2 1. A Brief Classical Introduction
polynomial of the form
q\x\,... )Xn) = y ^ ciijXiXj G ±* \x\,... ,xn\.
Because multiplication in F is commutative, in considering q(xi,..., xn)
as a function
— F without loss of generality we can (and will, from
now on) suppose that the coefficients of q satisfy the condition a^ = a^; just
replace each of these coefficients by lJ ——. (Note that here the underlying
assumption that char F ^ 2 is needed.) With this convention in force, q is
now given by the symmetric matrix A — (a^) G Mn(F). This symmetric
matrix is the Gram matrix of q.
Let a G F. We say the form q represents a over i?, denoted a —• g, if
there are Ai,..., An G R such that g(Ai,..., An) = a. More briefly: viewing
q as a function
— F, the statement a — q means a G range q. (A
more general notion of representation will come in Definition 2.8.)
An instance of the representation problem is this: Given a form g,
for which a G F is it true that a — ql
Example 1.2. Pythagorean triples, mentioned earlier, are precisely the
solutions for the representation 0 — x\ + x\ — x\.
Example 1.3. Brahmagupta (598-668) could generate Pythagorean triples,
and he studied representations by other quadratic forms as well. For in-
stance, from a given representation 1 —•
(with a G N) he could
produce infinitely many.
Example 1.4. (i) Fermat (1601-1665) studied sums of integer squares,
and here is one of his typical results, stated by him in 1640 in a private
communication—along with hints on his method of proof—and with a for-
mal proof published by Euler in 1754: If n G N has prime factorization
then n —
if and only if c^ = 0 (mod 2) for all i
such that pi = 3 (mod 4).
(ii) In 1967 Leahey [Le] returned to Fermat's result, but replacing Z with
a polynomial ring. Let h(x) G Fq[x]. (Here ¥q denotes a finite field with q
elements, with q odd.) Say h(x) = ep
with the Pi(x) prime
(monic and irreducible) polynomials and e G F*. Then h(x) is a sum of
two polynomial squares—that is, h(x) —
and only if c^ = 0
(mod 2) for all i such that dpi(x) = 1 (mod 2). (Here we use "9" for the