Consequently, only the integration over y1 ∈ ByY0 (ǫ) is non-negligible. To evaluate it, we apply
Theorem 4.7. We calculate the integral over the pull-back with respect to the exponential map of
our differential forms. We use the notations introduced before Theorem 3.10. After the change of
√
variables Z 7→ pZ, an estimate, similar to the one which bounded the integral over Q, Lemma
2.10 and the second part of Lemma 3.1, applied for n := m, we see that (5.2) holds for
X
 E,0 Y 
EP
E
Kn,m
Jk,a , IA−1 ,b ,
(5.47)
Jk,r
(Z, ZY′ ) :=
k,p

a+b=r

1
Ak,p )−1 as in Theorem 3.5. From the corwhere IAY −1 ,b are the polynomials associated to ( pn−m+k
k,p

E,0
responding statements about the parity of Jk,r
, IAY −1 ,r from Theorem 3.5 and Lemmas 3.1, 5.10,
k,p

E
we deduce that the parity of Jk,r
coincides with k + r. From Proposition 3.9 and (5.47), we deduce
that
n
o
E,0
E
deg(Jk,r
) ≤ max deg(Jk,a
) + deg(IAY −1 ,b ) ,
(5.48)
k,p

where the maximum is taken over a + b = r. From Proposition 3.9, Lemma 5.10, (5.35), (5.48)
R,0
and the bound on the degree of Jk,a
from Lemma 5.10, we deduce the needed bound on the degree
E
of Jk,r.

### Language ###
English

### Assessment ###
The text mentions a mathematical relationship about parity coinciding with k + r, which could be asked about.

### Query ###
What is the relationship between parity and k + r in mathematics?
What is the relationship between parity and k + r in mathematics?