Solving Diophantine equations is at the heart of mathematics. It has applications in many branches of mathematics and computer science. The basic problem in the Diophantine theory is to solve a system of linear Diophantine equations.
The system of linear Diophantine equations with integer coefficient matrix is
is the transpose of the vectorof variables. The goal is to find solutions in the ring of integers.
The system obviously has the zero solution . The fundamental problem in the Diophantine approximation theory [Ca] asks how small a nonzero integer solution can be. It is also related to the shortest vector problem (SVP) in computational lattice theory [MG]. Siegel showed the following bound.
Theorem 1.2 ([Si]).
System 1.1.1 has a nonzero integer solution such that
where is an upper bound for absolute coefficients in .
Working on the adèles, Bombieri and Vaaler applied the techniques from Geometry of Numbers, and improved Siegel’s bound.
Theorem 1.3 ([Bv]).
Suppose that has full row rank . System 1.1.1 has a nonzero integer solution such that
where is the greatest common divisor (G.C.D.) of determinants of all minors of .
They also proved a stronger version of Theorem 1.3, similar to Minkowski’s second theorem for successive minima.
Theorem 1.4 ([Bv]).
To compute directly, one has to find determinants of all square matrices which would take exponential time, e.g., when . In this paper, we will prove an efficient verison of Bombieri and Vaaler’s bounds. We work on lattices generated by and the corresponding kernel lattice, and resort only to the basic facts in linear algebra.
An (integer) lattice is an additive subgroup of . A family of vectors is called a basis of the lattice if every vector has the unique representation
Here, we call and the rank and the dimension of the lattice respectively. We also say that the lattice is generated by , and call the matrix with rows a basis matrix for .
The determinant of a lattice plays an important role in the lattice theory. It is an invariant of a lattice independently of the choices of basis matrices.
Let be a lattice with a basis matrix . The determinant of the lattice is defined to be
It is also called the volume of the lattice.
2. Our results
We now define the kernel lattice. One can contrast it with the better-known dual lattice.
The integer solutions of System 1.1.1 forms an additive group. It is called the kernel lattice of , and it is denoted by .
It is well known that the determinant of the dual lattice equals to the inverse of the determinant of the original lattice. Furthermore if we take the dual of a dual lattice, we will get back to the original lattice. However, it is not true for kernel lattices. In the other words, may not be equal to . For example, consider the lattice in generated by the vector . Its kernel lattice is generated by the vector , whose kernel lattice is generated by . Nevertheless it is always true that for any integer lattice ,
First we introduce some notations:
Let be an integer matrix. Denote the lattices generated by rows and columns of by and , respectively.
So the basis matrix of the lattice is if rows of are linear independent over . To study the determinant of the kernel lattice, we define
Let be the lattice with a basis matrix . The normalized determinant of , denoted by , is defined to be
It is easy to see that the definition of normalized determination is independent of the choices of basis matrix . Furthermore, the definition can be extended to with coefficients in rational numbers . The main contribution of this paper is to relate the determinant of kernel lattice with the normalized determinant.
Let be an integer lattice and be its kernel lattice. Then
So the normalized determinant can be viewed as the determinant of its kernel lattice. The following theorem shows that the normalized determinant is invariant under the kernel operation.
Let and be two integer matrices of full row rank such that . Then we have
Suppose that has full row rank . Denote by the G.C.D. of determinants of all minors of . Then we have
From the computational aspect, there is a polynomial time algorithm to compute , while computing takes exponential time as one needs to compute determinants of -minors of and then take the G.C.D.. Using G.C.D. of determinants of all full-rank minors of to bound the solution of 1.1.1 was rediscovered in [BR], which extends the special case where the G.C.D is one in [HB].
For any matrix , there is a square unimodular matrix such that is of the form:
is lower triangle, i.e., for all .
The leading coefficient of a nonzero column is positive and for all .
Zero columns are arranged on the right.
We call is the column Hermite Normal Form (HNF) of .
Let be the lattice with a basis matrix and be its kernel lattice. Suppose that columns of has the HNF , i.e.,
for some unimodular matrix , where denotes the zero matrix and blocks , respectively. Then we have
The determinant of column lattice is
Moreover, it is invariant under multiplication by an invertible integer matrix on the left of . So, it is independent of the choices of basis matrices for .
The matrix is a basis matrix of the kernel lattice of , and
Rows of form a basis for the lattice .
We call the normalized lattice of , whose basis matrix can be calculated as .
(1) From the definition of HNF, it is clear that columns of form a basis for the column lattice . So it follows the equality . For the later statement, one can deduce it from Theorem 2.7.
(2) One proof uses dual lattice, which explains where comes from. Since rows of form an -basis for the kernel space of , the kernel lattice of is
where denotes the -vector space generated by rows of . To determine the kernel lattice of , it only needs to decide which vector satisfies
The set of such vectors ’s forms the dual lattice of column lattice . And we know that columns in is a basis for . So by the formula for basis of dual lattice [MG], is a basis matrix for the dual lattice. Hence, is a basis matrix of the kernel lattice of , and
Another proof is more direct. Since , we have
where is the identity matrix. As is unimodular, rows of form a subset of a basis for lattice , which is called primitive in [Ca]. The lattice generated by rows of can not be a proper sublattice of another (integer) lattice in the kernel space of . So rows of form a basis of the kernel lattice of .
(3) Diophantine System 1.1.1 is equivalent to
Therefore, we get a -basis matrix for the solution space of
where denotes the identity matrix. Using the same argument in the proof of the statement (2), one can prove that
indeed form a basis matrix for the lattice . ∎
Proof of Theorem 2.4.
Without loss of generality, we may assume the first columns of are linearly independent. Suppose that is the column HNF of and is the unimodular transformation matrix. That is,
By Lemma 3.2(3) we get a basis matrix for the kernel lattice
From the assumption that , we have
The Sylvester’s determinant identity states that for any matrices and
The proof can be found in many places, e.g. [AAM].
Proof of Theorem 2.5.
Denote . By Theorem 2.4,
On the other hand, we have already showed that equals the determinant of the kernel lattice of by statements (1) and (2) of Lemma 3.2, i.e.,
Proof of Theorem 2.7.
For any integer matrix , let denote the G.C.D of the determinants of all minors. We claim that if an integer matrix is the result of an integer elementary column operation from , then . It is obviously true if the operation is a column switching, or a column multiplication (by an integer). The remaining case is a column addition. In this case, the determinant of a minor of is an integral linear combination of determinants of some minors of . So again we have .
For an integer matrix , after a sequence of invertible integer column operations, we can obtain of a matrix of form We have , and
So . ∎
Jun Zhang is supported by the National Natural Science Foundation of China No. 11601350, by Scientific Research Project of Beijing Municipal Education Commission under Grant No. KM201710028001, by Beijing outstanding talent training program No.2014000020124G140 and by China Scholarship Council. He would like to thank University of Oklahoma for the hospitality during his visit. Qi Cheng is supported by China 973 program No. 2013CB834201 and by US NSF No. CCF-1409294.
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