1,963 research outputs found
The Combinatorial Game Theory of Well-Tempered Scoring Games
We consider the class of "well-tempered" integer-valued scoring games, which
have the property that the parity of the length of the game is independent of
the line of play. We consider disjunctive sums of these games, and develop a
theory for them analogous to the standard theory of disjunctive sums of
normal-play partizan games. We show that the monoid of well-tempered scoring
games modulo indistinguishability is cancellative but not a group, and we
describe its structure in terms of the group of normal-play partizan games. We
also classify Boolean-valued well-tempered scoring games, showing that there
are exactly seventy, up to equivalence.Comment: 60 pages, 21 figure
Finite burden in multivalued algebraically closed fields
We prove that an expansion of an algebraically closed field by arbitrary
valuation rings is NTP, and in fact has finite burden. It fails to be
NIP, however, unless the valuation rings form a chain. Moreover, the incomplete
theory of algebraically closed fields with valuation rings is decidable.Comment: 40 page
Dp-finite fields VI: the dp-finite Shelah conjecture
We prove the dp-finite case of the Shelah conjecture on NIP fields. If K is a
dp-finite field, then K admits a non-trivial definable henselian valuation
ring, unless K is finite, real closed, or algebraically closed. As a
consequence, the conjectural classification of dp-finite fields holds.
Additionally, dp-finite valued fields are henselian. Lastly, if K is an
unstable dp-finite expansion of a field, then K admits a unique definable
V-topology.Comment: 26 page
Dp-finite fields V: topological fields of finite weight
We prove that unstable dp-finite fields admit definable V-topologies. As a
consequence, the henselianity conjecture for dp-finite fields implies the
Shelah conjecture for dp-finite fields. This gives a conceptually simpler proof
of the classification of dp-finite fields of positive characteristic.
For , we define a local class of "-topological fields",
generalizing V-topological fields. A -topology is the same thing as a
V-topology, and a -topology is some higher-rank analogue. If is an
unstable dp-finite field, then the canonical topology is a definable
-topology for . Every -topology has
between 1 and coarsenings that are V-topologies. If the given
-topology is definable in some structure, then so are the V-topological
coarsenings.Comment: Preliminary draft, comments welcom
On the proof of elimination of imaginaries in algebraically closed valued fields
We give a simplified proof of elimination of imaginaries (in the geometric
sorts) in ACVF, based on ideas of Hrushovski. This proof manages to avoid many
of the technical issues which arose in the original proof by Haskell,
Hrushovski, and Macpherson.Comment: New and improved coding for finite sets (section 4.3
A pathological o-minimal quotient
We give an example of a definable quotient in an o-minimal structure which
cannot be eliminated over any set of parameters, giving a negative answer to a
question of Eleftheriou, Peterzil, and Ramakrishnan. Equivalently, there is an
o-minimal structure M whose elementary diagram does not eliminate imaginaries.
We also give a positive answer to a related question, showing that any
imaginary in an o-minimal structure is interdefinable over an independent set
of parameters with a tuple of real elements. This can be interpreted as saying
that interpretable sets look "locally" like definable sets, in a sense which
can be made precise.Comment: Withdrawn; incorporated into arXiv:1911.1007
Circular Planar Resistor Networks with Nonlinear and Signed Conductors
We consider the inverse boundary value problem in the case of discrete
electrical networks containing nonlinear (non-ohmic) resistors. Generalizing
work of Curtis, Ingerman, Morrow, Colin de Verdiere, Gitler, and Vertigan, we
characterize the circular planar graphs for which the inverse boundary value
problem has a solution in this generalized non-linear setting. The answer is
the same as in the linear setting. Our method of proof never requires that the
resistors behave in a continuous or monotone fashion; this allows us to recover
signed conductances in many cases. We apply this to the problem of recovery in
graphs that are not circular planar. We also use our results to make a
frivolous knot-theoretic statement, and to slightly generalize a fact proved by
Lam and Pylyavskyy about factorization schemes in their electrical linear
group.Comment: 52 pages, 36 figure
Dp-finite fields I: infinitesimals and positive characteristic
We prove that NIP valued fields of positive characteristic are henselian.
Furthermore, we partially generalize the known results on dp-minimal fields to
dp-finite fields. We prove a dichotomy: if K is a sufficiently saturated
dp-finite expansion of a field, then either K has finite Morley rank or K has a
non-trivial Aut(K/A)-invariant valuation ring for a small set A. In the
positive characteristic case, we can even demand that the valuation ring is
henselian. Using this, we classify the positive characteristic dp-finite pure
fields.Comment: 89 pages; changed title, corrected typos, fixed some references,
improved formatting, added references to sequel
Forking and Dividing in Fields with Several Orderings and Valuations
We consider existentially closed fields with several orderings, valuations,
and -valuations. We show that these structures are NTP of finite burden,
but usually have the independence property. Moreover, forking agrees with
dividing, and forking can be characterized in terms of forking in ACVF, RCF,
and CF.Comment: 41 pages; dissertation chapte
A criterion for uniform finiteness in the imaginary sorts
Let be a theory. If eliminates , it need not follow
that eliminates , as shown by the example of the
-adics. We give a criterion to determine whether eliminates
. Specifically, we show that eliminates
if and only if is eliminated on all
interpretable sets of "unary imaginaries." This criterion can be applied in
cases where a full description of is unknown. As an application, we
show that eliminates when is a C-minimal
expansion of ACVF.Comment: 6 page
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