Suppose f, g ∈ Z [x 1 , ..., x n ]. Prove that

Singularities, Monodromy and Zeta Functions Blatt 3

Exercises for presentation in the exercise class on 8.11.2018

Aufgabe 1:

Suppose f, g ∈ Z [x ₁ , ..., x _n ]. Prove that

{ x ¯ ∈ Q ⁿ p |v(f (¯ x)) ≥ v(g(¯ x))}

is a semi-algebraic set.

Aufgabe 2:

A valuative ball in Q p is a set of the form

B(c; γ) := {x ∈ Q p |v(x − c) ≥ γ},

where c ∈ Q p is a centre of the ball and γ ∈ Z its valuative radius.

(a) Suppose that B is a valuative ball in Q p and x 6∈ B. Prove that y 7→ ac 1 (y − x) is constant on B.

(b) Show that for any x, y ∈ Q p with v(x − y) > v(x), we have ac 1 (x) = ac 1 (y).

(c) Generalising (b), show that for every l ≥ 1,

v(x − y) ≥ v(x) + l ⇒ ac _l (x) = ac _l (y).

Aufgabe 3:

Let f (t) = (t − c 1 )(t − c 2 ) ² ∈ Q p [t] with c 1 6= c 2 .

(a) Find λ ∈ Z , a 1 , a 2 , a 3 ∈ Z and µ 1 , µ 2 , µ 3 ∈ Z such that:

• For each i ∈ {1, 2}, we have that for all t ∈ Q p with v(t − c i ) > λ,

v(f (t)) = µ i + a i · v(t − c i );

• While for all t ∈ Q p with v(t − c 1 ) ≤ λ and v(t − c 2 ) ≤ λ,

v(f (t)) = µ 3 + a 3 · v(t − c 1 ).

(b) Find λ ∈ Z , a ₁ , a ₂ , a ₃ ∈ Z and b ₁ , b ₂ , b ₃ ∈ F p such that:

• For each i ∈ {1, 2}, we have that for all t ∈ Q p with v(t − c _i ) > λ,

ac ₁ (f (t)) = b _i · ac ₁ (t − c _i ) ^a

;

• For t ∈ Q ^p such that v(t − c 1 ) < λ,

ac 1 (f (t)) = b 3 · ac 1 (t − c 1 ) ^a

;

• The remainder of Q p is a disjoint union of finitely many valuative balls B j , each of valuative radius (λ + 1), with ac 1 ◦f constant on each B j .

(c) Formulate and prove a similar statement for ac 2 ; i.e. there exists a partition

Q p = A 1 ∪ A 2 ∪ A 3 ∪ B 1 ∪ ... ∪ B m

so that there is a simple formula for ac ₂ ◦f on each A _i and that ac ₂ ◦f is constant on each B _j .

Aufgabe 4:

Complete the proof of the Hensel-Rychlik-Newton Lemma.

Course website: http://reh.math.uni-duesseldorf.de/~internet/Zeta_WS18/