Singularities, Monodromy and Zeta Functions Blatt 11
Exercises for discussion in the class on 24.1.2019
Givenf ∈Z[x1, ..., xn]letbf ∈Q[s]be the Bernstein polynomial forf.
Aufgabe 1:
(a) Suppose thatf is non-constant. Prove thatbf(s)is divisible by(s+ 1).
Hint: considers=−1.
(b) What isbf whenf is constant?
Aufgabe 2:
Show that for any non-zeroa∈Z,
bf =baf.
Aufgabe 3:
Supposeg= (f◦A)for someA∈GLn(Q).
Show thatbf is also the Bernstein polynomial forg.
Aufgabe 4:
(*) Suppose thatf is non-constant and that there is noa∈Csuch that
f(a) = ∂f
∂x1(a) =· · ·= ∂f
∂xn(a)= 0.
Prove thatbf= (s+ 1).
Hint: Use Hilbert’s Nullstellensatz to find polynomialsai such that a0(x)f(x) +X
ai(x)∂f
∂xi
= 1.
Then use thoseai to specify a differential operator P (satisfying P fs+1=bffs) explicitly.
Aufgabe 5:
(*) (Warm-up for next lecture)
Considerf as a function fromCn to C.
Supposexis a point of the variety defined byf such thatf(x) = 0andgradf(x)6= 0. Show there exist neighbourhoods U ⊆Cn containingx and D ⊆ C containing 0 so that the restriction of f to U ∩f−1(D) is a topologically trivial fibration.
Course website:http://reh.math.uni-duesseldorf.de/~internet/Zeta_WS18/
