2 – Algebraic Functions
283
and in showing that the subset X of the graph of F located over B admits
a natural complex analytic structure (in other words, is a non-compact Riemann surface) for which p : (z, ζ) → z and F : (z, ζ) → ζ are holomorphic.
The compact Riemann surface sought will then be obtained by adjoining a
finite number of points to X, an operation analogous to the construction of
the Riemann sphere ˆ
C from C.
Let us first prove the following result :
Lemme 1 (Continuity of the roots of an algebraic equation). Let a
be a point of C and α ∈ C a multiple root of order p of P (a, ζ) = 0. There
exist numbers r > 0 and ρ > 0 such that, for all z satisfying 0 < |z − a| < r,
the equation P (z, ζ) = 0 has exactly p roots, all simple, in 0 < |ζ − α| < ρ.
As seen in Chap. VIII, n
◦ 5, (viii) in a more general context, if the given
ρ > 0 is sufficiently small, the number ν(z) of roots of P (z, ζ) = 0 satisfying
|ζ − α| < ρ is a continuous function, with respect to the topology of compact
convergence, of the function P z : ζ → P (z, ζ). But since P is a continuous
function of the couple (z, ζ), hence uniformly continuous on every compact
subset of C × C, P z is a continuous function of z with respect to the topology
of compact convergence; ν(z) is, therefore, a continuous function of z, and so
ν(z) = ν(a) in the neighbourhood of a. Hence, for sufficiently small |z−a| < r,
equation P (z, ζ) = 0 has exactly p not necessarily distinct roots such that
|ζ − α| < ρ. There being finitely many values of z for which the equation
P (z, ζ) = 0 has a multiple root, the only one of these values satisfying |z−a| <
r is a if r is sufficiently small. The p roots of P (z, ζ) = 0 such that |ζ − α| < ρ
are, therefore, simple if 0 < |z − a| < r, qed.
It will be seen later that F decompose into n uniform branches f k (z) in
every simply connected open subset U of B, a uniform branch in U being, by
definition, a genuine function f defined and (for the moment) holomorphic
on U , such that f (z) ∈ F(z) for all z ∈ U ; this is what had been proved in
§ 4 of Chap. IV for the pseudo-function Log z for U ⊂ C
∗ .
A local result first needs to be proved :
Lemma 2. Let P (X, Y ) be a polynomial with complex coefficients and E ⊂
C
2 the set of simple points
7 of the curve P (z, ζ) = 0. Then E is a submanifold
of C
2 = R
4 and, for all (a, α) ∈ E where D 2 P (a, α) = 0, there are open
neighbourhoods V of a and W of α such that E ∩ (V × W ) is the graph of a
function ζ = f (z) defined and holomorphic on V with values in W .
Equivalently, there exists a unique holomorphic function f on V satisfying
f (a) = α & P [z, f (z)] = 0 for all z ∈ V .
(2.3)
f is called a local uniform branch at a of the algebraic function defined by
P .
Since α is a simple root of P (a, ζ) = 0, there are (lemma 1) neighbourhoods V and W of a and α such that, for all z ∈ V , the equation P (z, ζ) = 0
7 i.e. points where D1P (z, ζ) and D2P (z, ζ) are not both zero.
283
and in showing that the subset X of the graph of F located over B admits
a natural complex analytic structure (in other words, is a non-compact Riemann surface) for which p : (z, ζ) → z and F : (z, ζ) → ζ are holomorphic.
The compact Riemann surface sought will then be obtained by adjoining a
finite number of points to X, an operation analogous to the construction of
the Riemann sphere ˆ
C from C.
Let us first prove the following result :
Lemme 1 (Continuity of the roots of an algebraic equation). Let a
be a point of C and α ∈ C a multiple root of order p of P (a, ζ) = 0. There
exist numbers r > 0 and ρ > 0 such that, for all z satisfying 0 < |z − a| < r,
the equation P (z, ζ) = 0 has exactly p roots, all simple, in 0 < |ζ − α| < ρ.
As seen in Chap. VIII, n
◦ 5, (viii) in a more general context, if the given
ρ > 0 is sufficiently small, the number ν(z) of roots of P (z, ζ) = 0 satisfying
|ζ − α| < ρ is a continuous function, with respect to the topology of compact
convergence, of the function P z : ζ → P (z, ζ). But since P is a continuous
function of the couple (z, ζ), hence uniformly continuous on every compact
subset of C × C, P z is a continuous function of z with respect to the topology
of compact convergence; ν(z) is, therefore, a continuous function of z, and so
ν(z) = ν(a) in the neighbourhood of a. Hence, for sufficiently small |z−a| < r,
equation P (z, ζ) = 0 has exactly p not necessarily distinct roots such that
|ζ − α| < ρ. There being finitely many values of z for which the equation
P (z, ζ) = 0 has a multiple root, the only one of these values satisfying |z−a| <
r is a if r is sufficiently small. The p roots of P (z, ζ) = 0 such that |ζ − α| < ρ
are, therefore, simple if 0 < |z − a| < r, qed.
It will be seen later that F decompose into n uniform branches f k (z) in
every simply connected open subset U of B, a uniform branch in U being, by
definition, a genuine function f defined and (for the moment) holomorphic
on U , such that f (z) ∈ F(z) for all z ∈ U ; this is what had been proved in
§ 4 of Chap. IV for the pseudo-function Log z for U ⊂ C
∗ .
A local result first needs to be proved :
Lemma 2. Let P (X, Y ) be a polynomial with complex coefficients and E ⊂
C
2 the set of simple points
7 of the curve P (z, ζ) = 0. Then E is a submanifold
of C
2 = R
4 and, for all (a, α) ∈ E where D 2 P (a, α) = 0, there are open
neighbourhoods V of a and W of α such that E ∩ (V × W ) is the graph of a
function ζ = f (z) defined and holomorphic on V with values in W .
Equivalently, there exists a unique holomorphic function f on V satisfying
f (a) = α & P [z, f (z)] = 0 for all z ∈ V .
(2.3)
f is called a local uniform branch at a of the algebraic function defined by
P .
Since α is a simple root of P (a, ζ) = 0, there are (lemma 1) neighbourhoods V and W of a and α such that, for all z ∈ V , the equation P (z, ζ) = 0
7 i.e. points where D1P (z, ζ) and D2P (z, ζ) are not both zero.