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§2. Irreducible Plane Algebraic Curves and Hypersurfaces 49

(2.4) Corollary. Let f and f be two nonzero irreducible homogeneous polynomials
in k[y 0 ,... , y n ]. If for some algebraically closed field L the sets H f (L) = H f (L),

then f = cf for some nonzero c in k, and H f (K) = H f (K) for all extension fields

Kof k.
2 2 2

For k the rational numbers and f (w, x, y,) = w + x + y and f (w, x, y) =
2
2
2
w + 2x + y the sets H f (R) = H f (R) are empty, but clearly f is not a constant

multiple of f . So the hypothesis that L is algebraically closed is essential in (2.3)
and (2.4).
Let k[y 0 ,... , y n ] d denote the vector space of homogeneous polynomials of de-
gree d. Any nonzero element defines a hypersurface of degree d, and two nonzero
elements define the same hypersurface if and only if they are in the same one-
dimensional space. Thus the projective space P(k[y 0 ,... , y n ] d ) parametrizes the
hypersurfaces over k of degree d in P n .
d
(2.5) Notation. Let H (k) denote the projective space P(k[y 0 ,... , y n ] d ) of hyper-
n
surfaces of degree d in P n defined over k. The coordinates of this projective space
are just the coefficients of the corresponding equation.
d
For k algebraically closed, the space H (k) is just the d-dimensional projective
1
space of d points, repetitions allowed, on the projective line P 1 (k).
(2.6) Proposition. The dimension of k[w, x, y] d over k is [(d + 1)(d + 2)]/2, and
d
the dimension of the projective space H is
2
d(d + 3) (d + 1)(d + 2)
= − 1.
2 2
a b c
Proof. We have to count the number of monomials w x y , where a + b + c = d
since they form a basis of k[w, x, y] d . For a fixed index a the number of monomials
a b c
w x y is d − a + 1. To obtain the dimension of the vector space in question, we
must sum a from 0 to d. The dimension of the projective space is one less than the
dimension of the vector space. We leave the computation to the reader.
We have the following table of values for these dimensions:

1 2 3 4 5 6 7 8
d
dim H (k) 2 5 9 14 20 27 35 44
2

(2.7) Remark. We can generalize the assertion that two points determine a line.
Since the requirement that a plane algebraic curve goes through a point in P 2 (k)
d
d
is a hyperplane in H (k), and since the intersection of m hyperplanes in H (k) for
2 2
m ≤ [d(d+3)]/2 is nonempty, it follows that there exists a curve of degree d through
m given points if m ≤ [d(d + 3)]/2.