Number Theory: An Introduction to Mathematics

4 Supplements 323

where f 0 (t),f 1 (t),...,fn(t)∈ F[t]. Assume that f 0 does not divide fjfor some
j∈{ 1 ,...,n}.Thend:=degf 0 >0 and we can write

fj(t)=gj(t)f 0 (t)+hj(t),

wheregj(t),hj(t)∈F[t] and deghj<d( 1 ≤j≤n).
Let

(x,y)={φ(x+y)−φ(x)−φ(y)}/ 2

be the symmetric bilinear form associated with the quadratic formφand put

f=(f 1 ,...,fn), g=(g 1 ,...,gn), h=(h 1 ,...,hn).

If

f 0 ∗={(g,g)−p}f 0 − 2 {(f,g)−pf 0 },f∗={(g,g)−p}f− 2 {(f,g)−pf 0 }g,

andf∗=(f 1 ∗,...,fn∗), then clearlyf 0 ∗,f 1 ∗,...,fn∗∈F[t]. Since(f,f)=pf 02 and
g=(f−h)/f 0 , we can also write

f 0 ∗=(h,h)/f 0 ,f∗={(h,h)f− 2 (f,h)h}/f 02.

It follows that degf 0 ∗<dand(f∗,f∗)=pf 0 ∗^2 .Alsof 0 ∗=0, sinceh=0andφis
anisotropic. Thus

p(t)=α 1 {f 1 ∗(t)/f 0 ∗(t)}^2 +···+αn{fn∗(t)/f 0 ∗(t)}^2.

Iff 0 ∗does not dividefj∗for somej∈{ 1 ,...,n}, we can repeat the process. After at
mostdsteps we must obtain a representation forp(t)of the required form.

It was already known to Hilbert (1888) that there is no analogue of Proposition 43
for polynomials in more than one variable. Motzkin (1967) gave the simple example

p(x,y)= 1 − 3 x^2 y^2 +x^4 y^2 +x^2 y^4 ,

which is a sum of 4 squares inR(x,y), but is not a sum of any finite number of squares
inR[x,y].
In the same paper in which he proved Proposition 43 Pfister introduced his
multiplicative forms. The quadratic formsfa, fa,bin§2 are examples of such forms.
Pfister (1966) used his multiplicative forms to obtain several new results on the
structure of the Witt ring and then (1967) to give a strong solution to Hilbert’s 17th
Paris problem. We restrict attention here to the latter application.
Letg(x),h(x)∈R[x] be polynomials innvariablesx =(ξ 1 ,...,ξn)with real
coefficients. The rational functionf(x)=g(x)/h(x)is said to bepositive definiteif
f(a)≥0foreverya∈Rnsuch thath(a)=0. Hilbert’s 17th problem asks if every
positive definite rational function can be represented as a sum of squares:

f(x)=f 1 (x)^2 +···+fs(x)^2 ,