \mu\bigl( P_i(x)^2\bigr)\,.
\end{equation}
\end{definition}

Suppose that $(\A,\B)$ is a pair of alphabets of cardinalities
$(n\plus 1,n)$, and let $\mu$ be the functional defined
in Equation (\ref{45}). The following result is a translation
to Schur functions of a well-known {\it normalization} property
of orthogonal polynomials.

\begin{lemma}\label{L100} With the above notation,
\begin{equation}\label{55.1}
\mu\bigl(S_{i^i}(\AB\moins x)^2\bigr)=
(-1)^i S_{(i\moins 1)^i}(\AB)S_{i^{i+1}}(\AB)\,.
\end{equation}
\end{lemma}
(For the reader's convenience, we recall a proof in Appendix \ref{A1}.)

Assume now, in addition, that the pair $(\A,\B)$ is general.
By combining the congruence (\ref{38}), Proposition \ref{P16} and
Lemma \ref{L100}, we infer the following congruence modulo the
ideal $(f(x),f(y))$:

\begin{equation}\label{56}
 \Bez(\A,\B) \equiv \phi(x)\phi(y)\, K(x,y)\,.
\end{equation}

This congruence suggests that the Bezoutian $\Bez(\A,\B)$ has a reproducing
property.
\begin{proposition}\label{P21}
Given a general pair of alphabets $(\A,\B)$ of respective cardinalities
$n\plus 1$ and $n$, and a polynomial $g(x)$, we have
\begin{equation}\label{57}
\mu\bigl( g(x)\, \Bez(\A,\B)\bigr)  \equiv \phi(y)^2\,  g(y)
          \ \mod f(y) \,.
\end{equation}
\end{proposition}
\proof \ Since 
$$
\Bez(\A,\B)=\partial_{xy}f(x) \phi(y)+
f(y)\partial_{xy}\phi(y)
$$
(cf. \cite{Lascoux} (7.1.9)), we have
\begin{equation}
\Bez(\A,\B) \equiv \phi(y)\, \partial_{xy} f(x) \ \ \hbox{mod} \ \  f(y)\,.
\end{equation}
Note that $\partial_{xy} f(x)=S_n(x+y-\A)$. By using \cite{Lascoux} (1.7.5),
(2.3.1), we have for any $i\ge 0$, modulo $f(y)$,
\begin{multline*}
\mu \bigl(x^i\, \phi(y)\, S_n(x+y-\A)\bigr) \\
=\phi(y)[S_{i+n}(\A\moins \B)+S_{i+n-1}(\A\moins \B)S_1(y\moins \A)+
\cdots+S_i(\A\moins \B)S_n(y\moins \A)] \\
\equiv \phi(y)S_{i+n}(\A-\B+y-\A)= \phi(y)^2 \, y^i\,.\qed
\end{multline*}

\smallskip

Finally, consider a general pair of alphabets $(\A,\B)$ of
the same cardinality $n$. We state only results and omit their proofs
which are similar to the ones in the previous case.
We recall (cf. Lemma \ref{L13}):
\begin{lemma}\label{L22} For $i=1,\ldots, n$,
\begin{equation}\label{58}
\cR_i(x) \equiv S_{i^{i-1}}(\AB\moins x)\, \phi(x)  \  \mod\ f(x)  \ .
\end{equation}
\end{lemma}

\begin{proposition}\label{P23} The Schur functions \
$S_{k^{k-1}}(\AB\moins x)$\,,
$k=1,\ldots,n$\,, form a unique (up to normalization) orthogonal basis
of polynomials of respective degrees $0,1,\ldots,n\moins 1$,
with respect to the following functional $\nu$ on $\C[x]$:
\begin{equation}\label{59}
\nu: x^i \mapsto S_{i\plus 1}(\AB)\,, \ \ \ \ \ \ i\in \N \,.
\end{equation}
\end{proposition}

\begin{proposition}\label{P24} For any polynomial $g(x)$, the following
congruence holds:
\begin{equation}\label{60}
\nu(g(x) \Bez(\A,\B)) \equiv \phi(y)^2\,  g(y)
             \ \mod f(y) \,.
\end{equation}
\end{proposition}


\section{Appendix: proofs of Proposition \ref{P16} and Lemma \ref{L100}}
\label{A1}

Proposition \ref{P16} can be proved as follows.
Fix $k=1,\ldots,n$. We have the following equality:
\begin{equation}\label{61}
S_{k^k}(\AB\moins x)x^i = S_{k^k;\, i}(\AB;\, x)\,.
\end{equation}
(We can pass from the right-hand side of Equation (\ref{61}) to its
left-hand side
by multiplying the $i$th row by $x$
and subtracting it from the $(i\moins 1)$st row, successively for
$i=2,3,\ldots,k\plus 1$.)

The functional $\mu$ acts only on the last column of the determinant
$$S_{k^k;\, i}(\AB;\, x)\,,$$ and sends it onto
$$S_{k^k;\, i}(\AB;\, \AB)\,.$$
When $i<k$, this last determinant $S_{k^k,i}(\AB)$ vanishes,
having two identical columns. In other words, the polynomial
$P_k=S_{k^k}(\AB\moins x)$ is orthogonal to $x^0,\ldots, x^{k-1}$.
This proves the proposition.

\bigskip

Lemma \ref{L100} admits the following justification.
Using Proposition \ref{P16} and appropriate Laplace expansion,
we have
$$
\mu\bigl(S_{i^i}(\AB\moins x) S_{i^i}(\AB\moins x)\bigr)
=\mu\bigl(S_{i^i}(\AB\moins x)(-x)^i S_{(i-1)^i}(\AB)\bigr)
$$
$$
= (-1)^i S_{(i-1)^i}(\AB)S_{i^{i+1}}(\AB)\,.
$$
This proves the lemma.

\smallskip

For more on symmetric function interpretation of orthogonal
polynomials, consult \cite{Lascoux}, Chap.~8.

\bigskip

\noindent
{\bf Acknowledgment.}
The authors wish to thank the referees for a number of comments
and suggestions that helped to ameliorate the exposition of the paper.


\begin{thebibliography}{1}

\bibitem{Bezout0} M. B\'ezout.
\emph{Recherches sur le degr\'e des \'equations r\'esultants de
l'evanouissement des inconnues, et sur les moyens qu'il convient
d'employer pour trouver ces \'equations},
M\'em. Acad. Roy. Sci. Paris, (1764), 288--338.

\bibitem{Bezout} M. B\'ezout.
\emph{Th\'eorie g\'en\'erale des \'equations alg\'ebriques},
Paris, (1779).

\bibitem{Brezinski1} C. Brezinski.
\emph{Pad\'e-type approximation and general orthogonal polynomials},
Birkh\"auser (1980).

\bibitem{Brezinski2} C. Brezinski.
\emph{History of continued fractions and Pad\'e approximants},
Springer (1991).

\bibitem{Brioschi1} F. Brioschi. \emph{Th\'eorie des d\'eterminants},
 appendix to the French edition, Paris, Mallet-Bachelier (1856).

\bibitem{Brioschi2} F. Brioschi. \emph{Sur les fonctions de Sturm},
Nouvelles Annales {\bf 13} (1854), 71--80.

\bibitem{Cayley} A. Cayley.
\emph{ Note sur la m\'ethode d'\'elimination de B\'ezout},
J. Reine Angew. Math. {\bf 53} (1857), 366--367.

\bibitem{Chebyshev} P. Chebyshev.
\emph{Oeuvres, 2 vols}, A. Markoff, N. Sonin eds.,
Chelsea, New York.

\bibitem{Collins} G. E. Collins.
\emph{Subresultants and reduced polynomial remainder sequences},
J. Assoc. Comp. Mach.  {\bf 14}, 128--142.

\bibitem{Diaz} G. M. Diaz-Toca, L. Gonz\'alez-Vega.
\emph{Various new expressions for subresultants and their
applications}, preprint, 25 pp.,
to appear in: Applicable Algebra in Engineering, Communication
and Computing (2004).

\bibitem{Euler} L. Euler.
\emph{Introductio to Analysin Infinitorum},
Tom 2, Lausanne (1748).

\bibitem{Fuhrmann} P. A. Fuhrmann.
\emph{A polynomial approach to linear algebra},
Springer, New-York (1996).

\bibitem{Gonzalez} L. Gonz\'alez-Vega.
\emph{Solving the real root counting problem via Bezoutians}
Preprint {\bf 12} (1993), 17 pp., Department of Mathematics, Statistics,
and Computation, University of Cantabria.

\bibitem{Heinig} G. Heinig, U. Jungnickel.
\emph{On the Routh-Hurwitz and Schur-Cohn problems for matrix
polynomials and generalized Bezoutians},
Math. Nachr. {\bf 116} (1984), 185--196.

\bibitem{Heinig0.1} G. Heinig, U. Jungnickel.
\emph{Hankel matrices},
Linear Algebra Appl. {\bf 76} (1986), 121--135.

\bibitem{Heinig1} G. Heinig, F. Hellinger.
\emph{On the Bezoutian structure of the Moore-Penrose inverses
of Hankel matrices},
SIAM J. Matrix Anal. Appl. {\bf 14} (1993), 629--645.

\bibitem{Helmke} U. Helmke, P.A. Fuhrmann.
\emph{Bezoutians}, Linear Algebra Appl. {\bf 122/123/124} (1989),
1039--1097.

\bibitem{Hermite} C. Hermite.
\emph{Oeuvres}, ed. E. Picard, Gauthier-Villars, Paris (1908).

\bibitem{Jacobi} C. G. J. Jacobi.
\emph{De eliminatione variablis e duabus aequatione algebraicis},
J. Reine Angew. Math. {\bf 15} (1836), 101--124.

\bibitem{Jury} I. Jury, Eliahou.
\emph{The roles of Sylvester and Bezoutian matrices in the historical
study of stability of linear discrete-time systems},
IEEE Trans. Circuits Systems I Fund. Theory Appl. {\bf 45} (1998),
1233--1251.

\bibitem{Lascoux} A. Lascoux.
\emph{Symmetric functions and combinatorial operators on polynomials},
CBMS/AMS Lectures Notes {\bf 99}, Providence (2003).

\bibitem{LascouxE} A. Lascoux.
\emph{Euclid algorithm and related topics},
preprint, 77 pp., Nankai University, Tjanjin (2003). Available on:

\noindent
{\tt http://phalanstere.univ-mlv.fr/$\sim$al}

\bibitem{LasPrag} A. Lascoux, P. Pragacz.
\emph{Double Sylvester sums for subresultants and multi-Schur functions},
J. Symb. Comp. {\bf 35} (2003), 689--710.

\bibitem{Loos} R. G. K. Loos
\emph{Generalized polynomial remainder sequences}, In: Buchberger et al.
(Eds.), Computer Algebra, Symbolic and Algebraic Computation,
Springer-Verlag, Vienna, New York, pp.~115--137.

\bibitem{Muir} T. Muir.
\emph{History of determinants},
5 volumes, Dover reprints (1960).
c
\bibitem{Szego} G. Szeg\"o.
\emph{Orthogonal polynomials},
Amer. Math. Soc., Providence (1939).

\bibitem{Sylvester0} J. J. Sylvester.
\emph{The collected mathematical papers},
Cambridge (1904) \ (see also Chelsea reprint, New York (1983)).

\bibitem{Sylvester53} J. J. Sylvester.
\emph{ A theory of the syzygetic relations of two rational integral
functions}, Phil. Trans. Royal Soc. London vol. CXLIII, Part III (1853)
407--548. Also: \emph{The collected mathematical papers}, Cambridge
(1904), vol.~1, paper 57, pp.~429--586.

\bibitem{Wimmer0} H. K. Wimmer.
\emph{Bezoutians of polynomial matrices and their
generalized inverses},
Linear Algebra Appl. {\bf 122/123/124} (1989), 475--487.

\bibitem{Wimmer} H. K. Wimmer.
\emph{On the history of the Bezoutian and the resultant matrix},
Linear Algebra Appl. {\bf 128} (1990), 27--34.

\bibitem{Yu} J. T. Yu.
\emph{An extremal problem for sets: a new approach via Bezoutians},
J. Combin. Theory Ser. A {\bf 62} (1993), 170--175.

\end{thebibliography}



\end{document}
