282
X – The Riemann Surface of an Algebraic Function
differential operator with respect to ζ. The classical result from algebra
5 –
that the set of these values of z is finite – follows. For the equation ζ
2 (z
2
−
1) − z
5 = 0, whose graph in R
2 has a cusp point at the origin, that is the case
if z = 0, 1 or +1 ; for z in the neighbourhood of 0, the equation obviously has
two roots in the neighbourhood of 0, and though they can be distinguished by
their sign in the real domain, this is impossible in the complex one. Indeed,
following by continuity one of the roots along a circle centered at 0, we end
up with the opposite root since the number z
5 /(z
2
− 1) with ζ as “ the ”
square root describes a curve around the origin. In the particularly simple
case when P (X, Y ) = Y
2
− X, which corresponds to ζ = z
1/2 , the curve
does not have any singularity at the origin, but its tangent at 0 is vertical
and the conclusion is the same: the “ function ” z
1/2 is not well-defined in the
neighbourhood of 0 as already explained in Chapter IV.
In what follows, S will denote the finite set of z ∈ C where equation
(1) has at most n distinct roots, either because its degree decreases (points
canceling P 0 ), or because it has multiple roots (critical points)).
If z and ζ are called canonical coordinates in C
2 , equation (1) defines a
complex algebraic curve in C
2 (in contrast to real algebraic curves in R
2 ) .
In set theoretic language (Chapter I, n
◦ 5), F is a correspondence
6 between
C and C whose curve is the graph. To transform F into a function F in
the strict sense, it suffices to consider the curve and to set F (z, ζ) = ζ, as
was done in § 4 du Chap. IV regarding the complex logarithm. Ignorance of
this simple procedure, and more generally of the abc of “ abstract ” set theory
has long confused matters, and not only about algebraic functions. Il explains
why somewhere, Dieudonn´ e qualified classical discourses on “ multiform functions ” as verbosity without, however, going as far as explaining to his readers
how to transform this verbosity into perfectly correct mathematical arguments. The main purpose of the theory is to construct a compact Riemann
surface on which z, the “ function ” ζ = F(z) and more generally any rational
expression in z and ζ become genuine meromorphic functions. The graph of
F is only a first approximation in the construction, which, as we will see, is
considerably more difficult.
The first step consists in considering the open set B = C − S defined by
the condition
z ∈ B ⇐⇒ Card F(z) = n
5 If P (Y ) and Q(Y ) are polynomials of degrees p and q with coefficients in an
integral ring, for example C[X], and if y is a common root of P and Q, successive
multiplication of P (y) by 1, y, . . . , y
q−1 and of Q(y) by 1, y, . . . , y
p−1 give p + q
linear equations homogeneous in y
n (0 ≤ n ≤ p + q − 1); as this system admits a
non-zero solution (since 1 = 0), its determinant, a polynomial in coefficients of
P and Q, must be zero. Do the calculations for p = q = 2.
6 Besides this term was used in algebraic geometry long before the invention or
propagation of set theory.
X – The Riemann Surface of an Algebraic Function
differential operator with respect to ζ. The classical result from algebra
5 –
that the set of these values of z is finite – follows. For the equation ζ
2 (z
2
−
1) − z
5 = 0, whose graph in R
2 has a cusp point at the origin, that is the case
if z = 0, 1 or +1 ; for z in the neighbourhood of 0, the equation obviously has
two roots in the neighbourhood of 0, and though they can be distinguished by
their sign in the real domain, this is impossible in the complex one. Indeed,
following by continuity one of the roots along a circle centered at 0, we end
up with the opposite root since the number z
5 /(z
2
− 1) with ζ as “ the ”
square root describes a curve around the origin. In the particularly simple
case when P (X, Y ) = Y
2
− X, which corresponds to ζ = z
1/2 , the curve
does not have any singularity at the origin, but its tangent at 0 is vertical
and the conclusion is the same: the “ function ” z
1/2 is not well-defined in the
neighbourhood of 0 as already explained in Chapter IV.
In what follows, S will denote the finite set of z ∈ C where equation
(1) has at most n distinct roots, either because its degree decreases (points
canceling P 0 ), or because it has multiple roots (critical points)).
If z and ζ are called canonical coordinates in C
2 , equation (1) defines a
complex algebraic curve in C
2 (in contrast to real algebraic curves in R
2 ) .
In set theoretic language (Chapter I, n
◦ 5), F is a correspondence
6 between
C and C whose curve is the graph. To transform F into a function F in
the strict sense, it suffices to consider the curve and to set F (z, ζ) = ζ, as
was done in § 4 du Chap. IV regarding the complex logarithm. Ignorance of
this simple procedure, and more generally of the abc of “ abstract ” set theory
has long confused matters, and not only about algebraic functions. Il explains
why somewhere, Dieudonn´ e qualified classical discourses on “ multiform functions ” as verbosity without, however, going as far as explaining to his readers
how to transform this verbosity into perfectly correct mathematical arguments. The main purpose of the theory is to construct a compact Riemann
surface on which z, the “ function ” ζ = F(z) and more generally any rational
expression in z and ζ become genuine meromorphic functions. The graph of
F is only a first approximation in the construction, which, as we will see, is
considerably more difficult.
The first step consists in considering the open set B = C − S defined by
the condition
z ∈ B ⇐⇒ Card F(z) = n
5 If P (Y ) and Q(Y ) are polynomials of degrees p and q with coefficients in an
integral ring, for example C[X], and if y is a common root of P and Q, successive
multiplication of P (y) by 1, y, . . . , y
q−1 and of Q(y) by 1, y, . . . , y
p−1 give p + q
linear equations homogeneous in y
n (0 ≤ n ≤ p + q − 1); as this system admits a
non-zero solution (since 1 = 0), its determinant, a polynomial in coefficients of
P and Q, must be zero. Do the calculations for p = q = 2.
6 Besides this term was used in algebraic geometry long before the invention or
propagation of set theory.