Bivariate Polynomial Matrix and Smith Form

Abstract

Matrix equivalence plays a pivotal role in multidimensional systems, which are typically represented by multivariate polynomial matrices. The Smith form of matrices is one of the important research topics in polynomial matrices. This article mainly investigates the Smith forms of several types of bivariate polynomial matrices and has successfully derived several necessary and sufficient conditions for matrix equivalence.

1. Introduction

In the multidimensional system theory, which finds extensive applications in sectors like image processing, linear multipass processes, geophysical exploration, and iterative learning control systems, the study of system equivalence stands as a pivotal research challenge. A significant aspect within this framework is the equivalence of multivariate polynomial matrices, further extending its relevance to multidimensional signal analysis and other related areas. A key rationale for converting a multivariate polynomial matrix into its Smith normal form is to simplify a multidimensional system into an equivalent representation with a reduced number of equations and unknowns [1,2,3,4,5,6,7].

Since the univariate polynomial ring is a principal ideal domain endowed with the Euclidean division property, it is always the case that univariate polynomial matrices are equivalent to their Smith forms. This essentially means the problem of equivalence for univariate polynomial matrices has been resolved. Despite the progress made in understanding the equivalence problem for multivariate polynomial matrices with two or more variables, as demonstrated by research on bivariate polynomial matrix equivalences, the broader issue is still unresolved when dealing with more than two variables. In fact, studies concerning systems with three or more variables have, to date, primarily focused on specific subclasses of such matrices, as illustrated in works like [8,9,10,11,12,13,14].

In 2006, Lin et al. in [15] presented necessary and sufficient conditions for a full-rank multivariate polynomial matrix $A\left(x\right)$ to be equivalent to its Smith form by introducing the addition of ZLP (zero left pseudo-inverse) or ZRP (zero right pseudo-inverse) matrices. However, for matrices with unequal row and column, the investigation shifts to the greatest common divisor (GCD) of their highest-degree principal minors. The study culminated in the derivation of several critical subclasses of specialized polynomial matrices, each accompanied by rigorous criteria that ensure their equivalence to their respective Smith forms. 2018, Liu et al. established that for a square matrix A with $det\left(A\right)={(x_1-f(x_2,...,x_n))}^t$, it holds that A can be transformed into a diagonal form diag$\{1,...,1,det(A\left)\right\}$ if and only if the ideal generated by $det\left(A\right)$ and all the minors of lower one order of A is a unit ideal [16]. In 2020, Liu et al. presented the necessary and sufficient conditions for a bivariate polynomial matrix to be equivalent to its Smith form (with $s-1$ ones on the main diagonal) when the greatest common divisor of its highest-degree minors is $p{\left(z\right)}^q$, where $p\left(z\right)$ is an irreducible polynomial in the single variable z [17]. Subsequently, the researchers further advanced their investigations into the equivalence of various classes of multivariate polynomial matrices, successfully deriving specific criteria to ascertain when these matrices are indeed equivalent to their Smith forms [15,18,19,20,21,22,23,24,25].

In this paper, our primary focus is the equivalence problem regarding bivariate polynomial matrices. This study primarily delves into the Smith forms of special types of polynomial matrices. That is, $F\in K^{l\times m}[x,y],(l\le m)$ is the row full rank, and its greatest common factor among all of its $l-1$ minors is $d\left(F\right)={(y-f\left(x\right))}^{q_1}p{\left(x\right)}^{q_2}$, where $q_1,q_2$ are non-negative integers, $p\left(x\right)$ is an irreducible polynomial, and $f\left(x\right)$ is arbitrary. We consider the problem of the necessary and sufficient condition for the equivalence of F and its Smith form.

The paper is organized as follows: we introduce foundational concepts pertaining to the equivalence of polynomial matrices. In Section 3, we delve into the pivotal conditions that are both necessary and sufficient for establishing the equivalence of polynomial matrices, concurrently encompassing their corresponding Smith forms.

2. Preliminaries

In this article, we adopt the following commonly used symbols.

The polynomial ring $\mathbb{R}$ over the field $\mathbb{K}$ is defined as the set of all polynomials in n variables, where these variables are $x_1,x_2,\cdots ,x_n$, denoted as: $R=K[x_1,x_2,\dots ,x_n]$. Furthermore, the set of $s\times t$ matrices with entries in $K[x_1,x_2,\dots ,x_n]$ are denoted as $K^{s\times t}[x_1,x_2,\dots ,x_n]$, and $K^{s\times t}\left[x\right]$ is the set of $s\times t$ matrices with their entriesin $K\left[x\right]$. The $s\times s$ identity matrix is represented as $I_s$ and $0_{s,t}$ refers to the $s\times t$ zero matrix. For $P\in K^{s\times t}[x_1,x_2,\dots ,x_n]$, we use $d_i\left(P\right)$ to denote the greatest common divisor for all the $i\times i$ minors of the matrix P, where $(i=1,...,s)$.

Subsequently, we will present several basic concepts. For detailed information regarding the following definition in this section, please refer to [15,16,17,18,19,20,21].

Definition 1.

Let $A\in K^{s\times t}\left[x\right]$. The largest order for all its nonzero minors is called the rank of A.

Definition 2.

For a matrix $A\in K^{s\times t}\left[x\right]$, let us denote its $i\times i$ minors for each i from 1 up to the rank r of A as $a_{i1},\dots ,a_{i{\beta }_i}$, where ${\beta }_i$ represents the number of such minors. The greatest common divisor among these minors is termed $d_i\left(A\right)$. Subsequently, the normalized or reduced ith order minors of A are expressed as follows:

$$b_{i1}\triangleq \frac{a_{i1}}{d_i\left(A\right)},\dots ,b_{i{\beta }_i}\triangleq \frac{a_{i{\beta }_i}}{d_i\left(A\right)}.$$

With respect to the provided definition, the ideal $J_i\left(F\right)$ signifies the one generated by the set $b_{i1},...,b_{i{\beta }_i}$, while we abbreviate $d\left(F\right)\triangleq d_r\left(F\right)$.

Definition 3.

Let $A_1$ and $A_2$ be two matrices in $R^{s\times t}$, $A_1$ and $A_2$ are said to be equivalent if there two invertible matrices $M\in R^{s\times s}$ and $N\in R^{t\times t}$ exist, such that $A_2\sim M{A}_{1}N$.

Definition 4.

Let $A\in K^{s\times s}[x,y]$, then A is said to be unimodular if det(A) is a unit in K.

Definition 5.

Let $M\in R^{s\times t}$ be a full row rank matrix; we say that M is zero left prime (zero right prime) if the $s\times s(t\times t)$ minors of M generate the unit ideal R.

If $M\in R^{s\times t}$ is zero left prime (zero right prime), then it can be concisely referred to as ZLP (ZRP). The Quillen–Suslin theorem [8] asserts that the matirx M exhibits the ZLP property if and only if an invertible polynomial matrix $N\in R^{t\times t}$ exists, which allows us to express the multiplication of M and N as $M·N=\left(I_s{0}_{s\times (s-t)}\right)$. The property of being ZLP (ZRP) in R is tantamount to saying that such a matrix can be completed in an unimodular (invertible) matrix.

In 1977, Suslin [5] and Quillen [6], independently arriving at affirmative solutions, confirmed Serre’s renowned conjecture, and identified a correspondence between ZLP matrices and unimodular matrices.

Theorem 1

(Quillen–Suslin Theorem). Let $F\in K^{s\times t}[x,y]$ be a ZLP matrix with $s<t$, then a unimodular matrix $U\in K^{t\times t}[x,y]$ can be constructed, such that F is its first l rows.

Various approaches to the Quillen–Suslin Theorem for a comprehensive treatment exist; we refer to [15,20,21,22,23,24,25].

Definition 6.

Let $A\in R^{s\times t}(s\le t)$, and ${\mathrm{\Phi }}_i$ is a polynomially defined as follows:

$${\mathrm{\Phi }}_i=\left\{\begin{array}{cc}{d}_i\left(A\right)/d_{i-1}\left(A\right),\hfill & \mathit{if}1\le i\le r\hfill \\ 0,\hfill & \mathit{if}r<i\le t\hfill \end{array}\right.$$

where r is the rank of A, $d_0\left(A\right)\equiv 1$, and ${\mathrm{\Phi }}_i$ satisfies:

$${\mathrm{\Phi }}_1|{\mathrm{\Phi }}_2|...|\mathrm{\Phi }r,$$

we define the Smith form of A as:

$$S=\left(diag\left\{{\mathrm{\Phi }}_i\right\}{0}_{s\times (t-s)}\right).$$

3. Main Results

Lemma 1

(see [16]). Let $F,F_1,F_2\in K^{l\times l}[x,y]$, $F=F_1{F}_2$. Then, the $l-1$ minors of $F_i$ have no common zeros in the algebraic closed field of K if the $l-1$ minors of F have no common zeros in the algebraic closed field of K, where $i=1,2.$

Throughout the subsequent discourse, we consistently employ and expound upon the properties of the following three matrices:

$$\left(\begin{array}{cc}{I}_{l-1}& 0\\ 0& p\left(x\right)\end{array}\right),\left(\begin{array}{cc}{I}_{l-1}& 0\\ 0& y-f\left(x\right)\end{array}\right),\left(\begin{array}{cc}{I}_{l-1}& 0\\ 0& (y-f(x\left)\right)p\left(x\right)\end{array}\right),$$

where $p\left(x\right)$ is an irreducible polynomial in the indeterminate x and $f\left(x\right)$ is an arbitrary polynomial in the same indeterminate. We denote these matrices as $P_1(x,y),P_2(x,y)$, and $P(x,y)$, respectively.

Lemma 2.

Let $F(x,y)\in K^{l\times l}[x,y]$, and $F(x,y)=P_1·$$V·$$P_2$, where $V\in K^{l\times l}\left[x\right]$ is a unimodular matrix. Then $F(x,y)\sim P(x,y),$ if all the $l-1$ minors of $F(x,y)$ generate a unit ideal.

Proof. 

We prove this by induction on l.

First, when $l=2$, let $V=\left(\begin{array}{cc}{v}_{11}& v_{12}\\ v_{21}& v_{22}\end{array}\right)$, then

$$F(x,y)=\left(\begin{array}{cc}{v}_{11}& v_{12}·(y-f\left(x\right))\\ v_{21}·p\left(x\right)& v_{22}·p\left(x\right)·(y-f\left(x\right))\end{array}\right).$$

We now prove that the column vector ${(v_{11}{v}_{21}·p\left(x\right))}^T$ is a unimodular column for $F(x,y)$.

If ${(v_{11}{v}_{21}·p\left(x\right))}^T$ is not a unimodular column, it follows that $v_{11}$ and $v_{21}·p\left(x\right)$ have some common zeros in the algebraic closed field, without a loss of generality, assume it to be a (an arbitrary element), then $(a,f(a\left)\right)$ is a common zero of $v_{11},v_{12}·(y-f\left(x\right)),v_{21}·p\left(x\right),v_{22}·p\left(x\right)·(y-f\left(x\right))$; this contradicts the condition that all $l-1$ minors of $F(x,y)$ generate a unit ideal.

From the theorem of $Quillen--Suslin$, there is a unimodular matrix $M_1$, such that

$$M_1·{(v_{11}{v}_{21}·p\left(x\right))}^T={\left(10\right)}^T,$$

as ${(v_{11}{v}_{21}·p\left(x\right))}^T$ is a unimodular column. So

$$M_1·F=\left(\begin{array}{cc}1& h_1(x,y)\\ 0& h_2(x,y)\end{array}\right).$$

Next,

let $M_2=\left(\begin{array}{cc}1& -h_1(x,y)\\ 0& 1\end{array}\right)$, clearly, $M_2$ is a unimodular matrix, and

$$M_1·F·M_2=\left(\begin{array}{cc}1& 0\\ 0& h_2(x,y)\end{array}\right),$$

by the properties of determinants, it follows that $F(x,y)\sim P(x,y).$

Assuming the conclusion holds for $l-1$, we demonstrate its validity for l. Let $V=\left(\begin{array}{cc}{V}_{11}& V_{12}\\ V_{21}& V_{22}\end{array}\right)$, $V_{11}\in K^{(l-1)\times (l-1)}\left[x\right]$, $V_{12}\in K^{(l-1)\times 1}\left[x\right]$, $V_{21}\in K^{1\times (l-1)}\left[x\right]$, $V_{22}\in K^{1\times 1}\left[x\right]$, then

$$F(x,y)=\left(\begin{array}{cc}{V}_{11}& V_{12}·(y-f\left(x\right))\\ V_{21}·p\left(x\right)& V_{22}·p\left(x\right)·(y-f\left(x\right))\end{array}\right).$$

Assume ${(v_{11}{v}_{21}\cdots v_{(l-1)1}{v}_{l1})}^T$ is the first column of matrix V; from Laplace’s Expansion, we have

$$v_{11}{a}_1+\cdots +v_{(l-1)1}{a}_{l-1}+v_{l1}{a}_l=detV\in K,$$

where $a_1,\cdots ,a_{l-1},a_l$ is the $l-1$ minor.

We demonstrate that the first column vector of the matrix $F(x,y)$, provided by ${(v_{11}{v}_{21}\cdots v_{(l-1)1}{v}_{l1}·q\left(x\right))}^T$, constitutes a unimodular column, which implies it has no common zeros.

If a common zero exists, then such a common zero must necessarily be of the form a, because

$$F(x,y)=\left(\begin{array}{cccc}{v}_{11}& \cdots & v_{1l-1}& v_{1l}·(y-f\left(x\right))\\ v_{21}& \cdots & v_{2l-1}& v_{2l}·(y-f\left(x\right))\\ \cdots & \cdots & \cdots & \cdots \\ p\left(x\right)v_{l1}& \cdots & p\left(x\right)v_{ll-1}& p\left(x\right)v_{ll}(y-f\left(x\right))\end{array}\right).$$

The $l-1$ minors of $F(x,y)$ can be categorized into two cases:

(1)

Those that involve the first column.

(2)

Those that do not involve the first column.

If the $l-1$ minor includes the first column, and assuming that the first column has a common zero a (an arbitrary element), then this particular $l-1$ minor will also have a common zero a.

If the $l-1$ minor does not involve the first column, it must necessarily include the last column, as $(a,f(a\left)\right)$ is a zero of the last column. Therefore, under this condition, these particular $l-1$ minors would have a common zero at $(a,f(a\left)\right)$.

In summary, based on the above analysis, all $l-1$ minors of $F(x,y)$ seem to have a common zero $(a,b)$, which contradicts the premise that these minors generate a unit ideal.

Therefore, the first column of matrix $F(x,y)$ constitutes a unimodular column.

Further, $N_1$ exist, such that

$$N_1·F=\left(\begin{array}{cc}1& Q_1(x,y)\\ 0_{l-1,1}& Q_2(x,y)\end{array}\right),$$

where $Q_2\in K^{(l-1)\times (l-1)}[x,y]$. Let $N_2=\left(\begin{array}{cc}1& -Q_1(x,y)\\ 0& I_{l-1}\end{array}\right)$, then

$$N=N_1·F·N_2=\left(\begin{array}{cc}1& 0_{1,l-1}\\ 0_{l-1,1}& Q_2(x,y)\end{array}\right),$$

From the induction hypothesis, unimodular matrices exist $B_1,B_2$ such that

$$B_1·Q_2·B_2=\left(\begin{array}{cc}{I}_{l-2}& 0\\ 0& p\left(x\right)(y-f(x\left)\right)\end{array}\right),$$

so

$$\left(\begin{array}{cc}1& 0\\ 0& B_1\end{array}\right)·N·\left(\begin{array}{cc}1& 0\\ 0& B_2\end{array}\right)=\left(\begin{array}{cc}{I}_{l-1}& 0\\ 0& p\left(x\right)(y-f(x\left)\right)\end{array}\right).$$

The proof is completed. □

Theorem 2.

Let $F\in K^{l\times l}[x,y]$ with $detF=(y-f(x\left)\right)p\left(x\right)$, where $p\left(x\right)$ is an irreducible polynomial, and $f\left(x\right)$ is an arbitrary polynomial. Then, F is the equivalent to its Smith form if and only if all the $l-1$ minors of F generate a unit ideal.

Proof. 

Necessity: If the matrix F is equivalent to its Smith form, denoted by P, then two unimodular matrices exist $U,V\in K^{l\times l}[x,y]$, such that $F=UPV$. Noticing that the $l-1$ minors of P have no common zeros, by Lemma 1, we deduce that the $l-1$ minors of F also do not have any common zero. Consequently, it follows that the $l-1$ minors of matrix F generate a unit ideal.

Sufficiency: If all the $l-1$ minors of F generate a unit ideal, then a decomposition exists:

$$U·F·V=\left(\begin{array}{cc}{I}_{l-1}& 0\\ 0& q\left(x\right)\end{array}\right)·G,$$

where the determinant of matrix G is a weak linear form in $(y-f(x\left)\right)$. Upon further factorization of G, we obtain

$$U·F·V=\left(\begin{array}{cc}{I}_{l-1}& 0\\ 0& q\left(x\right)\end{array}\right)·M_1·\left(\begin{array}{cc}{I}_{l-1}& 0\\ 0& y-f\left(x\right)\end{array}\right)·G_1,$$

where $G_1,M_1$ are unimodular matrices, and $M_1$ solely depends on the variable x. From Lemma 2, which asserts the equivalence under such conditions, this leads to the desired conclusion. □

Lemma 3

([17]). Let $D=\left(\begin{array}{cc}{I}_{l-1}& 0\\ 0& h(x,y)\end{array}\right)$ be a matrix in $K^{l\times l}[x,y]$, and $V\in K^{l\times l}[x,y]$ is a unimodular matrix. Then, $D^{s}V{D}^t$ is equivalent to $D^{s+t}$ if the all $l-1$ minors of $D^{s}V{D}^t$ generate a unit ideal, where $s,t$ are positive integers.

Theorem 3.

Let $F\in K^{l\times l}[x,y]$ with $detF={(y-f\left(x\right))}^{q}p{\left(x\right)}^q$, where q is a positive integer, $p\left(x\right)$ is an irreducible polynomial, and $f\left(x\right)$ is an arbitrary polynomial. Then. F is equivalent to its Smith form if and only if all the $l-1$ minors of F generate a unit ideal.

Proof. 

Sufficiency: Unimodular matrices $U_1,V_1\in K^{l\times l}[x,y]$ exist, such that

$$F=F_1{P}_1{V}_1{P}_2{U}_1,$$

where $det{F}_1={(y-f\left(x\right))}^{q-1}p{\left(x\right)}^{q-1}$, as all the $l-1$ minors of F generate a unit ideal. By lemma 1, the $l-1$ minors of $P_1{V}_1{P}_2$ can also generate a unit ideal. By Lemma 2, there are unimodular matrices $M_1,N_1\in K^{l\times l}[x,y]$, such that

$$F=F_1{M}_{1}P{N}_1{U}_1,$$

Then, the $l-1$ minors of $F_1$ generate a unit ideal from Lemma 1.

If $q\ge 2$, by repeating the above operation, we obtain

$$F_1=F_2{M}_{2}P{N}_2,$$

where $det{F}_2={(y-f\left(x\right))}^{q-2}p{\left(x\right)}^{q-2}$, then:

$$F=F_2{M}_{2}P{N}_2{M}_{1}P{N}_1{U}_1.$$

Furthermore, from Lemma 3, unimodular matrices $L_1,T_1\in K^{l\times l}[x,y]$ exist, such that $P{N}_2{M}_{1}P$ is equal to $L_1{P}^2{T}_1$, then

$$F=F_2{M}_2{L}_1{P}^2{T}_1{N}_1{U}_1.$$

Let $V_2=M_2{L}_1$, $U_2=T_1{N}_1{U}_1$, then $V_2,U_2$ are all unimodular matrices, and

$$F=F_2{V}_2{P}^2{U}_2.$$

If $q\ge 3$, by repeating the above operation, we can obtain

$$F=F_q{V}_q{P}^q{U}_q.$$

From the properties of the determinants, $det{F}_q=c,c\in K$, so $F_q$ is a unimodular matrix, and F is equivalent to its Smith form $P^q$.

Sufficiency: There are two unimodular matrices $U,V\in K^{l\times l}[x,y]$, such that $F=U{P}^{q}V$ if F is equivalent to its Smith form $P^q$.

Noting that the $l-1$ minors of $P^q$ have no common zeros, from Lemma 1, we infer that the $l-1$ minors of F also do not have any common zero. Consequently, this implies that the $l-1$ minors of F generate a unit ideal. □

Next, we consider the equivalence problem between $detF={(y-f\left(x\right))}^{q_1}p{\left(x\right)}^{q_2}$ and its Smith form, where $q_1,q_2$ are distinct positive integers.

Lemma 4.

Provided a matrix of $F\in K^{l\times l}[x,y]$ with $det\left(F\right)=p^t$, where p is an irreducible polynomial in $K\left[x\right]$ and t is a positive integer. Then, F is said to be equivalent to its Smith normal form if and only if the $J_i\left(F\right)=K[x,y]$ with $i=1,2,...,l$.

Theorem 4.

Let $F\in K^{l\times l}[x,y]$ with $detF={(y-f\left(x\right))}^{q_1}p{\left(x\right)}^{q_2}$, where $q_1,q_2$ are distinct positive integers, $p\left(x\right)$ is an irreducible polynomial, and $f\left(x\right)$ is an arbitrary polynomial, then F is equivalent to its Smith form $Q(x,y)=\left(\begin{array}{cc}{I}_{l-1}& 0\\ 0& {(y-f\left(x\right))}^{q_1}p{\left(x\right)}^{q_2}\end{array}\right)$ if and only if the $l-1$ minors generate a unit ideal.

Proof. 

Sufficiency:

(1)

$q_1=q_2=0$, then the conclusion evidently holds.

(2)

If $q_1$ or $q_2=0$, then from Lemma 4, the fact that each level of the invariant factors generates a unit ideal implies the equivalence between the provided matrix and its Smith normal form.

(3)

If $q_1,q_2$ are positive integers, $q_1>q_2$ or $q_2>q_1$, without a loss of generality, assume $q_1>q_2$, by Theorem 2, F is equivalent to

$$P^{q_2}V{P}_1^{q_1-q_2}=\left(\begin{array}{cc}{v}_{11}& v_{12}{P}_1^{q_1-q_2}\\ v_{21}·P_1^{q_2}{P}_2^{q_2}& v_{22}·P_1^{q_1}{P}_2^{q_2}\end{array}\right).$$

In fact, we can prove that the matrix $\left(v_{11}{v}_{12}{P}_1^{q_1-q_2}\right)$ is a ZLP matrix. If $c_1,...,c_l$ are all the $l-1$ minors of $\left(v_{11}{v}_{12}\right)$, then the $l-1$ minors of $\left(v_{11}{v}_{12}{P}_1^{q_1-q_2}\right)$ are:

$$c_1,c_2{P}_1^{q_1-q_2},...,c_l{P}_1^{q_1-q_2}.$$

Next, we prove that these minors have no common zeros.

If ${\alpha }_0$ is the common zero of $c_1,c_2{P}_1^{q_1-q_2},...,c_l{P}_1^{q_1-q_2}$, then ${\alpha }_0$ is a common zero of $c_1$ and $P_1$ as $c_1,...,c_l$ have no common zeros.

Observing that the last row of the matrix $P^{q_2}V{P}_1^{q_1-q_2}$, and $P_1$ is a common factor of it, it follows that its $l-1$ minors will either be constant multiples $c_1$ or contain $P_1$ as a common factor. Therefore, the $l-1$ minors of the matrix $P^{q_2}V{P}_1^{q_1-q_2}$ have common zeros. This is in contradiction with the condition that these minors should not have any common zeros. Thus, our initial assumption must be incorrect, and hence we can conclude that the $l-1$ minors of $\left(v_{11}{v}_{12}{P}_1^{q_1-q_2}\right)$ indeed do not share any common zeros and it is a ZLP matrix. From the Quillen–Suslin theorem, there is a unimodular matrix $N\in K^{l\times l}[x,y]$ such that

$$\left(v_{11}{v}_{12}{P}_1^{q_1-q_2}\right)N=\left(I_{l-1}{0}_{l-1,1}\right),$$

then

$$P^{q_2}V{P}_1^{q_1-q_2}N=\left(\begin{array}{cc}{I}_{l-1}& 0_{l-1,1}\\ H_1& H_2\end{array}\right).$$

By means of elementary row operations, a unimodular matrix exists $N_1\in K^{l\times l}[x,y]$ such that

$$N_1{P}^{q_2}V{P}_1^{q_1-q_2}N=\left(\begin{array}{cc}{I}_{l-1}& 0_{l-1,1}\\ 0_{1,l-1}& H_2\end{array}\right).$$

From the properties of determinants, $det{H}_2=u{(y-f\left(x\right))}^{q_1}p{\left(x\right)}^{q_2},u\in K$ as $N,N_1$ are all unimodular matrices, so F is equivalent to its Smith form Q.

Necessity: There are two unimodular matries $U,V\in K^{l\times l}[x,y]$, such that $F=UQV$ if F is equivalent to its Smith form Q. From Lemma 1, the $l-1$ minors of F have no common zeros as the $l-1$ minors of Q have no common zeros, so the $l-1$ minors of F can generate a unit ideal. □

Theorem 5.

Let $F\in K^{l\times m}[x,y],(l\le m)$ is row full rank, and its greatest common factor among all of its l minors is $d\left(F\right)={(y-f\left(x\right))}^{q_1}p{\left(x\right)}^{q_2}$, where $q_1,q_2$ are non-negative integers, $p\left(x\right)$ is an irreducible polynomial and $f\left(x\right)$ is arbitrary, then F is equivalent to its Smith normal form:

$$Q(x,y)=\left(B(x,y)0_{l\times (m-l)}\right)$$

if and only if the $l-1$ minors of F generate a unit ideal, where

$$B(x,y)=\left(\begin{array}{cc}{I}_{l-1}& 0\\ 0& d\left(F\right)\end{array}\right).$$

Proof. 

Sufficiency: From the ZLP decomposition theorem in [2], $G\in K^{l\times l}[x,y]$, $F_0\in K^{l\times m}[x,y]$ exists, such that $F=G·F_0$, where $detG=d\left(F\right)$, $F_0$ is a ZLP matrix. Combined with Theorem 3, there are unimodular matrices $V_1,V_2\in K^{l\times l}[x,y]$, such that $G=V_{1}B{V}_2$, then

$$F=V_{1}B{V}_2{F}_0.$$

It is clear that $V_2{F}_0$ is also a ZLP matrix, and from the Quillen–Suslin theorem, there is a unimodular matrix $U\in K^{m\times m}[x,y]$. such that $V_2{F}_0$ is its l row. So

$$F=V_1\left(B{0}_{l,m-l}\right)U=V_{1}QU.$$

Necessity: If F is equivalent to its Smith normal form Q, and the $l-1$ minors of Q have no common zeros, From Lemma 1, the $l-1$ minors of F also have no common zeros, which means the $l-1$ minors of F generate a unit ideal. □

4. Conclusions

This paper investigates the Smith forms of several types of bivariate polynomial matrices, establishing some important results. We have established a set of criteria that ascertain the equivalence between these matrices and their corresponding Smith forms. One can verify these criteria effortlessly by examining if the $l-1$ minors of the provided matrix generate a unit ideal. These results hold considerable theoretical significance and scientific value in the fields of multidimensional systems, circuit analysis, and symbolic computation.