Subresultants of Several Univariate Polynomials in Newton Basis¹¹footnotemark: 1

Weidong Wang weiyang020499@163.com Jing Yang yangjing0930@gmail.com [

Abstract

In this paper, we consider the problem of formulating the subresultant polynomials for several univariate polynomials in Newton basis. It is required that the resulting subresultant polynomials be expressed in the same Newton basis as that used in the input polynomials. To solve the problem, we devise a particular matrix with the help of the companion matrix of a polynomial in Newton basis. Meanwhile, the concept of determinant polynomial in power basis for formulating subresultant polynomials is extended to that in Newton basis. It is proved that the generalized determinant polynomial of the specially designed matrix provides a new formula for the subresultant polynomial in Newton basis, which is equivalent to the subresultant polynomial in power basis. Furthermore, we show an application of the new formula in devising a basis-preserving method for computing the gcd of several Newton polynomials.

keywords:

subresultant , Newton polynomial , companion subresultant , companion matrix , determinant polynomial

^†^†journal: Journal of Symbolic Computation^t1^t1footnotetext: This article is part of the volume titled “Computational Algebra and Geometry: A special issue in memory and honor of Agnes Szanto”.

url]https://jyangmath.github.io/

\affiliation

[1]organization=SMS–HCIC–School of Mathematics and Physics,
Center for Applied Mathematics of Guangxi,
Guangxi Minzu University,addressline=
188 Daxue East Road, city=Nanning, postcode=530006, state=Guangxi, country=China

1 Introduction

Resultant theory is a fundamental tool in computer algebra with numerous applications (e.g., Gonzalez-Vega et al. (1998); Szanto (2008); Imbach et al. (2017); Perrucci & Roy (2018); Roy & Szpirglas (2020)), such as polynomial system solving (e.g., Kapur et al. (1994); Wang (1998, 2000)) and quantifier elimination (e.g., Arnon et al. (1984); Collins & Hong (1991)). Due to its importance, extensive research has been carried out both in theoretical and practical aspects on resultants, subresultants, and their variants (see Sylvester (1853); Collins (1967); Barnett (1971); Lascoux & Pragacz (2003); Terui (2008); Bostan et al. (2017); Hong & Yang (2021); Cox & D’Andrea (2021)). One of the essential topics in resultant theory is the representation of resultants and subresultant polynomials. In resultant theory, the input polynomials are typically assumed to be expressed in power basis (i.e., standard basis), and so are the output subresultant polynomials. However, with the rising popularity of basis-preserving algorithms in various applications (see Farouki & Rajan (1987); Goodman & Said (1991); Carnicer & Peña (1993); Delgado & Peña (2003); Bini & Gemignani (2004); Amiraslani et al. (2006); Cheng & Labahn (2006); Marco & Martínez (2007); Aruliah et al. (2007)), people are more and more interested in resultants and subresultants for polynomials in non-standard basis. Among these bases, Newton basis is widely used and has many applications. A typical application is to formulate the interpolating polynomials. One may wonder whether the resulting Newton polynomials have common zeros or what their gcd looks like. These problems can be reduced to the computation of resultant/subresultants. One straightforward way is changing from Newton basis to power basis, then carrying out the required computations, and changing back to the initial basis afterwards. However, due to the numerical instability caused by basis transformation, it is desired to have a basis-preserving algorithm for computing resultant/subresultants that does not involve basis transformation. In other words, given two polynomials in Newton basis, we want to compute their gcd in the given basis, where basis transformation is not used. In this paper, we will focus on the problem of formulating a type of subresultant polynomials for multiple univariate polynomials in Newton basis, and we aim to ensure that the formulated subresultant polynomials are: (1) expressed in the given Newton basis and (2) the same as those formulated in power basis after expansion.

Subresultant polynomials are usually expressed in the form of determinant polynomials. This choice is because determinant polynomials exhibit nice algebraic properties, greatly facilitating theoretical developments and subsequent practical applications. Consequently, people have developed various formulas in the form of determinant polynomials for subresultant polynomials in power basis, including the Sylvester type (Sylvester (1853); Li (2006)), Bézout type (Hou & Wang (2000)), Barnett type (Barnett (1983); Diaz-Toca & Gonzalez-Vega (2002)), and other variants (Diaz-Toca & Gonzalez-Vega (2004)). Following this approach, we develop a determinant polynomial formula for subresultant polynomials of multiple univariate polynomials in Newton basis.

To develop the formula of subresultant polynomial for multiple univariate polynomials in Newton basis, we use the extended version of the well-known companion matrix of a polynomial in power basis to Newton basis. This extension is achieved based on the understanding that the companion matrix of a polynomial in power basis with degree $n$ can represent an endomorphism of the spaces consisting of polynomials with degree less than $n$ defined by the multiplication of the involved variable in the power basis. The companion matrix of a polynomial in Newton basis possesses a similar property. Their only difference is that we use Newton basis this time. This underlying idea has been conceived in Diaz-Toca & Gonzalez-Vega (2002, 2004). In Diaz-Toca & Gonzalez-Vega (2002), the authors reformulated Barnett’s theorem with respect to various non-standard basis and demonstrated that the reformulated matrices partially share several nice properties with Barnett’s matrices, e.g., their ranks are equal, and the indices for independent rows are the same. However, it remains unsolved how to express the gcd of the input polynomials in the given basis. In Diaz-Toca & Gonzalez-Vega (2004), the authors considered a specific Newton basis with nodes being the roots of one polynomial and constructed the subresultant polynomials of two polynomials. The constraint on nodes is removed in the formula presented in this paper.

With this essential concept of companion matrix extended to Newton basis, we proceed to construct a matrix that will be used to formulate the subresultant polynomials by utilizing a similar approach as adopted by Barnett (1969, 1970, 1971), Diaz-Toca & Gonzalez-Vega (2004), and Hong & Yang (2021). To express the subresultant polynomials in terms of the given Newton basis, we generalize the concept of determinant polynomial in power basis to that in Newton basis. This generalization allows the formulation of subresultant polynomials in the provided Newton basis. Furthermore, as an application of the new formula, we devise a method for computing the gcd of several numerical Newton polynomials, which does not involve basis transformation.

Compared with previous related works, the newly developed formula of subresultant polynomials for multiple polynomials has the following features. First, it can be viewed as a generalization of the Barnett-type subresultant polynomials by extending the basis from power basis to Newton basis. Second, it also serves as a generalization of the formula of subresultant polynomial based on roots proposed in Hong (2002) and Diaz-Toca & Gonzalez-Vega (2004) in the sense that the previous formulas use a specific Newton basis with nodes to be the roots of one of the given polynomials. In contrast, the Newton basis employed in this paper allows for an arbitrary choice of nodes. As there are infinitely many possibilities for the nodes of the Newton basis, an infinite number of formulas of subresultant polynomials can be obtained. Furthermore, the formula of subresultant polynomials developed in this paper applies to the multi-polynomial case.

The paper is structured as follows. Section 2 briefly reviews the subresultant theory for two polynomials. It is followed by a formal statement of the problem to be addressed by the current paper in Section 3. The main result of the paper provides a solution to the problem raised in the previous section and is elaborated in Section 4. The correctness of the main result is verified in Section 5. In Section 6, we show an application of the main result in computing the gcd of several numerical Newton polynomials. The paper is concluded in Section 7 with further remarks.

2 Preliminaries

This section reviews some well-known results in subresultant theory for two univariate polynomials. It is noted that the involved polynomials are all expressed in power basis. Throughout the paper, we assume $\mathbb{F}$ is the fraction field of an integral domain and $\mathbb{F}_{n}[x]$ is the vector space consisting of polynomials in $x$ with degree no greater than $n$ .

2.1 Sylvester subresultant polynomial of two polynomials

Subresultant polynomials for two univariate polynomials are usually defined with the minors of their Sylvester matrix. Thus we start by recalling the definition of Sylvester matrix below.

Consider $A,B\in\mathbb{F}[x]$ with the following form:

	$\displaystyle A(x)$	$\displaystyle=a_{n}x^{n}+\cdots+a_{1}x+a_{0},$		(1)
	$\displaystyle B(x)$	$\displaystyle=b_{m}x^{m}+\cdots+b_{1}x+b_{0},$		(2)

where $a_{n}b_{m}\neq 0$ , $a_{i},b_{j}\in\mathbb{F}$ ( $0\leq i\leq n,0\leq j\leq m$ ). Then the Sylvester matrix of $A$ and $B$ with respect to $x$ is defined as

{\rm{Syl}}(A,B):=\underbrace{\left[\begin{array}[]{*{20}{l}}{{a_{n}}}&\cdots&{{a_{0}}}&{}\\ &\ddots&&\ddots&{}\\ &{}&{{a_{n}}}&\cdots&{{a_{0}}}\\ \hline\cr{{b_{m}}}&\cdots&{{b_{0}}}&{}\\ &\ddots&&\ddots&{}\\ &{}&{{b_{m}}}&\cdots&{{b_{0}}}\end{array}\right]}_{(m+n)\ \text{columns}}\begin{array}[]{*{20}{l}}{\left.{\begin{array}[]{*{20}{c}}{}\hfil\\[1.0pt] {}\hfil\\[1.0pt] {}\hfil\end{array}}\right\}}{}~{}m\ \text{rows}\\[15.0pt] {\left.{\begin{array}[]{*{20}{c}}{}\hfil\\[1.0pt] {}\hfil\\[1.0pt] {}\hfil\end{array}}\right\}}{}~{}n\ \text{~{}rows}\end{array}

To define the subresultant polynomial of $A$ and $B$ , we introduce the concept of determinant polynomial (e.g., see Mishra (1993, Definition 7.5.1)).

Definition 1 (Determinant polynomial).

Given $M\in\mathbb{F}^{(n-k)\times n}$ , the determinant polynomial of $M$ in terms of $x$ is defined as

\operatorname*{detp}M:=\sum_{i=0}^{k}\det M_{i}\cdot x^{i},

where $M_{i}$ is the submatrix of $M$ consisting of the first $n-k-1$ columns and the $(n-i)$ th column.

With the above settings, the definition of subresultant polynomials of two univariate polynomials is presented below.

Definition 2 (Subresultant polynomial, Diaz-Toca & Gonzalez-Vega (2004)).

For $0\leq k\leq\min(m,n)-1$ , the $k$ th subresultant polynomial of $A$ and $B$ with respect to $x$ is defined as

S_{k}(A,B):=\operatorname*{detp}{\rm Syl}_{k}(A,B),

where ${\rm Syl}_{k}(A,B)$ is the submatrix of ${\rm Syl}(A,B)$ obtained by deleting the last $k$ rows from the upper block (consisting of the first $m$ rows) and the lower block (consisting of the last $n$ rows) respectively and the last $k$ columns.

Example 3.

Given

A(x)=4x^{3}-8x^{2}+5x-1,\quad B(x)=2x^{3}+3x^{2}-1,

we have

{\rm Syl}_{1}(A,B)=\begin{bmatrix}4&-8&\ \ 5&-1&\\ &\ \ 4&-8&\ \ 5&-1\\ 2&\ \ 3&\ \ 0&-1&\\ &\ \ 2&\ \ 3&\ \ 0&-1\\ \end{bmatrix}

and the first subresultant polynomial of $A$ and $B$ with respect to $x$ is

	$\displaystyle S_{1}(A,B)$	$\displaystyle={\rm detp}\ {\rm Syl}_{1}(A,B)$
		$\displaystyle=\det\begin{bmatrix}4&-8&\ \ 5&-1\\ &\ \ 4&-8&\ \ 5\\ 2&\ \ 3&\ \ 0&-1\\ &\ \ 2&\ \ 3&\ \ 0\\ \end{bmatrix}x+\det\begin{bmatrix}4&-8&\ \ 5&\\ &\ \ 4&-8&-1\\ 2&\ \ 3&\ \ 0&\\ &\ \ 2&\ \ 3&\ -1\\ \end{bmatrix}$
		$\displaystyle=576\,x-288.$

Apart from Sylvester subresultant polynomials, people also developed various alternative expressions for subresultant polynomials. Depending on the types of matrices used to formulate them, we can categorize the expressions into Sylvester type, Bézout type, Barnett type, etc. (see Diaz-Toca & Gonzalez-Vega (2004) for more details).

2.2 Companion matrix and Barnett matrix

Companion matrix has been proved to be a useful tool for formulating subresultant polynomials of two univariate polynomials (see Diaz-Toca & Gonzalez-Vega (2004)). We give a brief review of the companion matrix below.

Given a polynomial $A(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots+a_{0}\in\mathbb{F}[x]$ , the companion matrix of $A(x)$ is defined as an $n\times n$ matrix $\Delta_{A}$ of the following form

{\Delta_{A}}=\left[{\begin{array}[]{*{20}{c}}0&0&\cdots&0&{-{a_{0}}}\\ a_{n}&0&\cdots&0&{-{a_{1}}}\\ 0&a_{n}&\cdots&0&{-{a_{2}}}\\ \vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&\cdots&a_{n}&{-{a_{n-1}}}\end{array}}\right]_{n\times n}

With the help of the companion matrix above, we may construct the Barnett matrix of two univariate polynomials, whose minors give the coefficients of subresultant polynomials.

More explicitly, let $A$ and $B$ be as in (1) and (2), respectively. Then $B(\Delta_{A}/a_{n})$ is the Barnett matrix of $A$ and $B$ with respect to $x$ . Furthermore, let $B_{k}$ be the submatrix of $B(\Delta_{A}/a_{n})$ obtained by deleting the last $k$ columns and $B_{ik}$ be the submatrix of $B_{k}$ consisting of the last $n-k-1$ rows and the $(i+1)$ th row. Then by Diaz-Toca & Gonzalez-Vega (2004, Theorem 2.2-1),

S_{k}(A,B)=a_{n}^{m-k}\sum_{i=0}^{k}\det B_{ik}\cdot x^{i}

Example 4.

Consider the polynomials $A$ and $B$ in Example 3. Then with some calculations, we obtain

\Delta_{A}=\left[\begin{array}[]{rrr}0&0&1\\ 4&0&-5\\ 0&4&8\end{array}\right],\quad B(\Delta_{A}/a_{n})=\left[\begin{array}[]{rrr}-{\dfrac{1}{2}}&{\dfrac{7}{4}}&{\dfrac{23}{8}}\\[8.0pt] -{\dfrac{5}{2}}&-{\dfrac{37}{4}}&-{\dfrac{101}{8}}\\[8.0pt] 7&{\dfrac{23}{2}}&{\dfrac{55}{4}}\end{array}\right]

It follows that

S_{1}(A,B)=4^{3-1}\left(\det\left[\begin{array}[]{rrr}-{\dfrac{5}{2}}&-{\dfrac{37}{4}}\\[8.0pt] {7}&{\dfrac{23}{2}}\end{array}\right]x+\det\left[\begin{array}[]{rrr}-{\dfrac{1}{2}}&{\dfrac{7}{4}}\\[8.0pt] {7}&{\dfrac{23}{2}}\end{array}\right]\right)=576\,x-288

2.3 Subresultant polynomial in roots

The subresultant polynomials in the previous subsections are all expressed in coefficients. In Hong (1999, 2002) and D’Andrea et al. (2006), Hong et al. developed an equivalent formula in roots for subresultant polynomials, which inspires the authors in Hong & Yang (2021) with a promising way to extend the concept of subresultant polynomial from two polynomials to multiple polynomials.

Given $A,B\in\mathbb{F}[x]$ with degrees $n$ and $m$ , respectively, let $a_{n}$ be the leading coefficient of $A$ and $\alpha_{1},\ldots,\alpha_{n}$ be the $n$ roots of $A$ over the algebraic closure of $\mathbb{F}$ . Then

S_{k}(A,B)=\dfrac{a_{n}^{m-k}\cdot\det\left[\begin{array}[]{rcr}\alpha_{1}^{n-k-1}B(\alpha_{1})&\cdots&\alpha_{n}^{n-k-1}B(\alpha_{n})\\ \vdots{}{}{}{}{}{}{}{}&&\vdots{}{}{}{}{}{}{}{}\\ \alpha_{1}^{0}B(\alpha_{1})&\cdots&\alpha_{n}^{0}B(\alpha_{n})\\ \hline\cr\alpha_{1}^{k-1}(x-\alpha_{1})&\cdots&\alpha_{n}^{k-1}(x-\alpha_{n})\\ \vdots{}{}{}{}{}{}{}{}&&\vdots{}{}{}{}{}{}{}{}\\ \alpha_{1}^{0}(x-\alpha_{1})&\cdots&\alpha_{n}^{0}(x-\alpha_{n})\end{array}\right]_{n\times n}}{\det\begin{bmatrix}\alpha_{1}^{n-1}&\cdots&\alpha_{n}^{n-1}\\ \vdots&&\vdots\\ \alpha_{1}^{0}&\cdots&\alpha_{n}^{0}\end{bmatrix}_{n\times n}}

(3)

The above rational expression for $S_{k}$ in roots should be interpreted as follows; otherwise, the denominator will vanish when $A$ is not squarefree.

(1)

Treat $\alpha_{1},\ldots,\alpha_{n}$ as indeterminates and carry out the exact division, which results in a symmetric polynomial in terms of $\alpha_{1},\ldots,\alpha_{n}$ .
(2)

Evaluate the obtained polynomial with $\alpha_{1},\ldots,\alpha_{n}$ assigned the values of roots of $A$ .

Example 5.

Consider the polynomials $A$ and $B$ in Example 3. It is noted that the roots of $A$ are $(\alpha_{1},\alpha_{2},\alpha_{3})=(1/2,1/2,1)$ . With further calculation, we have

	$\displaystyle S_{1}(A,B)=$	$\displaystyle\dfrac{4^{3-1}\cdot\det\begin{bmatrix}\alpha_{1}^{1}B(\alpha_{1})&\alpha_{2}^{1}B(\alpha_{2})&\alpha_{3}^{1}B(\alpha_{3})\\ \alpha_{1}^{0}B(\alpha_{1})&\alpha_{2}^{0}B(\alpha_{2})&\alpha_{3}^{0}B(\alpha_{3})\\ \alpha_{1}^{0}(x-\alpha_{1})&\alpha_{2}^{0}(x-\alpha_{2})&\alpha_{3}^{0}(x-\alpha_{3})\end{bmatrix}}{\det\begin{bmatrix}\alpha_{1}^{2}&\alpha_{2}^{2}&\alpha_{3}^{2}\\ \alpha_{1}^{1}&\alpha_{2}^{1}&\alpha_{3}^{1}\\ \alpha_{1}^{0}&\alpha_{2}^{0}&\alpha_{3}^{0}\end{bmatrix}}$
	$\displaystyle=\,$	$\displaystyle 16\big{(}(\,{4\alpha_{{1}}^{2}}{\alpha_{{2}}^{2}}+4\,{\alpha_{{1}}^{2}}{\alpha_{{3}}^{2}}+4\,{\alpha_{{2}}^{2}}{\alpha_{{3}}^{2}}+4\,{\alpha_{{1}}^{2}}\alpha_{{2}}\alpha_{{3}}+4\,\alpha_{{1}}{\alpha_{{2}}^{2}}\alpha_{{3}}+4\,\alpha_{{1}}\alpha_{{2}}{\alpha_{{3}}^{2}}+6\,{\alpha_{{1}}^{2}}\alpha_{{2}}$
		$\displaystyle+6\,{\alpha_{{1}}^{2}}\alpha_{{3}}+6\,\alpha_{{1}}{\alpha_{{2}}^{2}}+6\,\alpha_{{1}}{\alpha_{{3}}^{2}}+6\,{\alpha_{{2}}^{2}}\alpha_{{3}}+6\,\alpha_{{2}}{\alpha_{{3}}^{2}}+12\,\alpha_{{1}}\alpha_{{2}}\alpha_{{3}}+9\,\alpha_{{1}}\alpha_{{2}}+9\,\alpha_{{1}}\alpha_{{3}}$
		$\displaystyle+9\,\alpha_{{2}}\alpha_{{3}}+2\,\alpha_{{1}}+2\,\alpha_{{2}}+2\,\alpha_{{3}}+3)x-(4\,{\alpha_{{1}}^{2}}{\alpha_{{2}}^{2}}\alpha_{{3}}+4\,{\alpha_{{1}}^{2}}\alpha_{{2}}{\alpha_{{3}}^{2}}+4\,\alpha_{{1}}{\alpha_{{2}}^{2}}{\alpha_{{3}}^{2}}$
		$\displaystyle+6\,{\alpha_{{1}}^{2}}\alpha_{{2}}\alpha_{{3}}+6\,\alpha_{{1}}{\alpha_{{2}}^{2}}\alpha_{{3}}+6\,\alpha_{{1}}\alpha_{{2}}{\alpha_{{3}}^{2}}+9\,\alpha_{{1}}\alpha_{{2}}\alpha_{{3}}+2\,{\alpha_{{1}}^{2}}+2\,{\alpha_{{2}}^{2}}+2\,{\alpha_{{3}}^{2}}$
		$\displaystyle+2\,\alpha_{{1}}\alpha_{{2}}+2\,\alpha_{{1}}\alpha_{{3}}+2\,\alpha_{{2}}\alpha_{{3}}+3\,\alpha_{{1}}+3\,\alpha_{{2}}+3\,\alpha_{{3}})\big{)}$

The evaluation of $S_{1}(A,B)$ with $(\alpha_{1},\alpha_{2},\alpha_{3})=(1/2,1/2,1)$ yields $S_{1}(A,B)=576\,x-288.$

3 Problem Statement

In this section, we formalize the problem to be addressed in the current paper. For this purpose, we begin with a review of the generalized subresultant polynomial of multiple univariate polynomials, developed in recent years by Hong & Yang (2021).

3.1 Subresultant polynomial of several univariate polynomials

We need the following notations to present the generalized subresultant polynomial for more than two polynomials.

Notation 6.

1.

$F=({F_{0}},{F_{1}},\ldots,{F_{t}})\subseteq\mathbb{F}[x]$ ;
2.

${d_{i}}=\deg{F_{i}}$ ;
3.

$\alpha_{1},\ldots,\alpha_{d_{0}}$ are the $d_{0}$ roots of $F_{0}$ in the algebraic closure of $\mathbb{F}$ ;
4.

$\delta=({\delta_{1}},{\delta_{2}},\ldots,{\delta_{t}})\in\mathbb{N}^{t}$ ;²²2It is assumed that $0\in\mathbb{N}$ .
5.

$\left|\delta\right|={\delta_{1}}+\cdots+{\delta_{t}}\leq{d_{0}}$ .

With the above notations, we recall the definition of the $\delta$ th subresultant polynomial for several univariate polynomials in terms of roots from Hong & Yang (2021, Definition 2 and Lemma 30), which can be viewed as the generalization of Equation (3) to multiple polynomials. We point out that the expression of subresultant polynomial we use in the current paper differs from that in Hong & Yang (2021) by a sign factor. The reason for the sign adjustment is that we want to make the sign match with the equivalent form of subresultant polynomials in coefficients as developed in a more recent work by Hong & Yang (2023), which naturally extends the concept of Sylvester subresultant polynomial from two polynomials to several polynomials.

Definition 7.

The $\delta$ th subresultant polynomial $S_{\delta}$ of $F$ is defined by

{S_{\delta}}(F):=a_{0d_{0}}^{\delta_{0}}\cdot\det M_{\delta}/\det V

(4)

where $a_{0d_{0}}$ is the leading coefficient of $F_{0}$ in terms of $x$ , and

1.

$M_{\delta}=\left[{\begin{array}[]{rcr}{\alpha_{1}^{{\delta_{1}}-1}{F_{1}}({\alpha_{1}})}&\cdots&{\alpha_{{d_{0}}}^{{\delta_{1}}-1}{F_{1}}({\alpha_{{d_{0}}}})}\\ \vdots{}{}{}{}{}{}{}{}&&\vdots{}{}{}{}{}{}{}{}\\ {\alpha_{1}^{0}{F_{1}}({\alpha_{1}})}&\cdots&{\alpha_{{d_{0}}}^{0}{F_{1}}({\alpha_{{d_{0}}}})}\\ \hline\cr\vdots{}{}{}{}{}{}{}{}&{}\hfil&\vdots{}{}{}{}{}{}{}{}\\ \vdots{}{}{}{}{}{}{}{}&{}\hfil&\vdots{}{}{}{}{}{}{}{}\\ \hline\cr{\alpha_{1}^{{\delta_{t}}-1}{F_{t}}({\alpha_{1}})}&\cdots&{\alpha_{{d_{0}}}^{{\delta_{t}}-1}{F_{t}}({\alpha_{{d_{0}}}})}\\ \vdots{}{}{}{}{}{}{}{}&&\vdots{}{}{}{}{}{}{}{}\\ {\alpha_{1}^{0}{F_{t}}({\alpha_{1}})}&\cdots&{\alpha_{{d_{0}}}^{0}{F_{t}}({\alpha_{{d_{0}}}})}\\ \hline\cr{\alpha_{1}^{\varepsilon-1}(x-{\alpha_{1}})}&\cdots&{\alpha_{{d_{0}}}^{\varepsilon-1}(x-{\alpha_{{d_{0}}}})}\\ \vdots{}{}{}{}{}{}{}{}&&\vdots{}{}{}{}{}{}{}{}\\ {\alpha_{1}^{0}(x-{\alpha_{1}})}&\cdots&{\alpha_{{d_{0}}}^{0}(x-{\alpha_{{d_{0}}}})}\end{array}}\right]_{d_{0}\times d_{0}}$
2.

$V=\left[{\begin{array}[]{*{20}{c}}{\alpha_{1}^{{d_{0}}-1}}&\cdots&{\alpha_{{d_{0}}}^{{d_{0}}-1}}\\ \vdots&{}\hfil&\vdots{}\\ {\alpha_{1}^{0}}&\cdots&{\alpha_{d_{0}}^{0}}\end{array}}\right]_{d_{0}\times d_{0}}$
3.

$\delta_{0}=\max(d_{1}+\delta_{1}-d_{0},\ldots,d_{t}+\delta_{t}-d_{0},1-|\delta|);$
4.

$\varepsilon={d_{0}}-|\delta|$ .

The coefficient of $S_{\delta}(F)$ in the term $x^{\varepsilon}$ is called the leading coefficient of $S_{\delta}(F)$ , denoted by $s_{\delta}(F)$ . If no ambiguity occurs, we can write $S_{\delta}(F)$ as $S_{\delta}$ for simplicity.

Remark 8.

(1)

It is emphasized that the rational expression (4) in Definition 7 should be interpreted the same way as that for interpreting Equation (3).
(2)

When $\delta_{i}=0$ for some $i>0$ , the block of $M_{\delta}$ involving $F_{i}$ will disappear and $S_{\delta}$ will not contain the information of $F_{i}$ .
(3)

It is noted that $\varepsilon$ captures the formal degree of $S_{\delta}$ in terms of $x$ .

The choice for $\delta_{0}$ in Definition 7 seems artificial. Now we justify why $\delta_{0}$ is chosen in this particular way. The reason is to make $S_{\delta}$ be a polynomial in terms of the coefficients of $F_{i}$ ’s with the smallest degree. A more detailed explanation is given below. Again, treating $\alpha_{1},\ldots,\alpha_{d_{0}}$ as indeterminates and carrying out the exact division for $\det M_{\delta}/\det V$ , we obtain a symmetric polynomial in $\alpha_{1},\ldots,\alpha_{d_{0}}$ which can always be converted into a polynomial in elementary symmetric polynomials (denoted by $e_{1},\ldots,e_{d_{0}}$ ) in terms of $\alpha_{1},\ldots,\alpha_{d_{0}}$ . By Vieta formulas, $e_{i}~{}(1\leq i\leq d_{0})$ is a rational function in the coefficients of $F_{0}$ with the denominator to be $a_{0d_{0}}$ . To make $S_{\delta}$ a polynomial in terms the coefficients of $F_{i}$ ’s, we need to cancel the denominators by multiplying $\det M_{\delta}/\det V$ with $a_{0d_{0}}^{p}$ where $p$ is the total degree of $\det M_{\delta}/\det V$ (viewed as polynomial in $e_{1},\ldots,e_{d_{0}}$ ) in terms of $e_{i}$ ’s. Next, we will figure out what $p$ is. Note that $e_{i}$ is linear in $\alpha_{1}$ . Thus the total degree of $\det M_{\delta}/\det V$ in terms of $e_{i}$ ’s is equal to its degree in terms of $\alpha_{1}$ . Combining the observations that the degree of $\det M_{\delta}$ in $\alpha_{1}$ is $\max(d_{1}+\delta_{1},\ldots,d_{t}+\delta_{t},\varepsilon+1)$ and that of $\det V$ is $d_{0}$ , we have

	$\displaystyle p$	$\displaystyle=\max(d_{1}+\delta_{1},\ldots,d_{t}+\delta_{t},\varepsilon+1)-d_{0}$
		$\displaystyle=\max(d_{1}+\delta_{1}-d_{0},\ldots,d_{t}+\delta_{t}-d_{0},1-\|\delta\|),$

which is exactly the value of $\delta_{0}$ in Definition 7.

Example 9.

Given

F=({F_{0}},{F_{1}},{F_{2}})=(4x^{3}-8x^{2}+5x-1,2x^{3}-3x^{2}+x,2x^{3}+x^{2}-x)

and $\delta=(1,1)$ , obviously, we have $(d_{0},d_{1},d_{2})=(3,3,3)$ . Moreover, it is easy to obtain by calculation that $(\alpha_{1},\alpha_{2},\alpha_{3})=(1/2,1/2,1)$ and

\delta_{0}=\max(3+1-3,3+1-3,1-(1+1))=1.

Now we construct

	$\displaystyle V=$	$\displaystyle\begin{bmatrix}\alpha_{1}^{2}&\alpha_{2}^{2}&\alpha_{3}^{2}\\ \alpha_{1}^{1}&\alpha_{2}^{1}&\alpha_{3}^{1}\\ \alpha_{1}^{0}&\alpha_{2}^{0}&\alpha_{3}^{0}\end{bmatrix},$
	$\displaystyle M_{\delta}=$	$\displaystyle\left[\begin{array}[]{lll}\alpha_{1}^{0}(2{\alpha_{1}^{3}}-3{\alpha_{1}^{2}}+\alpha_{1})&\alpha_{2}^{0}(2{\alpha_{2}^{3}}-3{\alpha_{2}^{2}}+\alpha_{2})&\alpha_{3}^{0}(2{\alpha_{3}^{3}}-3{\alpha_{3}^{2}}+\alpha_{3})\\ \alpha_{1}^{0}(2{\alpha_{1}^{3}}+{\alpha_{1}^{2}}-\alpha_{1})&\alpha_{2}^{0}(2{\alpha_{2}^{3}}+{\alpha_{2}^{2}}-\alpha_{2})&\alpha_{3}^{0}(2{\alpha_{3}^{3}}+{\alpha_{3}^{2}}-\alpha_{3})\\ \alpha_{1}^{0}(x-\alpha_{1})&\alpha_{2}^{0}(x-\alpha_{2})&\alpha_{3}^{0}(x-\alpha_{3})\end{array}\right].$

Further calculation yields

	$\displaystyle\det V=\,$	$\displaystyle\left(\alpha_{{2}}-\alpha_{{3}}\right)\left(\alpha_{{1}}-\alpha_{{3}}\right)\left(\alpha_{{1}}-\alpha_{{2}}\right),$
	$\displaystyle\det M_{\delta}=\,$	$\displaystyle 2\det V\cdot\left(\left(4e_{2}-2\,e_{1}+1\right)x-4\,e_{3}\right),$

where $e_{1}=\alpha_{{1}}+\alpha_{{2}}+\alpha_{{3}}$ , $e_{2}=\alpha_{{1}}\alpha_{{2}}+\alpha_{{1}}\alpha_{{3}}+\alpha_{{2}}\alpha_{{3}}$ , $e_{3}=\alpha_{{1}}\alpha_{{2}}\alpha_{{3}}$ . Therefore,

S_{\delta}(F)=4^{1}\cdot 2(\left(4e_{2}-2\,e_{1}+1\right)x-4\,e_{3}).

The evaluation of $S_{\delta}(F)$ with $(\alpha_{1},\alpha_{2},\alpha_{3})=(1/2,1/2,1)$ yields $S_{\delta}(F)=16x-8.$

The $\delta$ th subresultant polynomial provides a practical tool for tackling the problems of parametric gcds and parametric multiplicities because of the inherent connection it has with the incremental gcds of several univariate polynomials.

Definition 10 (Hong & Yang (2021)).

Given $F\subseteq\mathbb{F}[x]$ , let

\theta_{i}=\deg\gcd(F_{0},\ldots,F_{i-1})-\deg\gcd(F_{0},\ldots,F_{i})\quad\text{for}\quad i=1,\ldots,t

where $\gcd(F_{0}):=F_{0}$ . Then $\operatorname*{icdeg}(F):=(\theta_{1},\ldots,\theta_{t})$ is called the incremental cofactor degree of $F$ .

Theorem 11 (Hong & Yang (2021)).

Let $\theta=\max\limits_{s_{\delta}(F)\neq 0}\delta$ where $\delta$ ’s are as specified in Notation 6 and $\max$ is with respect to the ordering $\succ_{\operatorname*{glex}}$ . Then we have

\operatorname*{icdeg}(F)=\theta\ \ \ \text{and}\ \ \ \gcd(F)=S_{\theta}(F).

Remark 12.

The ordering $\succ_{\rm{glex}}$ is defined for two sequences, e.g., $\gamma$ and $\delta$ , in $\mathbb{N}^{t}$ . We say $\delta\succ_{\rm{glex}}\gamma$ if $|\delta|>|\gamma|$ , or $|\delta|=|\gamma|$ and there exists $i\leq t$ such that $\delta_{i}>\gamma_{i}$ and $\delta_{j}=\gamma_{j}$ for $j<i$ .

Example 13.

Consider $F=(F_{0},F_{1},F_{2})$ where $F_{i}$ ’s are as in Example 9. All the possible $\delta$ ’s for the specific $F$ form the following set:

\{\ (3,0),\ (2,1),\ (1,2),\ (0,3),\ (2,0),\ (1,1),\ (0,2),\ (1,0),\ (0,1),\ (0,0)\ \}

where the entries are ordered with respect to $\succ_{\rm glex}$ . With some calculations, we obtain

s_{(3,0)}(F)=s_{(2,1)}(F)=s_{(1,2)}(F)=s_{(0,3)}(F)=s_{(2,0)}(F)=0

and $s_{(1,1)}(F)\neq 0$ . Hence, $\gcd F=S_{(1,1)}(F)$ .

3.2 Determinant polynomial in Newton basis

In order to state the problem, we need to generalize the concept of determinant polynomial from power basis to Newton basis.

Definition 14 (Newton basis).

Let $\lambda=(\lambda_{1},\ldots,\lambda_{n})\in\mathbb{F}^{n}$ and $B^{\lambda}(x)=({B_{0}},{B_{1}},\ldots,{B_{n}})$ where

B_{i}=(x-\lambda_{1})\cdots(x-\lambda_{i})

(5)

with the convention $B_{0}:=1$ . We call $B^{\lambda}(x)$ ¹¹1 $B^{\lambda}(x)$ can be abbreviated as $B^{\lambda}$ if no ambiguity occurs. the Newton basis of $\mathbb{F}_{n}[x]$ with respect to $\lambda$ .

It should be pointed out that the classical definition of Newton basis requires that $\lambda_{i}$ ’s are distinct. In contrast, we discard this restriction in this paper because the logical correctness of the result does not rely on the restriction.

The following concept extends the determinant polynomial in power basis to an arbitrary polynomial set, which also covers Newton basis. The more general concept is introduced because it will play a certain role in the proofs later.

Definition 15 (Determinant polynomial with respect to a polynomial set).

Let $M\in\mathbb{F}^{{(n-k)\times n}}$ where $k\leq n$ and $P=(P_{0},\ldots,{P_{k}})\subseteq\mathbb{F}[x]$ . Then the determinant polynomial of $M$ with respect to $P$ is defined as

{\rm detp}_{P}M=\sum_{i=0}^{k}\det M_{i}\cdot P_{i}

where $M_{i}$ is the submatrix of $M$ consisting of the last $n-k-1$ columns and the $i$ th column.

Remark 16.

1.

When $P=(B_{0},\ldots,B_{k})$ , Definition 15 gives the formal definition of determinant polynomial with respect to Newton basis.
2.

The selection of columns for generating $M_{i}$ is slightly different from the usual way in Definition 1. We use the current version because it is more friendly for the statement of the main result.

3.3 Problem statement

Now we are ready to give a formal statement of the problem to be solved in the paper.

Problem 17.

Given:: $F=(F_{0},\ldots,F_{t})\subseteq\mathbb{F}[x]$ where $F_{i}$ ’s are expressed in the Newton basis $B^{\lambda}=(B_{0},B_{1},\ldots)$ with respect to $\lambda$ , and $\delta\in\mathbb{N}^{t}$ such that $|\delta|\leq d_{0}$ where $d_{0}=\deg F_{0}$
Find:: a matrix $N_{\lambda,\delta}(F)$ such that $S_{\delta}(F)=c\cdot{\rm detp}_{(B_{0},\ldots,B_{\varepsilon})}N_{\lambda,\delta}(F)$ for some constant $c$ where $\varepsilon=d_{0}-|\delta|$ .

4 Main Result

This section is devoted to presenting the main result of the paper, i.e., a particular solution to Problem 17. In order to construct the desired subresultant matrix, we introduce the companion matrix of a polynomial in Newton basis.

Definition 18 (Companion matrix in Newton basis, Corless & Litt (2001)).

Let $B^{{\lambda}}=(B_{0},\ldots,B_{n})$ be the Newton basis of $\mathbb{F}_{n}[x]$ with respect to $\lambda=(\lambda_{1},\ldots,\lambda_{n})\in\mathbb{F}^{n}$ . Let $P={p_{n}}{B_{n}}+\cdots+{p_{1}}B_{1}+{p_{0}B_{0}}\in\mathbb{F}[x]$ where $p_{n}\neq 0$ . Then the companion matrix of $P$ in $B^{{\lambda}}$ , denoted by $\Lambda_{\lambda,P}$ , is defined as

{\Lambda_{\lambda,P}}=p_{n}\left[{\begin{array}[]{*{20}{c}}{{\lambda_{1}}}&{}\hfil&{}\hfil&{}\hfil&{-{p_{0}}/{p_{n}}}\\ 1&{{\lambda_{2}}}&{}\hfil&{}\hfil&{-{p_{1}}/{p_{n}}}\\ {}\hfil&1&\ddots&{}\hfil&\vdots\\ {}\hfil&{}\hfil&\ddots&{{\lambda_{n-1}}}&{-{p_{n-2}}/{p_{n}}}\\ {}\hfil&{}\hfil&{}\hfil&1&{\lambda_{n}-{p_{n-1}}/{p_{n}}}\end{array}}\right]_{n\times n}

Similar to the companion matrix of a polynomial in power basis, the companion matrix ${\Lambda_{\lambda,P}}$ defines an endomorphism of $\mathbb{F}_{n-1}[x]$ given by the multiplication by $p_{n}x$ with respect to the Newton basis (as detailed by Proposition 25-(1) in Section 5).

Now we are ready to state the main result of this paper.

Theorem 19 (Main result).

When $\delta\neq(0,\ldots,0)$ , a solution to Problem 17 is

{N}_{\lambda,\delta}(F)=\begin{bmatrix}{R_{11}}&\cdots&R_{1\delta_{1}}&\cdots&\cdots&{R_{t1}}&\cdots&R_{t\delta_{t}}\end{bmatrix}^{T}

(6)

with $c=(-1)^{\sigma}\cdot a_{0d_{0}}^{\delta_{0}}$ , where

1.

$R_{ij}$ is the $j$ th column of $F_{i}({\Lambda_{{\lambda,F_{0}}}}/a_{0d_{0}})$ ,
2.

${\Lambda_{{\lambda,F_{0}}}}$ is the companion matrix of $F_{0}$ in the basis $(B_{0},\ldots,B_{d_{0}})$ ,
3.

$\sigma=\left(\sum_{i=1}^{t}\binom{\delta_{i}}{2}\right)+\binom{\varepsilon}{2}-\binom{d_{0}}{2}+(d_{0}-1)\varepsilon$ ,¹¹1When $m<p$ , $\binom{m}{p}:=0$ . and
4.

$d_{0}$ , $a_{0d_{0}}$ , $\varepsilon$ and $\delta_{0}$ are as in Definition 7.

Equivalently,

{S_{\delta}}(F)=(-1)^{\sigma}\cdot a_{0d_{0}}^{\delta_{0}}\cdot\operatorname*{detp}\nolimits_{(B_{0},\ldots,B_{\varepsilon})}{N}_{\lambda,\delta}(F)

We make a few observations on Theorem 19.

Remark 20.

(1)

Note that $N_{\lambda,\delta}$ is of order $|\delta|\times d_{0}$ . If $\delta_{i}=0$ , the block from $F_{i}({\Lambda_{{\lambda,F_{0}}}}/a_{0d_{0}})^{T}$ does not appear in ${{N}_{\lambda,\delta}}(F)$ .
(2)

When $\delta=(0,\ldots,0)$ , the matrix ${{N}_{\lambda,\delta}}(F)$ is null, and thus it cannot provide any information about the $\delta$ th subresultant polynomial. On the other hand, by Definition 7, we have

$S_{(0,\ldots,0)}(F)=a_{0d_{0}}^{\delta_{0}-1}F_{0}.$

Therefore, we assume that $\delta\neq(0,\ldots,0)$ in the rest of the paper.
(3)

From the construction of $N_{\lambda,\delta}(F)$ , it is seen that $F_{1},\ldots,F_{t}$ are not necessarily to be expressed in Newton basis.
(4)

When $F$ is specialized with $(A,B)$ where $\deg A=n$ and $\lambda$ is specialized with $(\beta_{1},\ldots,\beta_{n})$ which are the roots of $A$ , ${S_{(n-k)}}$ derived from Theorem 19 is equivalent to $S_{k}$ derived from the formula in Diaz-Toca & Gonzalez-Vega (2004, Proposition 5.1) and Hong (1999). Both can be viewed as subresultant polynomials in terms of the roots of the given polynomials.

Example 21.

Consider

F=({F_{0}},{F_{1}},{F_{2}})=(4B_{3}-8B_{2}+9B_{1},2B_{3}-3B_{2}+3B_{1},2B_{3}-B_{2}-B_{1}+2B_{0})

and $\delta=(1,1)$ , where

B^{\lambda}=(B_{0},B_{1},B_{2},B_{3})=(1,\ x-1,\ (x-1)(x+1),\ x(x-1)(x+1))

and $\lambda=(1,-1,0)$ . It is easy to verify that when converted into expressions in power basis, $F_{i}$ ’s are the same as those in Example 9. We construct the companion matrix of $F_{0}$ in $B^{\lambda}$ and obtain

\Lambda_{\lambda,F_{0}}=4\begin{bmatrix}1&\ \ 0&\ \ 0\\ 1&-1&-\dfrac{9}{4}\\[5.0pt] 0&\ \ 1&\ \ 2\end{bmatrix}

It follows that

F_{1}(\Lambda_{\lambda,F_{0}}/4)=\left[\begin{array}[]{ccc}\ \ 0&\ \ 0&\ \ 0\\[2.0pt] -{\dfrac{3}{2}}&{\ \ \dfrac{3}{4}}&\ \ {\dfrac{9}{8}}\\[7.0pt] \ \ 1&-{\dfrac{1}{2}}&-{\dfrac{3}{4}}\end{array}\right]\quad\text{and}\quad F_{2}(\Lambda_{\lambda,F_{0}}/4)=\left[\begin{array}[]{ccc}\ \ 2&\ \ 0&\ \ 0\\[2.0pt] -{\dfrac{7}{2}}&-{\dfrac{9}{4}}&-{\dfrac{27}{8}}\\[7.0pt] \ \ 5&\ \ {\dfrac{3}{2}}&{\ \ \dfrac{9}{4}}\end{array}\right]

Given $\delta=(1,1)$ , by Theorem 19,

\displaystyle{N}_{\lambda,\delta}(F)=

\displaystyle\begin{bmatrix}\ \ 0&\ \ 2\\[2.0pt] -{\dfrac{3}{2}}&{-\dfrac{7}{2}}\\[7.0pt] \ \ 1&\ \ 5\end{bmatrix}^{T}=\begin{bmatrix}0&-\dfrac{3}{2}&1\\[7.0pt] 2&-\dfrac{7}{2}&5\end{bmatrix}

Hence

	$\displaystyle\varepsilon=\,$	$\displaystyle 3-(1+1)=1$
	$\displaystyle\sigma=\,$	$\displaystyle\binom{1}{2}+\binom{1}{2}+\binom{1}{2}-\binom{3}{2}+(3-1)\cdot 1=-1$
	$\displaystyle\delta_{0}=$	$\displaystyle\max(3+1-3,3+1-3,1-(1+1))=1$
	$\displaystyle S_{\delta}(F)=\,$	$\displaystyle c\cdot{\rm detp}_{(B_{0},B_{1})}{N}_{\lambda,\delta}(F)=(-1)^{-1}\cdot 4^{1}\cdot(-4B_{1}-2B_{0})=16x-8$

5 Proof of the Main Result (Theorem 19)

For readers to better understand the proof details, we give a brief sketch of ideas for proving Theorem 19 above. The three key steps for verifying the theorem are listed as follows.

(1)

Convert the $\delta$ th subresultant polynomial in roots from power to Newton basis (see Lemma 23). The key in this step is the transition matrix between the power and Newton bases.
(2)

Convert the $\delta$ th subresultant polynomial in Newton basis from a rational expression in roots to a determinant expression in coefficients (see Lemma 26). To achieve the goal, we use the companion matrix of a polynomial in Newton basis as the bridge to connect the expression in roots and that in coefficients together.
(3)

Convert the resulting determinant expression in coefficients to its equivalent form of determinant polynomial as in Theorem 19 (see Lemma 28).

It is seen that Lemmas 23, 26 and 28 are the key ingredients for proving the main theorem of the paper. Meanwhile, they are interesting on their own. We hope that they can be useful for tackling other problems in the future.

For the sake of simplicity, we introduce the following short-hands which will be frequently used in the proofs of the key lemmas.

Notation 22.

1.

$\bar{x}_{i}=(x^{i-1},\ldots,x^{0})$ ;
2.

$\bar{x}_{i}(\alpha_{j})=(\alpha_{j}^{i-1},\ldots,\alpha_{j}^{0})$ ;
3.

$\tilde{B}^{\lambda}_{i}=(B_{0},\ldots,B_{i-1})$ ;
4.

$\operatorname*{diag}F_{i}(\alpha)=\begin{bmatrix}F_{i}(\alpha_{1})&&\\ &\ddots&\\ &&F_{i}(\alpha_{d_{0}})\end{bmatrix}$ .

Lemma 23 presented below is targeted at fulfilling the goal in Step (1).

Lemma 23.

We have

S_{\delta}(F)=(-1)^{\sigma^{\prime}}a_{0d_{0}}^{\delta_{0}}\cdot\det M_{\delta}^{\lambda}\big{/}\det\tilde{B}^{\lambda}(\alpha)

(7)

where

1.

$M_{\delta}^{\lambda}=\left[\begin{array}[]{lcl}(\tilde{B}^{\lambda}_{\delta_{1}}(\alpha_{1}))^{T}F_{1}(\alpha_{1})&\cdots&(\tilde{B}^{\lambda}_{\delta_{1}}(\alpha_{d_{0}}))^{T}F_{1}(\alpha_{d_{0}})\\ ~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\vdots&&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\vdots\\ (\tilde{B}^{\lambda}_{\delta_{t}}(\alpha_{1}))^{T}F_{t}(\alpha_{1})&\cdots&(\tilde{B}^{\lambda}_{\delta_{t}}(\alpha_{d_{0}}))^{T}F_{t}(\alpha_{d_{0}})\\ (\tilde{B}^{\lambda}_{\varepsilon}(\alpha_{1}))^{T}(x-\alpha_{1})&\cdots&(\tilde{B}^{\lambda}_{\varepsilon}(\alpha_{d_{0}}))^{T}(x-\alpha_{d_{0}})\end{array}\right]$ ,
2.

$\tilde{B}^{\lambda}(\alpha)=\left[{\begin{array}[]{*{20}{c}}{\tilde{B}^{\lambda}_{d_{0}}({\alpha_{1}})}\\ \vdots\\ {\tilde{B}^{\lambda}_{d_{0}}({\alpha_{d_{0}}})}\end{array}}\right]^{T}$
3.

$\sigma^{\prime}=\left(\sum_{i=1}^{t}\binom{\delta_{i}}{2}\right)+\binom{\varepsilon}{2}-\binom{d_{0}}{2}$ , and
4.

$\delta_{0}$ and $\varepsilon$ are as in Definition 7.

Remark 24.

It is noted that the rational expression (7) in Lemma 23 should be interpreted the same way as that for interpreting Equation (3) and the rational expression (4) for $S_{\delta}$ in Definition 7. Otherwise, it would make no sense when $\alpha_{i}=\lambda_{j}$ for some $i,j$ .

Proof.

Recall $S_{\delta}(F)=a_{0d_{0}}^{\delta_{0}}\cdot\det M_{\delta}\big{/}\det V$ . We will show

\det\tilde{B}^{\lambda}(\alpha)=(-1)^{\binom{d_{0}}{2}}\det V\quad\text{and}\quad\det M_{\delta}^{\lambda}=(-1)^{\sum_{i=1}^{t}\binom{\delta_{i}}{2}+\binom{\varepsilon}{2}}\det M_{\delta}

respectively, from which the lemma is followed. For this purpose, we assume $U$ is the transition matrix from $\bar{x}_{d_{0}}$ to $\tilde{B}^{\lambda}_{d_{0}}$ , i.e., $\tilde{B}^{\lambda}_{d_{0}}=\bar{x}_{d_{0}}\cdot U$ . It is easy to see that $U$ is a skew lower-triangular matrix with its anti-diagonal entries to be all ones.

(1) Note that

\left[{\begin{array}[]{*{20}{c}}{\tilde{B}^{\lambda}_{d_{0}}({\alpha_{1}})}\\ \vdots\\ {\tilde{B}^{\lambda}_{d_{0}}(\alpha_{d_{0}})}\end{array}}\right]=\left[{\begin{array}[]{*{20}{c}}{\bar{x}_{d_{0}}(\alpha_{1})}\\ \vdots\\ {\bar{x}_{d_{0}}(\alpha_{d_{0}})}\end{array}}\right]\cdot U=V^{T}\cdot U

Taking determinants on the left-hand and right-hand sides of the above equation and observing that $U$ is a skew-lower-triangle matrix with anti-diagonal entries to be all ones, we immediately get

\det\tilde{B}^{\lambda}(\alpha)=\det V^{T}\cdot\det U=(-1)^{\binom{d_{0}}{2}}\det V

(2) To show $\det M_{\delta}=(-1)^{\sum_{i=1}^{t}\binom{\delta_{i}}{2}+\binom{\varepsilon}{2}}\det M_{\delta}^{\lambda}$ , we partition $M_{\delta}$ and $M_{\delta}^{\lambda}$ into $t+1$ parts, that is,

M_{\delta}=\begin{bmatrix}M_{1}\\ \vdots\\ M_{t}\\ X_{\varepsilon}\end{bmatrix}\quad\text{and}\quad M_{\delta}^{\lambda}=\begin{bmatrix}M_{1}^{\lambda}\\ \vdots\\ M_{t}^{\lambda}\\ X_{\varepsilon,\lambda}\end{bmatrix}

where

	$\displaystyle M_{i}$	$\displaystyle=\begin{bmatrix}\bar{x}_{\delta_{i}}(\alpha_{1}){F_{i}}(\alpha_{1})&\ \cdots&{\bar{x}_{\delta_{i}}(\alpha_{d_{0}})F({\alpha_{{d_{0}}}})}\end{bmatrix}$
	$\displaystyle X_{\varepsilon}$	$\displaystyle=\begin{bmatrix}{\bar{x}_{\varepsilon}(\alpha_{1})(x-{\alpha_{1}})}&\cdots&{\bar{x}_{\varepsilon}(\alpha_{d_{0}})(x-{\alpha_{{d_{0}}}})}\end{bmatrix}$
	$\displaystyle M_{i}^{\lambda}$	$\displaystyle=\begin{bmatrix}{(\tilde{B}^{\lambda}_{\delta_{i}}({\alpha_{1}}))^{T}{F_{i}}({\alpha_{1}})}&\ \,\cdots\ &{(\tilde{B}^{\lambda}_{\delta_{i}}({\alpha_{{d_{0}}}}))^{T}{F_{i}}({\alpha_{{d_{0}}}})}\end{bmatrix}$
	$\displaystyle X_{\varepsilon}^{\lambda}$	$\displaystyle=\begin{bmatrix}{(\tilde{B}^{\lambda}_{\varepsilon}({\alpha_{1}}))^{T}(x-{\alpha_{1}})}&\cdots&{(\tilde{B}^{\lambda}_{\varepsilon}({\alpha_{{d_{0}}}}))^{T}(x-{\alpha_{{d_{0}}}})}\\ \end{bmatrix}$

Note that

	$\displaystyle M_{i}^{\lambda}$	$\displaystyle=\begin{bmatrix}{(\tilde{B}^{\lambda}_{\delta_{i}}({\alpha_{1}}))^{T}}&\cdots&{(\tilde{B}^{\lambda}_{\delta_{i}}({\alpha_{{d_{0}}}}))^{T}}\end{bmatrix}\cdot\operatorname*{diag}F_{i}(\alpha)$
		$\displaystyle=\begin{bmatrix}I_{\delta_{i}}&0_{d_{0}-\delta_{i}}\end{bmatrix}\cdot\begin{bmatrix}{(\tilde{B}^{\lambda}_{d_{0}}({\alpha_{1}}))^{T}}&\cdots&{(\tilde{B}^{\lambda}_{d_{0}}({\alpha_{{d_{0}}}}))^{T}}\end{bmatrix}\cdot\operatorname*{diag}F_{i}(\alpha)$
		$\displaystyle=\begin{bmatrix}I_{\delta_{i}}&0_{d_{0}-\delta_{i}}\end{bmatrix}\cdot U^{T}\cdot\begin{bmatrix}\bar{x}_{d_{0}}(\alpha_{1})&\cdots&\bar{x}_{d_{0}}(\alpha_{d_{0}})\end{bmatrix}\cdot\operatorname*{diag}F_{i}(\alpha)$

where $I_{\delta_{i}}$ is the identity matrix of order $\delta_{i}$ and $0_{d_{0}-\delta_{i}}$ is the zero matrix of order $\delta_{i}\times{(d_{0}-\delta_{i}})$ . Since $U$ is skew lower-triangular, so is $U^{T}$ , which inspires us to partition $U^{T}$ in the following manner

U^{T}=\begin{bmatrix}&U_{\delta_{i}}\\ U_{\delta_{i}}^{\prime}&\cdot\end{bmatrix}

where $U_{\delta_{i}}$ and $U_{\delta_{i}}^{\prime}$ are skew lower-triangular matrices of orders $\delta_{i}$ and $d_{0}-\delta_{i}$ , respectively. Then

	$\displaystyle M_{i}^{\lambda}$	$\displaystyle=\begin{bmatrix}I_{\delta_{i}}&0_{d_{0}-\delta_{i}}\end{bmatrix}\cdot\begin{bmatrix}&U_{\delta_{i}}\\ U_{\delta_{i}}^{\prime}&\cdot\end{bmatrix}\cdot\begin{bmatrix}\bar{x}_{d_{0}}(\alpha_{1})&\cdots&\bar{x}_{d_{0}}(\alpha_{d_{0}})\end{bmatrix}\cdot\operatorname*{diag}F_{i}(\alpha)$
		$\displaystyle=U_{\delta_{i}}\cdot\begin{bmatrix}\bar{x}_{\delta_{i}}(\alpha_{1})&\cdots&\bar{x}_{\delta_{i}}(\alpha_{d_{0}})\end{bmatrix}\cdot\operatorname*{diag}F_{i}(\alpha)$
		$\displaystyle=U_{\delta_{i}}M_{i}$

With the similar technique, one can derive that $X_{\varepsilon}^{\lambda}=U_{\varepsilon}X_{\varepsilon}$ . Assembling $M_{1}^{\lambda}$ , $\ldots,M_{t}^{\lambda}$ and $X_{\varepsilon}^{\lambda}$ together, we have

\begin{bmatrix}M_{1}^{\lambda}\\ \vdots\\ M_{t}^{\lambda}\\ X_{\varepsilon}^{\lambda}\end{bmatrix}=\begin{bmatrix}U_{\delta_{1}}M_{1}\\ \vdots\\ U_{\delta_{t}}M_{t}\\ U_{\varepsilon}X_{\varepsilon}\end{bmatrix}=\operatorname*{diag}\begin{bmatrix}U_{\delta_{1}}&\cdots&U_{\delta_{t}}&U_{\varepsilon}\end{bmatrix}\cdot\begin{bmatrix}M_{1}\\ \vdots\\ M_{t}\\ X_{\varepsilon}\end{bmatrix}

Taking the determinants on the left-hand and right-hand sides of the above equation and noting that $\det U_{k}=(-1)^{\binom{k}{2}}$ , we immediately get

\det M_{\delta}^{\lambda}=\left(\prod_{i=1}^{t}\det U_{\delta_{i}}\right)\cdot\det U_{\varepsilon}\cdot\det M_{\delta}=(-1)^{\sum_{i=1}^{t}\binom{\delta_{i}}{2}+\binom{\varepsilon}{2}}\det M_{\delta}

The proof is completed. ∎

Next, we convert the subresultant polynomial in Newton basis from an expression in roots to that in coefficients. To achieve the goal, we introduce the following proposition, which shows that the companion matrix of a polynomial in Newton basis represents an endomorphism of $\mathbb{F}_{n-1}[x]$ defined by the multiplication by $cx$ for some $c$ with respect to the Newton basis (Fuhrmann (1996, Proposition 5.1.8)), which captures the very essential property of companion matrices. Proposition 25 provides us with a bridge to connect the subresultant polynomial expression in roots and that in coefficients together.

Proposition 25.

Given $\lambda\in\mathbb{F}^{n}$ , assume $P=\sum_{i=0}^{n}p_{i}B_{i}$ where $B_{i}$ ’s are as in (14). Let ${\Lambda_{{\lambda},P}}$ be the companion matrix of $P$ in $\tilde{B}^{\lambda}=(B_{0},B_{1},\ldots,B_{n-1})$ . Then we have

(1)

$p_{n}x\cdot\tilde{B}^{\lambda}{\equiv_{P}}\tilde{B}^{\lambda}\cdot{\Lambda_{{\lambda},P}}$ , and
(2)

$Q\cdot\tilde{B}^{\lambda}{\equiv_{P}}\tilde{B}^{\lambda}\cdot Q({\Lambda_{\lambda,P}}/p_{n})$ for any $Q\in\mathbb{F}[x]$ ,

where ${\equiv_{P}}$ denotes the modulo operation of $P$ .

With the help of Proposition 25, we now prove the following lemma, which allows us to convert the subresultant polynomial in Newton basis from an expression in roots to that in coefficients.

Lemma 26.

Let ${{N}_{\lambda,\delta}}(F)$ be as in (6) and

X_{B^{\lambda},\varepsilon}=\begin{bmatrix}x-\lambda_{1}&-1&&&&&&\\ &x-\lambda_{2}&-1&&&&&\\ &&\ddots&\ddots&&&&\\ &&&x-\lambda_{\varepsilon}&-1&&&\end{bmatrix}_{\varepsilon\times d_{0}}

where $\varepsilon=d_{0}-|\delta|$ . Then

S_{\delta}(F)=(-1)^{\sigma^{\prime}}\cdot a_{0d_{0}}^{\delta_{0}}\cdot\det\begin{bmatrix}{{N}_{\lambda,\delta}}(F)\\ X_{B^{\lambda},\varepsilon}\end{bmatrix}_{d_{0}\times d_{0}}

where $\sigma^{\prime}=\left(\sum_{i=1}^{t}\binom{\delta_{i}}{2}\right)+\binom{\varepsilon}{2}-\binom{d_{0}}{2}$ .

Proof.

Recall $M_{\delta}^{\lambda}$ in Lemma 23. We only need to show that

\det M_{\delta}^{\lambda}=\det\begin{bmatrix}{{N}_{\lambda,\delta}}(F)\\ X_{B^{\lambda},\varepsilon}\end{bmatrix}\cdot\det\tilde{B}^{\lambda}(\alpha)=\det\begin{bmatrix}{{N}_{\lambda,\delta}}(F)\cdot\tilde{B}^{\lambda}(\alpha)\\ X_{B^{\lambda},\varepsilon}\cdot\tilde{B}^{\lambda}(\alpha)\end{bmatrix}

(8)

This inspires us to partition $M_{\delta}^{\lambda}$ in the following way: $M_{\delta}^{\lambda}=\begin{bmatrix}D_{1}\\ D_{2}\end{bmatrix}$ where $D_{1}$ consists of the first $t$ blocks of $M_{\delta}^{\lambda}$ and $D_{2}$ is the block involving $x$ . The remaining part will be dedicated to showing that

D_{1}={{N}_{\lambda,\delta}}(F)\cdot\tilde{B}^{\lambda}(\alpha)\ \ \text{and}\ \ D_{2}=X_{B^{\lambda},\varepsilon}\cdot\tilde{B}^{\lambda}(\alpha),

respectively.

Recall that

{{{N}_{\lambda,\delta}}(F)}=\begin{bmatrix}{R_{11}}&\cdots&R_{1\delta_{1}}&\cdots&\cdots&{R_{t1}}&\cdots&R_{t\delta_{t}}\end{bmatrix}^{T}

where $R_{ij}$ is the $j$ th column of $F_{i}({\Lambda_{{\lambda,F_{0}}}}/a_{0d_{0}})$ and $a_{0d_{0}}$ is the leading coefficient of $F_{0}$ . The expansion of ${{N}_{\lambda,\delta}}(F)\cdot\tilde{B}^{\lambda}({\alpha})$ yields

{{N}_{\lambda,\delta}}(F)\cdot\tilde{B}^{\lambda}({\alpha})=\left[{\begin{array}[]{*{20}{c}}\tilde{M}_{1}^{\lambda}\\ \vdots\\ \tilde{M}_{t}^{\lambda}\end{array}}\right]

where

\tilde{M}_{i}^{\lambda}=\begin{bmatrix}{\tilde{B}^{\lambda}({\alpha_{1}})R_{i1}}&\cdots&{\tilde{B}^{\lambda}({\alpha_{{d_{0}}}})R_{i1}}\\ \vdots&{}&\vdots\\ {\tilde{B}^{\lambda}({\alpha_{1}})R_{i{\delta_{i}}}}&\cdots&{\tilde{B}^{\lambda}({\alpha_{{d_{0}}}})R_{i{\delta_{i}}}}\end{bmatrix}_{\delta_{i}\times d_{0}}

(9)

Denote $F_{i}({\Lambda_{{\lambda,F_{0}}}}/a_{0d_{0}})$ with $R_{i}$ , i.e., $R_{i}=\begin{bmatrix}R_{i1}&\cdots&R_{id_{0}}\end{bmatrix}$ . By Proposition 25-(2),

\tilde{B}^{\lambda}(x)\cdot F_{i}(x){\equiv_{F_{0}}}\tilde{B}^{\lambda}(x)\cdot R_{i}

(10)

Substituting $x=\alpha_{j}$ into (10) and noting that $F_{0}(\alpha_{j})=0$ , we get

\tilde{B}^{\lambda}({\alpha_{j}})\cdot{F_{i}}({\alpha_{j}})=\tilde{B}^{\lambda}({\alpha_{j}})\cdot R_{i}

Taking the $k$ th columns of both sides of the above equation, we have

{B_{k-1}}({\alpha_{j}})\cdot{F_{i}}({\alpha_{j}})=\tilde{B}^{\lambda}({\alpha_{j}})\cdot R_{ik}

The application of the above relation on (9) immediately yields

\tilde{M}_{i}^{\lambda}=\begin{bmatrix}{\tilde{B}^{\lambda}_{\delta_{i}}({\alpha_{1}}){F_{i}}({\alpha_{1}})}&\cdots&{\tilde{B}^{\lambda}_{\delta_{i}}({\alpha_{{d_{0}}}}){F_{i}}({\alpha_{{d_{0}}}})}\end{bmatrix}

which is exactly the $i$ th block of $M_{\delta}^{\lambda}$ for $i=1,\ldots,t$ . Hence

{{N}_{\lambda,\delta}}(F)\cdot\tilde{B}^{\lambda}({\alpha})=D_{1}

(11)

Let $X_{s}$ be the $s$ th row of $X_{B^{\lambda},\varepsilon}$ , i.e.,

	$\displaystyle{X_{s}}$	$\displaystyle=\left[{\begin{array}[]{*{20}{c}}0&\cdots&0&{x-{\lambda_{s}}}&{-1}&0&\cdots&0\end{array}}\right]$
		$\displaystyle\hskip 65.00009pt{\begin{subarray}{c}\uparrow\\ s\text{th column}\end{subarray}}$

where $s\in\{1,2,\ldots,\varepsilon\}$ . Simple calculation yields

{X_{s}}\cdot\tilde{B}^{\lambda}({\alpha_{i}})^{T}=(x-{\lambda_{s}})\cdot{B_{s-1}}({\alpha_{i}})-{B_{s}}({\alpha_{i}})={B_{s-1}}({\alpha_{i}})(x-{\alpha_{i}})

It follows that $X_{B^{\lambda},\varepsilon}\tilde{B}^{\lambda}({\alpha_{i}})^{T}={\tilde{B}^{\lambda}_{\varepsilon}}({\alpha_{i}})(x-{\alpha_{i}}).$ Thus

	$\displaystyle X_{B^{\lambda},\varepsilon}\cdot\tilde{B}^{\lambda}({\alpha})$	$\displaystyle=X_{B^{\lambda},\varepsilon}\cdot\left[{\begin{array}[]{*{20}{c}}{\tilde{B}^{\lambda}({\alpha_{1}})}^{T}&\cdots&{\tilde{B}^{\lambda}({\alpha_{{d_{0}}}})}^{T}\end{array}}\right]$		(13)
		$\displaystyle=\left[{\begin{array}[]{*{20}{c}}{\tilde{B}^{\lambda}_{\varepsilon}}({\alpha_{1}})(x-{\alpha_{1}})&\cdots&{\tilde{B}^{\lambda}_{\varepsilon}}({\alpha_{d_{0}}})(x-{\alpha_{d_{0}}})\end{array}}\right]$		(15)

Combining (11) and (15), we have $\begin{bmatrix}{{N}_{\lambda,\delta}}(F)\\ X_{B^{\lambda},\varepsilon}\end{bmatrix}\cdot\tilde{B}^{\lambda}(\alpha)=M_{\delta}^{\lambda}$ . Applying Lemma 23, we finally attain

S_{\delta}(F)=(-1)^{\sigma^{\prime}}\cdot a_{0d_{0}}^{\delta_{0}}\det M_{\delta}^{\lambda}/\det\tilde{B}^{\lambda}(\alpha)=(-1)^{\sigma^{\prime}}\cdot a_{0d_{0}}^{\delta_{0}}\cdot\det\begin{bmatrix}{{N}_{\lambda,\delta}}(F)\\ X_{B^{\lambda},\varepsilon}\end{bmatrix}

where $\sigma^{\prime}=\left(\sum_{i=1}^{t}\binom{\delta_{i}}{2}\right)+\binom{\varepsilon}{2}-\binom{d_{0}}{2}$ . ∎

Remark 27.

Lemma 26 is a generalized version of Hong & Yang (2021, Theorem 27-2). The latter can be obtained by specializing $\lambda_{1}=\cdots=\lambda_{\varepsilon}=0$ in Lemma 26.

The final step is to convert the generalized subresultant polynomial from a determinant expression to an equivalent determinant polynomial expression. For this purpose, we present a more general result than what we need. The more general result is presented in the hope that it would be useful for other related problems.

Lemma 28.

Given

M=\left[{\begin{array}[]{*{20}{c}}{{a_{1,1}}}&\cdots&{{a_{1,n}}}\\ \vdots&{}\hfil&\vdots\\ {{a_{n-k,1}}}&\cdots&{{a_{n-k,n}}}\end{array}}\right]_{(n-k)\times n}

and $P=(P_{0},P_{1},\ldots,P_{k})\subseteq\mathbb{F}[x]$ , we have

\operatorname*{detp}\nolimits_{P}M=(-1)^{(n-1)k}\cdot P_{0}\cdot\det\begin{bmatrix}M\\ X_{P}\end{bmatrix},

where

X_{P}=\left[\begin{array}[]{cccccccc}P_{1}/P_{0}&-1&&&&&&\\ &P_{2}/P_{1}&-1&&&&&\\ &&\ddots&\ddots&&&&\\ &&&P_{k}/P_{k-1}&-1&&&\end{array}\right]_{k\times n}

Proof.

By Definition 15,

\operatorname*{detp}\nolimits_{P}M=\sum_{i=0}^{k}\det\left[{\begin{array}[]{*{20}{c}}a_{1,i+1}&{{{a_{1,k+2}}}}&\cdots&{{{a_{1,n}}}}\\ a_{2,i+1}&{{{a_{2,k+2}}}}&\cdots&{{a_{2,n}}}\\ \vdots&{}\hfil&&\vdots\\ a_{n-k,i+1}&{{{a_{n-k,k+2}}}}&\cdots&{{a_{n-k,n}}}\end{array}}\right]_{(n-k)\times(n-k)}\cdot{P_{i}}

Making use of the multi-linearity of determinants, we have

	$\displaystyle\operatorname*{detp}\nolimits_{P}M$	$\displaystyle=\sum_{i=0}^{k}\det\left[{\begin{array}[]{*{20}{c}}a_{1,i+1}P_{i}&{{{a_{1,k+2}}}}&\cdots&{{{a_{1,n}}}}\\ a_{2,i+1}P_{i}&{{{a_{2,k+2}}}}&\cdots&{{a_{2,n}}}\\ \vdots&{}\hfil&&\vdots\\ a_{n-k,i+1}P_{i}&{{{a_{n-k,k+2}}}}&\cdots&{{a_{n-k,n}}}\end{array}}\right]_{(n-k)\times(n-k)}$
		$\displaystyle=\det\left[{\begin{array}[]{*{20}{c}}\sum_{i=0}^{k}a_{1,i+1}P_{i}&{{{a_{1,k+2}}}}&\cdots&{{{a_{1,n}}}}\\ \sum_{i=0}^{k}a_{2,i+1}P_{i}&{{{a_{2,k+2}}}}&\cdots&{{a_{2,n}}}\\ \vdots&{}\hfil&&\vdots\\ \sum_{i=0}^{k}a_{n-k,i+1}P_{i}&{{{a_{n-k,k+2}}}}&\cdots&{{a_{n-k,n}}}\end{array}}\right]_{(n-k)\times(n-k)}$

The next step is tricky, but it is the key to relating the determinant polynomial with the desired determinant. We first insert $k$ columns, that is, the first $k+1$ columns of $M$ with the first column excluded, between the first two columns of the matrix on the right-hand side of the above equation and then add $k$ rows of the special shape as given in the following matrix. Then we have

\operatorname*{detp}\nolimits_{P}M=(-1)^{\tau}\det\left[\begin{array}[]{c|cccc|ccc}\sum_{i=0}^{k}a_{1,i+1}P_{i}&a_{1,2}&a_{1,3}&\cdots&a_{1,k+1}&{{{a_{1,k+2}}}}&\cdots&{{{a_{1,n}}}}\\ \sum_{i=0}^{k}a_{2,i+1}P_{i}&a_{2,2}&a_{2,3}&\cdots&a_{2,k+1}&{{{a_{2,k+2}}}}&\cdots&{{a_{2,n}}}\\ \vdots&\vdots&&\vdots&\vdots&&\vdots\\ \sum_{i=0}^{k}a_{n-k,i+1}P_{i}&a_{n-k,2}&a_{n-k,3}&\cdots&a_{n-k,k+1}&{{{a_{n-k,k+2}}}}&\cdots&{{a_{n-k,n}}}\\ \hline\cr&-1&&&&&&\\ &P_{2}/P_{1}&-1&&&&&\\ &&\ddots&\ddots&&&&\\ &&&P_{k}/P_{k-1}&-1&&&\\ \end{array}\right]_{n\times n}

where

\tau=k+\sum_{i=1}^{k}(1+i)+\sum_{i=1}^{k}(n-k+i)\equiv(n-1)k\mod 2.

The equation holds because there is only one non-zero minor in the last $k$ rows, which is bounded by the delimitations.

By subtracting the $i$ th column multiplied by $P_{i-1}$ for $i=2,\ldots,k+1$ from the first column and then taking the factor $P_{0}$ from the first column, we obtain

\operatorname*{detp}\nolimits_{P}M=(-1)^{(n-1)k}\cdot P_{0}\cdot\det\begin{bmatrix}M\\ X_{P}\end{bmatrix}.

The proof is completed. ∎

Remark 29.

By Lemma 28, the generalized subresultant polynomial developed in D’Andrea & Krick (2023) and D’Andrea et al. (2006, Equation (7)) for two univariate polynomials corresponding to an arbitrary set of monomials can be rewritten into a single determinant formula, which may bring some convenience for analyzing the properties of the generalized subresultant polynomials. In the Appendix section, we provide a detailed derivation for the formula.

Now we are ready to prove the main result of the paper.

Proof of Theorem 19.

By Lemma 26,

S_{\delta}(F)=(-1)^{\sigma^{\prime}}\cdot a_{0d_{0}}^{\delta_{0}}\cdot\det\begin{bmatrix}{{N}_{\lambda,\delta}}(F)\\ X_{B^{\lambda},\varepsilon}\end{bmatrix}_{d_{0}\times d_{0}}

where $\sigma^{\prime}=\left(\sum_{i=1}^{t}\binom{\delta_{i}}{2}\right)+\binom{\varepsilon}{2}-\binom{d_{0}}{2}$ . Next we specialize $M$ and $P$ in Lemma 28 with ${{N}_{\lambda,\delta}}(F)$ and $(B_{0},\ldots,B_{\varepsilon})$ , respectively. Then $n$ and $k$ in Lemma 28 are specialized with $d_{0}$ and $\varepsilon$ accordingly. Meanwhile, it is observed that $B_{i}/B_{i-1}=x-\lambda_{i}$ and $B_{0}=1$ . Therefore, we can achieve the following

S_{\delta}(F)=(-1)^{\sigma}\cdot a_{0d_{0}}^{\delta_{0}}\cdot\operatorname*{detp}\nolimits_{(B_{0},\ldots,B_{\varepsilon})}{{N}_{\lambda,\delta}}(F)

where $\sigma=\sigma^{\prime}+(d_{0}-1)\varepsilon$ . The proof is completed. ∎

6 Application

The $\delta$ th subresultant polynomial was originally proposed to solve the problem of parametric gcd for multiple polynomials. When combining Theorems 11 and 19, it can also be used to compute the gcd of numerical polynomials. In this section, we devise a method for computing the gcd of several univariate polynomials in Newton basis as an application of Theorem 19.

The method relies on the following proposition.

Proposition 30.

Let $\theta={\rm icdeg}F$ . Then the columns of the matrix $N_{\lambda,\theta}^{T}$ , i.e.,

{R_{11}},\ldots,R_{1\theta_{1}},\ldots,{R_{t1}},\ldots,R_{t\theta_{t}}

form a maximal set of independent columns of the matrix

\begin{bmatrix}{F_{1}({\Lambda_{{\lambda,F_{0}}}}/a_{0d_{0}})}&\cdots&F_{t}({\Lambda_{{\lambda,F_{0}}}}/a_{0d_{0}})\end{bmatrix}

Proof.

We first show that ${R_{11}},\ldots,R_{1\theta_{1}},\ldots,{R_{t1}},\ldots,R_{t\theta_{t}}$ are linearly independent. Otherwise, ${N}_{\lambda,\theta}(F)$ is rank-deficient and thus $\operatorname*{detp}\nolimits_{(B_{0},\ldots,B_{\varepsilon})}{N}_{\lambda,\theta}(F)=0$ where $\varepsilon=d_{0}-|\theta|$ . By Theorem 19,

S_{\theta}=c\cdot\operatorname*{detp}\nolimits_{(B_{0},\ldots,B_{\varepsilon})}{N}_{\lambda,\theta}(F)=0

which contradicts with Theorem 11.

Next we show that $R_{ij}$ for $j>\theta_{i}$ can be written as a linear combination of ${R_{11}},\ldots,R_{1\theta_{1}},\ldots,$ ${R_{t1}},\ldots,R_{t\theta_{t}}$ , which is equivalent to

\operatorname*{detp}\nolimits_{(B_{0},\ldots,B_{\varepsilon-1})}\,\begin{bmatrix}N_{\lambda,\theta}\\ R_{ij}^{T}\end{bmatrix}=0

Assume $R_{ij}=(R_{ij,1},\ldots,R_{ij,d_{0}})^{T}$ . By the multi-linearity of determinant,

	$\displaystyle\operatorname*{detp}\nolimits_{(B_{0},\ldots,B_{\varepsilon-1})}\begin{bmatrix}N_{\lambda,\theta}\\ R_{ij}^{T}\end{bmatrix}=$	$\displaystyle\det\begin{bmatrix}\sum^{\varepsilon-1}_{k=0}R_{11,k+1}\cdot B_{k}&R_{11,\varepsilon+1}&\cdots&R_{11,d_{0}}\\[-2.0pt] \vdots&\vdots&&\vdots\\ \sum^{\varepsilon-1}_{k=0}R_{1\theta_{1},k+1}\cdot B_{k}&R_{1\theta_{1},\varepsilon+1}&\cdots&R_{1\theta_{1},d_{0}}\\[-2.0pt] \vdots&\vdots&&\vdots\\ \sum^{\varepsilon-1}_{k=0}R_{t1,k+1}\cdot B_{k}&R_{t1,\varepsilon+1}&\cdots&R_{t1,d_{0}}\\[-2.0pt] \vdots&\vdots&&\vdots\\ \sum^{\varepsilon-1}_{k=0}R_{t\theta_{t},k+1}\cdot B_{k}&R_{t\theta_{t},\varepsilon+1}&\cdots&R_{t\theta_{t},d_{0}}\\ \sum^{\varepsilon-1}_{k=0}R_{ij,k+1}\cdot B_{k}&R_{ij,\varepsilon+1}&\cdots&R_{ij,d_{0}}\end{bmatrix}$
	$\displaystyle=$	$\displaystyle\det\begin{bmatrix}\sum^{d_{0}-1}_{k=0}R_{11,k+1}\cdot B_{k}&R_{11,\varepsilon+1}&\cdots&R_{11,d_{0}}\\[-2.0pt] \vdots&\vdots&&\vdots\\[-2.0pt] \sum^{d_{0}-1}_{k=0}R_{1\theta_{1},k+1}\cdot B_{k}&R_{1\theta_{1},\varepsilon+1}&\cdots&R_{1\theta_{1},d_{0}}\\[-2.0pt] \vdots&\vdots&&\vdots\\[-2.0pt] \sum^{d_{0}-1}_{k=0}R_{t1,k+1}\cdot B_{k}&R_{t1,\varepsilon+1}&\cdots&R_{t1,d_{0}}\\[-2.0pt] \vdots&\vdots&&\vdots\\[-2.0pt] \sum^{d_{0}-1}_{k=0}R_{t\theta_{t},k+1}\cdot B_{k}&R_{t\theta_{t},\varepsilon+1}&\cdots&R_{t\theta_{t},d_{0}}\\ \sum^{d_{0}-1}_{k=0}R_{ij,k+1}\cdot B_{k}&R_{ij,\varepsilon+1}&\cdots&R_{ij,d_{0}}\end{bmatrix}$		(16)

Note that $\sum^{d_{0}-1}_{k=0}R_{pq,k+1}\cdot B_{k}=\tilde{B}^{\lambda}\cdot R_{pq}$ . By Proposition 25-(2), $F_{i}\cdot\tilde{B}^{\lambda}{\equiv_{F_{0}}}\tilde{B}^{\lambda}\cdot F_{i}({\Lambda_{\lambda,F_{0}}}/a_{0d_{0}})$ . Comparing the $j$ th elements of both sides, we obtain $\tilde{B}^{\lambda}\cdot R_{pq}\equiv_{F_{0}}F_{p}B_{q-1}$ , which implies that

\tilde{B}^{\lambda}\cdot R_{pq}=F_{p}B_{q-1}+C_{pq}F_{0}

The substitution of the above relation into (16) yields

\operatorname*{detp}\nolimits_{(B_{0},\ldots,B_{\varepsilon-1})}\,\begin{bmatrix}N_{\lambda,\theta}\\ R_{ij}^{T}\end{bmatrix}=\det\begin{bmatrix}F_{1}B_{0}+C_{11}F_{0}&R_{11,\varepsilon+1}&\cdots&R_{11,d_{0}}\\[-2.0pt] \vdots&\vdots&&\vdots\\[-2.0pt] F_{1}B_{\theta_{1}-1}+C_{1\theta_{1}}F_{0}&R_{1\theta_{1},\varepsilon+1}&\cdots&R_{1\theta_{1},d_{0}}\\[-2.0pt] \vdots&\vdots&&\vdots\\[-2.0pt] F_{t}B_{0}+C_{t1}F_{0}&R_{t1,\varepsilon+1}&\cdots&R_{t1,d_{0}}\\[-2.0pt] \vdots&\vdots&&\vdots\\[-2.0pt] F_{t}B_{\theta_{t}-1}+C_{t\theta_{t}}F_{0}&R_{t\theta_{t},\varepsilon+1}&\cdots&R_{t\theta_{t},d_{0}}\\ F_{i}B_{\theta_{i}-1}+C_{ij}F_{0}&R_{ij,\varepsilon+1}&\cdots&R_{ij,d_{0}}\end{bmatrix}

which indicates that $\operatorname*{detp}\nolimits_{(B_{0},\ldots,B_{\varepsilon-1})}\begin{bmatrix}N_{\lambda,\theta}\\ R_{ij}^{T}\end{bmatrix}$ belongs to the ideal generated by $F$ . However,

\deg\left(\operatorname*{detp}\nolimits_{(B_{0},\ldots,B_{\varepsilon-1})}\begin{bmatrix}N_{\lambda,\theta}\\ R_{ij}^{T}\end{bmatrix}\right)\leq\varepsilon-1=\deg\gcd F-1<\deg\gcd F

Therefore,

\operatorname*{detp}\nolimits_{(B_{0},\ldots,B_{\varepsilon-1})}\begin{bmatrix}N_{\lambda,\theta}\\ R_{ij}^{T}\end{bmatrix}=0

Hence $R_{ij}^{T}$ is a linear combination of the rows in $N_{\lambda,\theta}$ . It immediately follows that ${R_{11}},\ldots,R_{1\theta_{1}},$ $\ldots,{R_{t1}},\ldots,R_{t\theta_{t}}$ form a maximal set of independent columns of the matrix $\left[{F_{1}({\Lambda_{{\lambda,F_{0}}}}/a_{0d_{0}})}\ \ \cdots\ \ \right.$ $\left.F_{t}({\Lambda_{{\lambda,F_{0}}}}/a_{0d_{0}})\right]$ . ∎

Now we describe the method for computing the gcd of several polynomials in Newton basis, where $F$ is assumed to be as in Notation 6 and expressed in the Newton basis $B^{\lambda}(x)$ .

(S1)

Determine the independent columns of ${F_{1}({\Lambda_{{\lambda,F_{0}}}}/a_{0d_{0}})}$ incrementally. More explicitly, add one column each time to test its linear dependency with the previous columns until we identify the first column which is linearly dependent with the previous columns. If the column is the $j$ th column of ${F_{1}({\Lambda_{{\lambda,F_{0}}}}/a_{0d_{0}})}$ , let $\theta_{1}=j-1$ and go to (S2).
(S2)

For $i=2,\ldots,t$ , determine the columns of ${F_{i}({\Lambda_{{\lambda,F_{0}}}}/a_{0d_{0}})}$ linearly independent with ${R_{11}},\ldots,$ $R_{1\theta_{1}},\ldots,{R_{i1}},\ldots,R_{i\theta_{i}}$ in an incremental way. More explicitly, add one column of ${F_{i}({\Lambda_{{\lambda,F_{0}}}}/a_{0d_{0}})}$ each time to test its linear dependency with the previous columns until we identify the first column which is linearly dependent with the previous columns. If the column is the $j$ th column of ${F_{i}({\Lambda_{{\lambda,F_{0}}}}/a_{0d_{0}})}$ , then let $\theta_{i}=j-1$ and go to the next loop.
(S3)

Construct

$N_{\lambda,\theta}=\begin{bmatrix}R_{11}&\cdots&R_{1\theta_{1}}&\cdots&\cdots&{R_{t1}}&\cdots&R_{t\theta_{t}}\end{bmatrix}^{T}$

Let $\varepsilon=d_{0}-|\theta|$ . Then the gcd of $F$ in Newton basis is $\operatorname*{detp}\nolimits_{(B_{0},\ldots,B_{\varepsilon})}N_{\lambda,\theta}$ .

We discuss the correctness of the above method below.

(1)

Apply Proposition 30 to $(F_{0},F_{1})$ . Then we have that a maximal set of linearly independent columns of ${F_{1}({\Lambda_{{\lambda,F_{0}}}}/a_{0d_{0}})}$ is ${R_{11}},\ldots,R_{1\theta_{1}}$ where $\theta_{1}=\deg F_{0}-\deg\gcd(F_{0},F_{1})$ . So the first $j$ such that the $j$ th column is linearly dependent with the previous $j-1$ columns must satisfy that $j=\theta_{1}+1$ . In other words, $\theta_{1}=j-1$ .
(2)

Assume we have a maximal set of linearly independent columns of ${F_{1}({\Lambda_{{\lambda,F_{0}}}}/a_{0d_{0}})},$ $\ldots$ , ${F_{i-1}({\Lambda_{{\lambda,F_{0}}}}/a_{0d_{0}})}$ , say ${R_{11}},\ldots,R_{1\theta_{1}},\cdots{R_{i1}},\ldots,R_{(i-1)\theta_{i-1}}$ where

$\theta_{j}=\deg\gcd(F_{0},\ldots,F_{j-1})-\deg\gcd(F_{0},\ldots,F_{j})$

It is noted that adding any other $R_{jk}$ with $j<i$ and $k>\theta_{j}$ will destroy the linearly independence. Now we add the columns of ${F_{i}({\Lambda_{{\lambda,F_{0}}}}/a_{0d_{0}})}$ into the linearly independent set one by one until we identify the first column which is linearly dependent with the previous columns. If the column is the $j$ th column of ${F_{i}({\Lambda_{{\lambda,F_{0}}}}/a_{0d_{0}})}$ , by Proposition 30, $j=\theta_{i}+1$ and thus $\theta_{i}=j-1$ .
(3)

By Theorem 11, $\gcd F=S_{\theta}(F)$ ; by Theorem 19, $S_{\theta}(F)=c\cdot\operatorname*{detp}_{(B_{0},\ldots,B_{\varepsilon})}N_{\lambda,\theta}$ . Hence $\operatorname*{detp}\nolimits_{(B_{0},\ldots,B_{\varepsilon})}N_{\lambda,\theta}$ is the gcd of $F$ .

Example 31 (Continued from Example 21).

We show $S_{(1,1)}$ obtained in Example 21 is exactly the gcd of $F$ . Recall that

F_{1}(\Lambda_{\lambda,F_{0}}/4)=\left[\begin{array}[]{ccc}\ \ 0&\ \ 0&\ \ 0\\[2.0pt] -{\dfrac{3}{2}}&{\ \ \dfrac{3}{4}}&\ \ {\dfrac{9}{8}}\\[7.0pt] \ \ 1&-{\dfrac{1}{2}}&-{\dfrac{3}{4}}\end{array}\right]\quad\text{and}\quad F_{2}(\Lambda_{\lambda,F_{0}}/4)=\left[\begin{array}[]{ccc}\ \ 2&\ \ 0&\ \ 0\\[2.0pt] -{\dfrac{7}{2}}&-{\dfrac{9}{4}}&-{\dfrac{27}{8}}\\[7.0pt] \ \ 5&\ \ {\dfrac{3}{2}}&{\ \ \dfrac{9}{4}}\end{array}\right]

It is noted that there is only one linearly independent column in $F_{1}(\Lambda_{\lambda,F_{0}}/4)$ which is the first column $R_{11}$ . Thus $\theta_{1}=1$ . With further calculation, we find that the first column $R_{21}$ of $F_{2}(\Lambda_{\lambda,F_{0}}/4)$ is linearly independent with $R_{11}$ ; however, the second column $R_{22}$ is linearly dependent with $(R_{11},R_{21})$ . Thus $\theta_{2}=1$ and $\theta=(1,1)$ . We construct $N_{\lambda,\theta}(F)$ as follows

{N}_{\lambda,\theta}(F)=\begin{bmatrix}\ \ 0&\ \ 2\\[2.0pt] -{\dfrac{3}{2}}&{-\dfrac{7}{2}}\\[7.0pt] \ \ 1&\ \ 5\end{bmatrix}^{T}=\begin{bmatrix}0&-\dfrac{3}{2}&1\\[7.0pt] 2&-\dfrac{7}{2}&5\end{bmatrix}

Then its determinant polynomial ${\rm detp}_{(B_{0},B_{1})}{N}_{\lambda,\theta}(F)=-4B_{1}-2B_{0}$ provides an expression for the gcd of $F$ and $S_{\theta}$ only differs from it by a constant factor.

7 Conclusion

In this paper, we propose a method for formulating subresultant polynomials of multiple univariate polynomials in Newton basis where the resulting subresultants polynomials are also expressed in the given Newton basis. The formulation does not require basis transformation and thus can avoid the numerical instability problem caused by basis change. To achieve this goal, we extend the determinant polynomial in power basis to that in Newton basis. By making use of the companion matrix in Newton basis, we construct a Barnett-type matrix whose determinant polynomial in Newton basis yields the subresultant polynomial of multiple polynomials in Newton basis.

It is noted that one may have infinite many choices for Newton interpolating nodes $\lambda$ . For each choice of $\lambda$ , we can write the input polynomials in the corresponding Newton basis and formulate the subresultant polynomial in the same basis. Therefore, one can express a subresultant polynomial as infinitely many determinant forms, depending on which Newton basis is used. Then a natural question arises: does there exist $\lambda$ ’s such that the constructed subresultant matrix is interesting? For example, is there a particular way to choose the value of $\lambda$ for a specific polynomial set such that the matrix $N_{\lambda,\delta}$ exhibits nice structures? Another related question worthy of further investigation is how to compute the matrix $N_{\lambda,\delta}$ and its determinant polynomial in an efficient way.

Acknowledgements. The authors would like to thank the anonymous reviewers for their helpful comments and insightful suggestions. The authors’ work was supported by the National Natural Science Foundation of China (Grant No. 12261010), the Natural Science Foundation of Guangxi (Grant No. 2023GXNSFBA026019), and the Natural Science Cultivation Project of GXMZU (Grant No. 2022MDKJ001).

Appendix

The Appendix is devoted to deriving a determinant formula for the generalized subresultant polynomial defined in D’Andrea et al. (2006) and D’Andrea & Krick (2023).

Consider $A,B\in\mathbb{F}[x]$ of the following form

A=a_{n}x^{n}+\cdots+a_{0},\quad B=b_{m}x^{m}+\cdots+b_{0},

where $a_{n}b_{m}\neq 0.$ For $P=(x^{\gamma_{0}},x^{\gamma_{1}},\ldots,x^{\gamma_{k}})\subseteq\mathbb{F}_{\ell}[x]$ where $m\leq\ell\leq m+n-1$ and $k=m-\max(0,\ell-n+1)$ , without loss of generality, we assume $0\leq\gamma_{0}<\gamma_{1}<\ldots<\gamma_{k}$ . Then the subresultant matrix for $A$ and $B$ of order $\ell+1$ is constructed as follows

M=\begin{bmatrix}a_{0}&\cdots&a_{n}\\ &\ddots&&\ddots\\ &&a_{0}&\cdots&a_{n}\\ b_{0}&\cdots&b_{m}\\ &\ddots&&\ddots\\ &&b_{0}&\cdots&b_{m}\\ \end{bmatrix}\begin{array}[]{l}\left.\begin{array}[]{c}\\ \\ \\ \end{array}\right\}~{}\max(0,\ell-n+1)\text{~{}rows}\\[20.0pt] \left.\begin{array}[]{c}\\ \\ \\ \end{array}\right\}~{}\ell-m+1\text{~{}rows}\end{array}.

which has the size $\left(\max(0,\ell-n+1)+(\ell-m+1)\right)\times(\ell+1)$ .

For the sake of simplicity, we partition $M$ by columns, that is,

M=\begin{bmatrix}M_{0}&M_{1}&\cdots&M_{\ell}\end{bmatrix}.

By D’Andrea & Krick (2023) and D’Andrea et al. (2006, Equation (7)), the generalized subresultant polynomial of $A$ and $B$ with respect to $P$ is

s(x)=\sum_{j=0}^{k}(-1)^{\sigma_{j}}\det\widehat{M_{j}}\cdot x^{\gamma_{j}}

where $\sigma_{j}=\sum_{\begin{subarray}{c}0\leq i\leq k\\ i\neq j\end{subarray}}{(i+\gamma_{i})}$ and $\widehat{M_{j}}$ is the submatrix of $M$ obtained by deleting the columns $M_{\gamma_{0}}$ , $M_{\gamma_{1}}$ , $\ldots$ , $M_{\gamma_{k}}$ but keeping $M_{\gamma_{j}}$ . In order to write $s(x)$ in the form of a polynomial determinant, we need to permute the columns of $\widehat{M_{j}}$ so that $M_{\gamma_{j}}$ becomes the first column. Denote the resulting matrix with $\widetilde{M_{j}}$ . Then $\det\widehat{M_{j}}=(-1)^{j+\gamma_{j}}\det\widetilde{M_{j}}$ and hence

	$\displaystyle s(x)=$	$\displaystyle\sum_{j=0}^{k}(-1)^{\sigma_{j}+j+\gamma_{j}}\det\widetilde{M_{j}}\cdot x^{\gamma_{j}}$
	$\displaystyle=$	$\displaystyle\sum_{j=0}^{k}(-1)^{\sum\limits_{0\leq i\leq k}(i+\gamma_{i})}\det\widetilde{M_{j}}\cdot x^{\gamma_{j}}$
	$\displaystyle=$	$\displaystyle(-1)^{\sum\limits_{0\leq i\leq k}(i+\gamma_{i})}\sum_{j=0}^{k}\det\widetilde{M_{j}}\cdot x^{\gamma_{j}}$
	$\displaystyle=$	$\displaystyle(-1)^{\sum\limits_{0\leq i\leq k}(i+\gamma_{i})}\operatorname{detp}_{P}\begin{bmatrix}M_{\gamma_{0}}&\cdots&M_{\gamma_{k}}&M_{0}&M_{1}&\cdots&M_{\gamma_{0}-1}&M_{\gamma_{0}+1}&\cdots\end{bmatrix}$

By Lemma 28, $s(x)$ can be written as

	$\displaystyle s(x)=$	$\displaystyle(-1)^{\sum\limits_{0\leq i\leq k}(i+\gamma_{i})+\ell k}\cdot x^{\gamma_{0}}$
		$\displaystyle\cdot\det\begin{bmatrix}M_{\gamma_{0}}&\cdots&M_{\gamma_{k}}&M_{0}&M_{1}&\cdots&M_{\gamma_{0}-1}&M_{\gamma_{0}+1}&\cdots\\ x^{\gamma_{1}-\gamma_{0}}&-1\\ &\ddots&\ddots\\ &&x^{\gamma_{k}-\gamma_{k-1}}&-1\end{bmatrix}$

which gives us a determinant representation for the generalized subresultant polynomial $s(x)$ .

References

Amiraslani et al. (2006) A. Amiraslani, D. A. Aruliah, and R. M. Corless. The Rayleigh quotient iteration for generalized companion matrix pencils. Preprint, 2006.
Arnon et al. (1984) D. S. Arnon, G. E. Collins, and S. McCallum. Cylindrical algebraic decomposition I: The basic algorithm. SIAM Journal on Computing, 13(4):865–877, 1984.
Aruliah et al. (2007) D. A. Aruliah, R. M. Corless, L. Gonzalez-Vega, and A. Shakoori. Geometric applications of the Bezout matrix in the Lagrange basis. In Proceedings of the 2007 International Workshop on Symbolic-Numeric Computation, SNC ’07, pages 55–64, 2007.
Barnett (1969) S. Barnett. Degrees of greatest common divisors of invariant factors of two regular polynomial matrices. In Mathematical Proceedings of the Cambridge Philosophical Society, volume 66, pages 241–245. Cambridge University Press, 1969.
Barnett (1970) S. Barnett. Degrees of greatest common divisors of invariant factors of two regular polynomial matrices. Linear Algebra and its Applications, 3(1):7–9, 1970.
Barnett (1971) S. Barnett. Greatest common divisor of several polynomials. In Mathematical Proceedings of the Cambridge Philosophical Society, volume 70, pages 263–268. Cambridge University Press, 1971.
Barnett (1983) S. Barnett. Polynomials and Linear Control Systems. Marcel Dekker, Inc., USA, 1983.
Bini & Gemignani (2004) D. A. Bini and L. Gemignani. Bernstein-Bezoutian matrices. Theoretical Computer Science, 315(2-3):319–333, 2004.
Bostan et al. (2017) A. Bostan, C. D’Andrea, T. Krick, A. Szanto, and M. Valdettaro. Subresultants in multiple roots: an extremal case. Linear Algebra and its Applications, 529:185–198, 2017.
Carnicer & Peña (1993) J. M. Carnicer and J. M. Peña. Shape preserving representations and optimality of the Bernstein basis. Advances in Computational Mathematics, 1(2):173–196, 1993.
Cheng & Labahn (2006) H. Cheng and G. Labahn. On computing polynomial GCD in alternate bases. In Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC ’06, pages 47–54, 2006.
Collins (1967) G. E. Collins Subresultants and reduced polynomial remainder sequences. Journal of the ACM (JACM), 14(1):128–142, 1967.
Collins & Hong (1991) G. E. Collins and H. Hong. Partial cylindrical algebraic decomposition for quantifier elimination. Journal of Symbolic Computation, 12(3):299–328, 1991.
Corless & Litt (2001) R. M. Corless and G. Litt. Generalized companion matrices for polynomials not expressed in monomial bases. Preprint, 2001.
Cox & D’Andrea (2021) D. A. Cox and C. D’Andrea. Subresultants and the Shape Lemma. Mathematics of Computation, 92:2355–2379, 2021.
D’Andrea & Krick (2023) C. D’Andrea and T. Krick. Corrigendum to “Multivariate subresultants in roots” [J. Algebra Appl. 302(1) (2006) 16–36]. Journal of Algebra, 635:838–839, 2023.
D’Andrea et al. (2006) C. D’Andrea, T. Krick, and A. Szanto. Multivariate subresultants in roots. Journal of Algebra, 302(1):16–36, 2006.
Delgado & Peña (2003) J. Delgado and J. M. Peña. A shape preserving representation with an evaluation algorithm of linear complexity. Computer Aided Geometric Design, 20(1):1–10, 2003.
Diaz-Toca & Gonzalez-Vega (2002) G. M. Diaz-Toca and L. Gonzalez-Vega. Barnett’s theorems about the greatest common divisor of several univariate polynomials through Bezout-like matrices. Journal of Symbolic Computation, 34(1):59–81, 2002.
Diaz-Toca & Gonzalez-Vega (2004) G. M. Diaz-Toca and L. Gonzalez-Vega. Various new expressions for subresultants and their applications. Applicable Algebra in Engineering, Communication and Computing, 15(3):233–266, 2004.
Farouki & Rajan (1987) R. T. Farouki and V. T. Rajan. On the numerical condition of polynomials in Bernstein form. Computer Aided Geometric Design, 4(3):191–216, 1987.
Fuhrmann (1996) P. A. Fuhrmann. A Polynomial Approach to Linear Algebra. Springer-Verlag, Berlin, Heidelberg, 1996.
Gonzalez-Vega et al. (1998) L. Gonzalez-Vega, T. Recio, H. Lombardi, and M.-F. Roy. 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), pages 300–316. Springer, 1998.
Goodman & Said (1991) T. N. T. Goodman and H. B. Said. Shape preserving properties of the generalised Ball basis. Computer Aided Geometric Design, 8(2):115–121, 1991.
Ho (1989) C.-J. Ho. Topics in Algebraic Computing: Subresultants, GCD, Factoring and Primary Ideal Decomposition. PhD thesis, USA, 1989.
Hong (1999) H. Hong. Subresultants in Roots. Technical report, Department of Mathematics. North Carolina State University, 1999.
Hong (2002) H. Hong. Subresultants in roots. In Proceedings of 8th International Conference on Applications of Computer Algebra (ACA’2002). Volos, Greece, 2002.
Hong & Yang (2021) H. Hong and J. Yang. Subresultant of several univariate polynomials. arXiv:2112.15370, 2021.
Hong & Yang (2023) H. Hong and J. Yang. Computing greatest common divisor of several parametric univariate polynomials via generalized subresultant polynomials. arXiv:2401.00408, 2023.
Hou & Wang (2000) X. R. Hou & D. M. Wang. Subresultants with the Bézout matrix. In Computer Mathematics, pages 19–28. World Scientific, 2000.
Imbach et al. (2017) R. Imbach, G. Moroz, and M. Pouget. A certified numerical algorithm for the topology of resultant and discriminant curves. Journal of Symbolic Computation, 80:285–306, 2017.
Kapur et al. (1994) D. Kapur, T. Saxena, and L. Yang. Algebraic and geometric reasoning using Dixon resultants. In Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC ’94, pages 99–107, 1994.
Lascoux & Pragacz (2003) A. Lascoux and P. Pragacz. Double Sylvester sums for subresultants and multi-Schur functions. Journal of Symbolic Computation, 35(6):689–710, 2003.
Li (2006) Y. B. Li. A new approach for constructing subresultants. Applied Mathematics and Computation, 183(1):471–476, 2006.
Marco & Martínez (2007) A. Marco and J.-J. Martínez. Bernstein–Bezoutian matrices and curve implicitization. Theoretical computer science, 377(1-3):65–72, 2007.
Mishra (1993) B. Mishra. Algorithmic Algebra. Springer-Verlag, Berlin, Heidelberg, 1993.
Perrucci & Roy (2018) D. Perrucci and M.-F. Roy. A new general formula for the Cauchy index on an interval with subresultants. Journal of Symbolic Computation, 109:465–481, 2018.
Roy & Szpirglas (2020) M.-F. Roy and A. Szpirglas. Sylvester double sums, subresultants and symmetric multivariate Hermite interpolation. Journal of Symbolic Computation, 96:85–107, 2020.
Sylvester (1853) J. Sylvester. On a theory of syzygetic relations of two rational integral functions, comprising an application to the theory of Sturm’s functions, and that of the greatest algebraic common measure. Phil. Trans, 143:407–548, 1853.
Szanto (2008) A. Szanto. Solving over-determined systems by subresultant methods (with an appendix by Marc Chardin). Journal of Symbolic Computation, 43(1):46–74, 2008.
Terui (2008) A. Terui. Recursive polynomial remainder sequence and its subresultants. Journal of Algebra, 320(2):633–659, 2008.
Wang (1998) D. M. Wang. Decomposing polynomial systems into simple systems. Journal of Symbolic Computation, 25(3):295–314, 1998.
Wang (2000) D. M. Wang. Computing triangular systems and regular systems. Journal of Symbolic Computation, 30(2):221–236, 2000.

Subresultants of Several Univariate Polynomials in Newton Basis11footnotemark: 1

Abstract

keywords:

1 Introduction

2 Preliminaries

2.1 Sylvester subresultant polynomial of two polynomials

Definition 1 (Determinant polynomial).

Definition 2 (Subresultant polynomial, Diaz-Toca & Gonzalez-Vega (2004)).

Example 3.

2.2 Companion matrix and Barnett matrix

Example 4.

2.3 Subresultant polynomial in roots

Example 5.

3 Problem Statement

3.1 Subresultant polynomial of several univariate polynomials

Notation 6.

Definition 7.

Remark 8.

Example 9.

Definition 10 (Hong & Yang (2021)).

Theorem 11 (Hong & Yang (2021)).

Remark 12.

Example 13.

3.2 Determinant polynomial in Newton basis

Definition 14 (Newton basis).

Definition 15 (Determinant polynomial with respect to a polynomial set).

Remark 16.

3.3 Problem statement

Problem 17.

4 Main Result

Definition 18 (Companion matrix in Newton basis, Corless & Litt (2001)).

Theorem 19 (Main result).

Remark 20.

Example 21.

5 Proof of the Main Result (Theorem 19)

Notation 22.

Lemma 23.

Remark 24.

Proof.

Proposition 25.

Lemma 26.

Proof.

Remark 27.

Lemma 28.

Proof.

Remark 29.

Proof of Theorem 19.

6 Application

Proposition 30.

Proof.

Example 31 (Continued from Example 21).

7 Conclusion

Appendix

References

Subresultants of Several Univariate Polynomials in Newton Basis¹¹footnotemark: 1