550 XIII Connections with Number Theory
where all unwritten terms have degree>1. Sincea,bare uniquely determined byf,
we can define thetangentto the affine curveCat the non-singular point (x,y)tobe
the affine line
(X,Y)=aX+bY−(ax+by).It is easily seen that these definitions do not depend on the choice of polynomial within
an equivalence class{λf:0=λ∈K}.
The study of the asymptotes of an affine plane curve leads one to consider also
its ‘points at infinity’, the asymptotes being the tangents at these points. We will now
make this precise.
If the polynomialf(X,Y)has degreed,then
F(X,Y,Z)=Zdf(X/Z,Y/Z)is a homogeneous polynomial of degreedsuch that
f(X,Y)=F(X,Y, 1 ).Furthermore, ifF(X,Y,Z)is any homogeneous polynomial such thatf(X,Y)=
F(X,Y, 1 ),thenF(X,Y,Z)=ZmF(X,Y,Z)for some non-negative integerm.
We d e fi n e aprojective plane curveover a fieldKto be a homogeneous polyno-
mialF(X,Y,Z)of degreed>0 in three indeterminates with coefficients fromK,but
we regard two homogeneous polynomialsF(X,Y,Z)andF∗(X,Y,Z)as defining the
same projective curve ifF∗=λFfor some nonzeroλ∈K. The projective curve is
said to be aprojective line, conicorcubicifFhas degree 1,2 or 3 respectively.
IfCis an affine plane curve, defined by a polynomialf(X,Y)of degreed>0, the
projective plane curveC ̄, defined by the homogeneous polynomialZdf(X/Z,Y/Z)
of the same degree, is called theprojective completionofC. Thus the projective
completion of an affine line, conic or cubic is respectively a projective line, conic
or cubic.
LetC ̄be a projective plane curve over the fieldK, defined by the homogeneous
polynomialF(X,Y,Z). We say that(x,y,z) ∈ K^3 is apoint,orK -point,ofC ̄
if(x,y,z)=( 0 , 0 , 0 )andF(x,y,z)=0, but we regard two triples (x,y,z)and
(x∗,y∗,z∗) as defining the sameK-point if
x∗=λx,y∗=λy,z∗=λz for some nonzeroλ∈K.IfC ̄is the projective completion of the affine plane curveC, then a point (x,y,z)
ofC ̄withz=0 corresponds to a point (x/z,y/z)ofC, and a point (x,y,0) ofC ̄
corresponds to apoint at infinityofC.
TheK-point (x,y,z) of the projective plane curve defined by the homogeneous
polynomialF(X,Y,Z)is said to benon-singularif there exista,b,c∈ K, not all
zero, such that
F(x+X,y+Y,z+Z)=aX+bY+cZ+···,where all unwritten terms have degree>1. Sincea,b,care uniquely determined by
F, we can define thetangentto the projective curve at the non-singular point (x,y,z)