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
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
, it holds that
A can be transformed into a diagonal form diag
if and only if the ideal generated by
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
ones on the main diagonal) when the greatest common divisor of its highest-degree minors is
, where
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, is the row full rank, and its greatest common factor among all of its minors is , where are non-negative integers, is an irreducible polynomial, and 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 over the field is defined as the set of all polynomials in n variables, where these variables are , denoted as: . Furthermore, the set of matrices with entries in are denoted as , and is the set of matrices with their entriesin . The identity matrix is represented as and refers to the zero matrix. For , we use to denote the greatest common divisor for all the minors of the matrix P, where .
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 . The largest order for all its nonzero minors is called the rank of A.
Definition 2. For a matrix , let us denote its minors for each i from 1 up to the rank r of A as , where represents the number of such minors. The greatest common divisor among these minors is termed . Subsequently, the normalized or reduced ith order minors of A are expressed as follows: With respect to the provided definition, the ideal signifies the one generated by the set , while we abbreviate .
Definition 3. Let and be two matrices in , and are said to be equivalent if there two invertible matrices and exist, such that .
Definition 4. Let , then A is said to be unimodular if det(A) is a unit in K.
Definition 5. Let be a full row rank matrix; we say that M is zero left prime (zero right prime) if the minors of M generate the unit ideal R.
If
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
exists, which allows us to express the multiplication of
M and
N as
. 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 be a ZLP matrix with , then a unimodular matrix 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 , and is a polynomially defined as follows:where r is the rank of A, , and satisfies:we define the Smith form of A as: 3. Main Results
Lemma 1 (see [
16]).
Let , . Then, the minors of have no common zeros in the algebraic closed field of K if the minors of F have no common zeros in the algebraic closed field of K, where Throughout the subsequent discourse, we consistently employ and expound upon the properties of the following three matrices:
where
is an irreducible polynomial in the indeterminate
x and
is an arbitrary polynomial in the same indeterminate. We denote these matrices as
, and
, respectively.
Lemma 2. Let , and , where is a unimodular matrix. Then if all the minors of generate a unit ideal.
Proof. We prove this by induction on l.
First, when
, let
, then
We now prove that the column vector is a unimodular column for .
If is not a unimodular column, it follows that and have some common zeros in the algebraic closed field, without a loss of generality, assume it to be a (an arbitrary element), then is a common zero of ; this contradicts the condition that all minors of generate a unit ideal.
From the theorem of
, there is a unimodular matrix
, such that
as
is a unimodular column. So
Next,
let
, clearly,
is a unimodular matrix, and
by the properties of determinants, it follows that
Assuming the conclusion holds for
, we demonstrate its validity for
l. Let
,
,
,
,
, then
Assume
is the first column of matrix
V; from Laplace’s Expansion, we have
where
is the
minor.
We demonstrate that the first column vector of the matrix , provided by , 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
The minors of can be categorized into two cases:
- (1)
Those that involve the first column.
- (2)
Those that do not involve the first column.
If the minor includes the first column, and assuming that the first column has a common zero a (an arbitrary element), then this particular minor will also have a common zero a.
If the minor does not involve the first column, it must necessarily include the last column, as is a zero of the last column. Therefore, under this condition, these particular minors would have a common zero at .
In summary, based on the above analysis, all minors of seem to have a common zero , which contradicts the premise that these minors generate a unit ideal.
Therefore, the first column of matrix constitutes a unimodular column.
Further,
exist, such that
where
. Let
, then
From the induction hypothesis, unimodular matrices exist
such that
so
The proof is completed. □
Theorem 2. Let with , where is an irreducible polynomial, and is an arbitrary polynomial. Then, F is the equivalent to its Smith form if and only if all the 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 , such that . Noticing that the minors of P have no common zeros, by Lemma 1, we deduce that the minors of F also do not have any common zero. Consequently, it follows that the minors of matrix F generate a unit ideal.
Sufficiency: If all the
minors of
F generate a unit ideal, then a decomposition exists:
where the determinant of matrix
G is a weak linear form in
. Upon further factorization of
G, we obtain
where
are unimodular matrices, and
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 be a matrix in , and is a unimodular matrix. Then, is equivalent to if the all minors of generate a unit ideal, where are positive integers. Theorem 3. Let with , where q is a positive integer, is an irreducible polynomial, and is an arbitrary polynomial. Then. F is equivalent to its Smith form if and only if all the minors of F generate a unit ideal.
Proof. Sufficiency: Unimodular matrices
exist, such that
where
, as all the
minors of
F generate a unit ideal. By lemma 1, the
minors of
can also generate a unit ideal. By Lemma 2, there are unimodular matrices
, such that
Then, the
minors of
generate a unit ideal from Lemma 1.
If
, by repeating the above operation, we obtain
where
, then:
Furthermore, from Lemma 3, unimodular matrices
exist, such that
is equal to
, then
Let
,
, then
are all unimodular matrices, and
If
, by repeating the above operation, we can obtain
From the properties of the determinants,
, so
is a unimodular matrix, and
F is equivalent to its Smith form
.
Sufficiency: There are two unimodular matrices , such that if F is equivalent to its Smith form .
Noting that the minors of have no common zeros, from Lemma 1, we infer that the minors of F also do not have any common zero. Consequently, this implies that the minors of F generate a unit ideal. □
Next, we consider the equivalence problem between and its Smith form, where are distinct positive integers.
Lemma 4. Provided a matrix of with , where p is an irreducible polynomial in and t is a positive integer. Then, F is said to be equivalent to its Smith normal form if and only if the with .
Theorem 4. Let with , where are distinct positive integers, is an irreducible polynomial, and is an arbitrary polynomial, then F is equivalent to its Smith form if and only if the minors generate a unit ideal.
Proof. Sufficiency:
- (1)
, then the conclusion evidently holds.
- (2)
If or , 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
are positive integers,
or
, without a loss of generality, assume
, by Theorem 2,
F is equivalent to
In fact, we can prove that the matrix
is a ZLP matrix. If
are all the
minors of
, then the
minors of
are:
Next, we prove that these minors have no common zeros.
If is the common zero of , then is a common zero of and as have no common zeros.
Observing that the last row of the matrix
, and
is a common factor of it, it follows that its
minors will either be constant multiples
or contain
as a common factor. Therefore, the
minors of the matrix
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
minors of
indeed do not share any common zeros and it is a ZLP matrix. From the Quillen–Suslin theorem, there is a unimodular matrix
such that
then
By means of elementary row operations, a unimodular matrix exists
such that
From the properties of determinants,
as
are all unimodular matrices, so
F is equivalent to its Smith form
Q.
Necessity: There are two unimodular matries , such that if F is equivalent to its Smith form Q. From Lemma 1, the minors of F have no common zeros as the minors of Q have no common zeros, so the minors of F can generate a unit ideal. □
Theorem 5. Let is row full rank, and its greatest common factor among all of its l minors is , where are non-negative integers, is an irreducible polynomial and is arbitrary, then F is equivalent to its Smith normal form:if and only if the minors of F generate a unit ideal, where Proof. Sufficiency: From the ZLP decomposition theorem in [
2],
,
exists, such that
, where
,
is a ZLP matrix. Combined with Theorem 3, there are unimodular matrices
, such that
, then
It is clear that
is also a ZLP matrix, and from the Quillen–Suslin theorem, there is a unimodular matrix
. such that
is its
l row. So
Necessity: If F is equivalent to its Smith normal form Q, and the minors of Q have no common zeros, From Lemma 1, the minors of F also have no common zeros, which means the minors of F generate a unit ideal. □