• Nie Znaleziono Wyników

# On the estimation of certain exponential sums by

N/A
N/A
Protected

Share "On the estimation of certain exponential sums by"

Copied!
30
0
0

Pełen tekst

(1)

q

n

qn

n

n

K

K

1k

0

p

n

0

kn/ p

n

n

n

n

1

n

n

n

x∈V (kn)

n

n

n

1

n

n

n

n

r i=1

i

n

s j=1

j

n

i

j

n

w/2

[329]

(2)

w

w

i

j

n

2d w=0

w

w

nw/2

2d

n

n

n

n

n

n

2d

2d

kn/ p

p

n

n

n

p

2d

p

n

n

w

w

w

w

n

n

d/2

w

w

3

3

(3)

w

w

n

i

j

j

i

j

m

m

m

m

m

m

m

λ

−1

K

1K

λd−1

λ

λ

λd−1

λd−1

n

n

d−3/2

(4)

1K

n

3n/2

λ

λd−1

λ j

K f

1K

λ

K

λ

0

λ0

λd−1

1

λ0

V0

λ

1

X

0

0

λ0

λ0

λ0

λd−1

0

(5)

1

K

l

c2d−1

K

l

c

n

n

d

n

d−1

n

n

d−1

0

0

n

0

n

n

d−1

n

2d i=0

i

n

ci

K

l

ci

n

d

n

d−1

n

n

d−1

n

d−1

(6)

n

d−1

K

K

1K

K

λ

−1

K

K

−1

1K

c0

1K

2

!

l

1K

c2

1K

1

!

l

c3

l

c1

1K

2

!

l

c1

1K

2

!

l

2

!

l

l

c1

1K

l

0

l

c1

K

l

1

1

l

1

1K

l

c3

l

c2

1K

1

!

l

c2

1K

1

!

l

0

1K

1

!

l

0

1K

1

!

l

0

1K

1

!

l

0

1K

1

!

l

1

λ

l

π1(P1K−S)

1

λ

l

π1(P1K−S)

l

0

λ

l

π1(P1K−S)

l

0

λ

l

l

λ

l

l

λ

l

π1(P1K−S)

l

λ

l

π1(P1

K−S)

(7)

λ

K

l

K

l

1

K

l

Nk

u

u

u

u

u

u

Nk

Nk

0

0

u

u

u

0

n

n

u

n

n

u∈U (kn)

n

u

u

u

n

n

n

n

n

N −1k

n

n

N −1k

n

n

N −1

n

n

N +d−2

n

u

u∈U (kn)

n

u

u∈U0(kn)

n

u

u6∈U0(kn)

n

u

0

n

N +d−2

(8)

n

u

n

u

n

d−1

n

d−2

n

n

d−2

0

n

u

u∈U0(kn)

n

u

n

0

n

d−1

n

d−2

n

n

N +d−2

n

0

n

Nk

n

N −1

n

n

N +d−1

n

N +d−2

n

n

N +d−2

n

n

l∈kn

n

2

n

λ

λ

−1

λ

n

λ∈kn

n

n

λ

n

−n

n

n

n

λ∈kn

n

λ

n

2

n

n

−1

1k

1k

n

2

n

2d−2

n

n

2d−2

(9)

λd−1

n

n

λ

n

d−1

n

d−2

n

λd−1

n

d−2

n

n

d

n

d−1

n

n

d−1

n

λ

λ

1k

λd−1

λd−1

n

λd−1

n

n

λ

n

n

λ

n

n

d−2

n

λ

n

n

d−3/2

n

n

2d−2

n

n

λ

n

λ

2

n

λ

n

λ

n

2

n

λ

n

λ

2

n

2

n

n

n

2

n

n

2

n

n

2d−2

n

n

(10)

n

2d−2

2d−2

n

2d−2

n

l∈kn

n

2

l∈ p

n

2

l∈ p

i

i

n

j

j

n 2

n

p

n

p

n

n

l

w

w

w

w

p

2d−2

2d−2

n

2d−2

2

n

2d−2

2d−2

2d−2

2

2

n

n

2d−2

n

n

2d−3/2

n

n

d−3/2

n

n

λ

n

d−1

λ

n

n

2d−1

λd−1

n

λd−1

n

i

i

n/n(λ)

i

n(λ)

n

n

2d−3

λ∈kn

i

n/n(λ)

2

n

2d−2

(11)

n0

n

0

n0

n

n0

n

2d−2

0

n

n

n(λ)

ψ0◦f

K

p

K

0

ψ0

1K

p

1

K

2d

i=0

i

ci

K

ψ0◦f

1K

q

!

l

K

1K

2p,q

2p,q

cp

1K

q

!

l

ψ0

c

K

ψ0◦f

1

p+q

2p,q

cp+q

K

ψ0◦f

2p,q

q

!

l

q

!

l

ψ0

2p,q

2p,2d−2

2p,2d−3

2p,2d−4

(12)

c2

22,q

c2

1K

q

!

l

ψ0

c2

q

!

l

ψ0

22,q

0

q

!

l

2d−2

!

l

c

1K

ψ0

2∗,2d−2

21,2d−3

c1

1K

2d−3

!

l

ψ0

c1

2d−3

!

l

ψ0

2d−3

!

l

wt≤2d−4

2d−3

!

l

wt=2d−3

wt=2d−3

2d−3

!

l

wt=2d−3

c1

wt≤2d−4

ψ0

c1

wt=2d−3

ψ0

wt=2d−3

1K

wt=2d−3

x∈A1K−W

wt=2d−3

ψ0

x

c1

wt=2d−3

ψ0

c1

1K

wt=2d−3

ψ0

c1

1K

wt=2d−3

ψ0

wt=2d−3

c1

1K

ψ0

c1

wt=2d−3

ψ0

(13)

N

1

l

1

m

N

1

l

j

i

j

i

i

N

i

i

i

i

i

i

0

0

N +1+L+2L0

N1

Nk

1

k

N

i

IJ

l i=1

i

i

j∈J

j

j

µ

µ

i∈I

i

i

i

n

N +1+l+|J|+2|I|

IJ

1

l

j

i

n

N +1+l+|J|+2|I|

IJ

n

l+|I|+|J|

0

n

µ

µ

i

n

N +1

n

i

j

i

i

i

i

i

(14)

i

i

µ

µ

µ

µ

n

n

0

n

µ

µ

n

n

IJ

IJ

i

j

i

n

N +1+l+|J|+2|I|

IJ

n

l+|I|+|J|

n

n

IJ

I

N

i

j

i

n

I

n

I∅

n

IM

n

I6=∅

|I|−1

n

I

n

n

−1

I6=∅

|I|−1

n

−l−|I|

n

N +1+l+2|I|

I∅

n

−m

n

N +1+l+m+2|I|

IM

0

n

−n

−2n

−(N +1+m)n

0

−n

n

−n

I6=∅

N +1+l+2|I|

I∅

N +1+l+m+2|I|

IM

i

j

i

i

i

i

n

n

(15)

i

j

j

i

2N +2+2l+t2+2mt

N +1+l+m+2t

n

n

n

2m

1

λ

−1

λ

λ

λ

(16)

λ

λ

λ

1K

x

λ−1

2

2

12

121

2

2

3

2

3

2

2

3

3

3

2

32

23

2

2

3

32

2

3

−2

2

−1

2

3

2

2

−2

8

2

(17)

x

λ

1K

x

λ

x

λ

λ

x

x

λ

n

1

2m

3n/2

n

3

3

3

3

3

n

3

3

3n/2

r

r

n

N

r

n

N

r

N

n

N/2

3

n

3

3

3n/2

3

λ

3

λ

λ

## has a triple point for every λ,

Cytaty

Powiązane dokumenty

We can say that in…nity is not a safe place for hiding, at least for curves of Type + : In particular, there are no smooth handsome annuli of Type + : The reader will see that the

6.5 Anti-differentiation with a boundary condition to determine the constant of integration; de nite integrals; area of the region enclosed by a curve and the x-axis or y-axis

This is the first nontrivial discrepancy bound for parts of the period of inversive congruential pseudo- random numbers with prime-power modulus.. An analogous result for prime

In general, even when there is a critical point of multiplicity d, a sharper upper bound than (1.16) is available by applying our result for pure exponential sums, Theorem 2.1,

We study the distribution of rational points on certain K3 surfaces defined over an algebraic number field k of finite degree, namely the Kummer surfaces S/k attached to

S z´ek e l y, Crossing numbers and hard Erd˝os problems in Discrete

• ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXI (1979) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO. Séria I:

[r]