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Lattices and the Geometry of Numbers

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Lattices and the Geometry of Numbers Lattices and the Geometry of Numbers Lattices and the Geometry of Numbers Sourangshu Ghosha, aUndergraduate Student ,Department of Civil Engineering , Indian Institute of Technology Kharagpur, West Bengal, India 1. ABSTRACT In this paper we discuss about properties of lattices and its application in theoretical and algorithmic number theory. This result of Minkowski regarding the lattices initiated the subject of Geometry of Numbers, which uses geometry to study the properties of algebraic numbers. It has application on various other fields of mathematics especially the study of Diophantine equations, analysis of functional analysis etc. This paper will review all the major developments that have occurred in the field of geometry of numbers. In this paper we shall first give a broad overview of the concept of lattice and then discuss about the geometrical properties it has and its applications. 2. LATTICE Before introducing Minkowski’s theorem we shall first discuss what is a lattice. Definition 1: A lattice 흉 is a subgroup of 푹풏 such that it can be represented as 휏 = 푎1풁 + 푎2풁 + . + 푎푚풁 풏 Here {푎푖} are linearly independent vectors of the space 푹 and 푚 ≤ 푛. Here 풁 is the set of whole numbers. We call these vectors {푎푖} the basis of the lattice. By the definition we can see that a lattice is a subgroup and a free abelian group of rank m, of the vector space 푹풏. The rank and dimension of the lattice is 푚 and 푛 respectively, the lattice will be complete if 푚 = 푛. This definition is not only limited to the vector space 푹풏. It can be extended to any arbitrary field 푭, in which the basis vectors {푎푖} will belong to the field 푭. In this article we shall discuss about both complete or full-rank lattices and incomplete lattices. We will now define another terms known as the fundamental mesh. Definition 2: The set of elements which can be denoted as 휑(푎) = {푎1풗ퟏ + 푎2풗ퟐ + . +푎푚풗풎|푣푖 ∈ 푅, 0 ≤ 푣푖 ≤ 1} is called a fundamental mesh of the lattice. A very important thing to notice is that not every given set of vectors {푎푖} forms the basis of a given lattice 흉. In the next lemma we shall state the condition for a given set of vectors {푎푖} to form the basis of a given lattice 흉. Lemma 1: A given set of vectors {푎푖} form the basis of a given lattice 흉 if it satisfies the following condition: 휑(푎) ∩ 흉 = {ퟎ} Proof: The lattice 흉 is the set of all their integer combinations of the basis vectors {푎푖}.We also know that 휑(푎) is the set of linear combinations of basis vectors {푎푖} with the coefficients 푣푖 ∈ 푅, 0 ≤ 푣푖 ≤ 1. Therefore the only element in common is the zero vector ퟎ. Let us now state a lemma regarding the properties of the lattice 흉. This proof can be found in many articles and books. We state here the elegant proof as stated and given by Comeaux1 and Neukirch2 Sourangshu Ghosh Page 1 Lattices and the Geometry of Numbers Lemma 2: A lattice 흉 in 푽 is complete if and only if there exists a bounded subset 푴 ⊆ 푽 such that the collection of all translates 푴 + 휸, 휸 ∈ 흉 covers the whole space 푽. Proof: Comeaux1 proves the theorem by proving it in the other direction. Let us first assume that 푴 is a bounded subset such that the collection of all translates 푴 + 휸, 휸 ∈ 흉 covers the whole space 푽. Let us also denote the subspace spanned by 흉 as 푷. Now to prove this theorem, we have to prove 푷 = 푽 or in other words if we can show that every element 푣 ∈ 푽, also belongs to 푷 we are done. Now we note that 푉 = ⋃휸∈흉(푴 + 휸), we also note that 푣 ∈ 푽 and 푎푣 ∈ 푴 and 훾푣 ∈ 흉. But 흉 is itself a subset of 푷. We can therefore write the following expression: 푣푣 = 푎푣 + 휸풗 Dividing both sides of the equation by 풗 and taking the limit of 풗 as infinity, we get 푎 휸 휸 푣 = lim( 푣) + lim( 풗) = lim( 풗) ∈ 푉′ 푣→∞ 푣 푣→∞ 푣 푣→∞ 푣 We will now define another term known as the determinant of the lattice 흉, or the volume of the parallelepiped spanned by the fundamental mesh. Before going to the technical definition let us discuss a more intuitive way to understand the notion of determinant of the lattice 흉, the determinant of the lattice 흉 can be expressed as: 푣표푙(푘) det(흉) = lim 푘→∞ 푁푢푚푏푒푟 표푓 푙푎푡푡푖푐푒 푝표푖푛푡푠 푖푛푠푖푑푒 Where 푘 is the radius of the ball. Definition 3: The determinant of the lattice 흉, or the volume of the parallelepiped spanned by the fundamental mesh spanned by the basis vectors {푎푖} is given as det(흉) = √det(푨푻푨) If the lattice if full-rank, the expression reduces to det(흉) = |det (푨)| We have discussed above what a lattice is. An important thing to notice is that it is not necessary that two distinct ′ ′ basic vectors {푎푖} and {푎푖} shall span distinct lattices. The span of two different basic vectors {푎푖} and {푎푖} can give us the same lattice. We shall now state another lemma regarding this question as given by Oded Regev3. ′ ′ Lemma 3: The span of two different basic vectors {푎푖} and {푎푖} will give us the same lattice if and only if 푨 = 푨푼, where 푼 is a uni-modular matrix. Proof: A uni-modular matrix is matrix whose determinant is ±1.. We first assume that the span of two different ′ basic vectors {푎푖} and {푎푖} is equal. From this point onwards we shall use {푎푖} and 푨 interchangeably. Now note ′ that each column of the matrix 푨 which is 푎푖 belongs to the span of 푨 , which tells us that there must exist a uni- modular U for which 푨′ = 푨푼. Similarly we can otherwise say that there must exist a uni-modular 푼′ for which 푨 = 푨′푼′. Substituting the later expression into the first one we get 푨′ = 푨푼 = 푨′푼′푼 Therefore we can also say that 푨′푻푨′ = (푼′푼)′(푨′푻푨′)푼′푼 Sourangshu Ghosh Page 2 Lattices and the Geometry of Numbers Now if we take determinant on both sides of the equation we get 풅풆풕(푨′푻푨′) = 풅풆풕(푼′푼)ퟐ풅풆풕(푨′푻푨′) This gives us 푑푒푡(푈′푈)2 = 1 i.e 푑푒푡(푈′푈) = ±1 or in other words 푑푒푡(푈′)푑푒푡 (푈) = ±1. Now notice that both 푈′ and 푈 is integer matrices and hence the determinant shall also be integers. Therefore we get 푑푒 푡(푈) = ±1 We shall now discuss about the method of Gram-Schmidt Orthogonalization. This again can be found in many textbooks but for the present purposes we shall be stating it as done by Oded Regev3. It is a method of ortho- normalizing (such that the new set of basis vectors 푎̃푖 satisfy 〈푎̃푖, 푎̃푗〉 = 0) a given set of linearly independent basis vectors {푎푖}, which may be a part of an inner product space equipped with a given inner product. Definition 4: For a sequence of 푛 linearly independent basis vectors {푎푖}. We now define the Gram-Schmidt orthogonalization as the sequence of vectors {푎̃푖} defined by 푖−1 〈푎푖,푎̃푗,〉 푎̃푖 = 푎푖 − ∑푗=1 휇푖,푗 푎̃푗 where 휇푖,푗 = 〈푎푗,푎̃푗〉 One should note that {푎̃푖} do not necessarily form a basis of 흉 and the order of the linearly independent basis vectors {푎푖} matters due to which we should consider it as a “sequence of vectors” rather than a “set of vectors”. Geometrically speaking in this method the basis vector 푎̃푖 is defined as difference between the basis vector 푎푖 and its projection onto the subspace which is generated by 푎1, 푎2, … , 푎푖−1 which shall be by definition same as the subspace generated by 푎̃1, 푎̃2, … , 푎̃푖−1. By subtracting the projection from the vector we are making sure that it will be orthogonal to subspace generated by 푎̃1, 푎̃2, … , 푎̃푖−1. Let us now state another important property of the Gram-Schmidt Orthogonalization due to Oded Regev3. Let there be n linearlly independent vectors denoted as {푎푖}. Therefore we get the ortho-normal basis vectors given by Gram- Schmidt Orthogonalization as {푎̃푖/‖푎̃푖‖}. Then we can represent the new basis vectors {푎̃푖} in terms of basis vectors {푎푖} as given as the columns of the 푚 × 푛 matrix. ‖푎̃1‖ 휇2,1‖푎̃1‖ … 휇푛,1‖푎̃1‖ 0 ‖푎̃2‖ 휇푛,2‖푎̃2‖ : [ ] 0 … 0 ‖푎̃푛‖ 0 … 0 0 : : : 0 … 0 0 This matrix will be a full-rank lattice if 푚 = 푛, in which case the matrix will become a upper-triangular matrix. The 푛 volume of the parallelopoid 푃({푎푖} ) will be equal to 푑푒푡(퐿({푎푖})) which in turn is same as ∏푖=1‖푎̃푖‖. 3. SUCCESIVE MINIMA Having discussed about the lattice and its properties, we shall now discuss about the concept of Minkowski Theory. Before that we shall state some definitions regarding the properties of subset. Definition 5: A subset 푋 ∈ 푉 is defined to be as centrally symmetric, if for every point 푥 ∈ 푋, −푥 ∈ 푋 also holds Definition 6: A subset 푋 ∈ 푉 is defined to be as convex, if for every two point 푥, 푦 ∈ 푋, the line segment defined as Sourangshu Ghosh Page 3 Lattices and the Geometry of Numbers {푡푦 + (1 − 푡)푥|0 ≤ 푡 ≤ 1} Also is contained inside the set 푉. One very important parameter of interest in the study of lattices is the determination of the length of the shortest nonzero vector in the lattice which we call as λ1.The other successive minima distances are written as, λ2, .
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