RePoSS #11: The Mathematics of Niels Henrik Abel: Continuation ...

342 Chapter 18. Tools in ABEL’s research on elliptic transcendentals
and because R did not contain any square factors and p and q could be assumed rela-
tively prime, ABEL found m0 = m1 = · · · = mn = 1 and obtained the factorization
R = R1
n
∏ (x + ak) = R1N.
k=0
with R1 an entire function. With this result, ABEL had found a reduced characteriza-
tion in the form
p 2 1 N − q2 R1 = 1 and M
N
= p1q dR
dx
�
+ 2 p dq
dx
�
− qdp R1.
dx
Considerations of degrees. ABEL’S next step was to investigate the consequences of
the first part of the characterization obtained above,
p 2 1 N − q2 R1 = 1. (18.10)
As he remarked, the equation could be solved by the method of indeterminate coeffi-
cients but this approach would be extremely cumbersome and not lead to any general
conclusion. Instead, he proposed a different approach. Before embarking on his novel
approach, ABEL introduced the notations δP to denote the degree of the (rational)
function P and EP to denote the entire part of P, i.e.
u = Eu + u ′ with δu ′ < 0.
Judging from the detailed introduction of these concepts, ABEL did not assume them
to be familiar to his readers. Concerning these new concepts, ABEL easily proved the
following lemma:
Lemma 3 If the functions u, v, z are related by
and δz < δv, then
u 2 = v 2 + z
Eu = ±Ev. ✷
ABEL now returned to the equation p 2 1 N − q2 R1 = 1 and applied his new result
(lemma 3). ABEL immediately obtained
�
δ p 2 1N �
and consequently
�
= δ q 2 �
R1
2δp1 + δN = 2δq + δR1, i.e.
δ (NR1) = 2 (δq + δR1 − δp1)