By Kiran S. Kedlaya

During the last 50 years the speculation of p-adic differential equations has grown into an energetic region of study in its personal correct, and has very important purposes to quantity idea and to laptop technological know-how. This publication, the 1st complete and unified advent to the topic, improves and simplifies present effects in addition to together with unique fabric. in response to a direction given via the writer at MIT, this contemporary remedy is on the market to graduate scholars and researchers. workouts are incorporated on the finish of every bankruptcy to aid the reader evaluate the cloth, and the writer additionally presents distinct references to the literature to assist additional learn.

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**Extra resources for p-adic Differential Equations**

**Example text**

N are not linearly independent over Q, the valuation vα is not uniquely determined by (a). ) (9) Let E be the completion of an infinite extension of Q p which is not discretely valued. Let α1 , α2 , . . ∈ E be any sequence of elements such that |α1 |, |α2 |, . . form a strictly decreasing sequence with positive limit. ) Prove that the sequence of discs {z ∈ E : |z − α1 − · · · − αi | ≤ |αi |} (10) (11) (12) (13) is decreasing, but its intersection is empty. ) Deduce as a corollary that E is not spherically complete.

For every x1 ∈ V1 , there exists x ∈ V lifting x1 with |x|V = |x1 |V1 . Hence V1 again satisfies (b). We may now apply the induction hypothesis to V1 to produce a basis m 2,1 , . . , m n,1 of V1 defining a supremum norm | · |V1 for which c−1/3 |x1 |V1 ≤ |x1 |V1 ≤ c1/3 |x1 |V1 (x1 ∈ V1 ). For i = 2, . . , n, choose m i ∈ V lifting m i,1 such that |m i |V ≤ c1/3 |m i,1 |V1 ; then |m i |V ≤ c1/3 |m i,1 |V1 ≤ c2/3 |m i,1 |V1 = c2/3 . Let | · |V be the supremum norm defined by m 1 , . . , m n .

N Pi T i ∈ R[T ], draw the set of points Given a polynomial P(T ) = i=0 {(−i, v(Pi )) : i = 0, . . , n; v(Pi ) < +∞} ⊂ R2 , then form the lower convex hull of these points. ) The boundary of this region is called the Newton polygon of P. The slopes of P are the slopes of this open polygon, viewed as a multiset in which each slope r counts with multiplicity equal to the horizontal width of the segment of the Newton polygon of slope r (or equal to zero if there is no such segment); the latter can also be interpreted as the width of P under | · |e−r .