CHAPTER 1

From Quadratic Forms to Sphere Packings and

Coverings

In this chapter we trace the origin of the lattice sphere packing problem back to

the study of Diophantine quadratic equations. We briefly introduce and survey on

known results of the sphere packing, the sphere covering and the sphere packing-

covering problem. For further reading on these topics and on detailed proofs for the

reviewed facts we refer to the books [206], [124], [234], [69], [271], [174], [121],

and for the history to [219] and [116].

1.1. Positive definite quadratic forms

1.1.1. Quadratic forms. In general, a quadratic form Q : V → K is a homo-

geneous polynomial of degree 2, defined for a field K and a vector space V over K.

We restrict ourselves in this book to real or rational quadratic forms Q defined over

K = R or K = Q. V is throughout a vector space of dimension d, hence V = Rd or

V = Qd. The prefixes “real” or “rational” will not be used, unless necessary. We

speak of an integral quadratic form Q if its coeﬃcients are integers.

Examples, as the left hand sides of the Diophantine equations

a2

+

b2

−

c2

= 0 or

a2

+

b2

+

c2

+

d2

= n

have almost symbolic character and occur throughout the history of mathematics.

To find integral solutions for these and other Diophantine equations has fascinated

mathematicians from time immemorial, and inspired them to develop beautiful

and deep mathematics. It was Fermat, who revived the arithmetic studies of the

Greeks and who found many significant results. For example, we attribute him

the complete characterization of numbers which are the sum of two squares. The

solution to the corresponding three square and four square problem were known to

him from the work of Diophantus, but proofs were given much later by Lagrange

[161] and Legendre [166] (cf. [116], [219]).

Hermite (cf. [135]) initiated the systematic arithmetic study of quadratic forms

in d variables x = (x1, . . . , xd)t ∈ Rd. We write

Q[x] =

d

i,j=1

qij xixj

with coeﬃcients qij ∈ R. By assuming, without loss of generality, that qij = qji,

we simply identify the quadratic form Q with the real symmetric matrix Q =

(qij )i,j=1,...,d. The space of all real quadratic forms in d variables is identified with

the space

Sd

= Q ∈

Rd×d

:

Qt

= Q

of real symmetric d × d matrices. Using matrix notations we have Q[x] =

xtQx.

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http://dx.doi.org/10.1090/ulect/048/01