2,533 research outputs found
Definability of linear equation systems over groups and rings
Motivated by the quest for a logic for PTIME and recent insights that the
descriptive complexity of problems from linear algebra is a crucial aspect of
this problem, we study the solvability of linear equation systems over finite
groups and rings from the viewpoint of logical (inter-)definability. All
problems that we consider are decidable in polynomial time, but not expressible
in fixed-point logic with counting. They also provide natural candidates for a
separation of polynomial time from rank logics, which extend fixed-point logics
by operators for determining the rank of definable matrices and which are
sufficient for solvability problems over fields. Based on the structure theory
of finite rings, we establish logical reductions among various solvability
problems. Our results indicate that all solvability problems for linear
equation systems that separate fixed-point logic with counting from PTIME can
be reduced to solvability over commutative rings. Moreover, we prove closure
properties for classes of queries that reduce to solvability over rings, which
provides normal forms for logics extended with solvability operators. We
conclude by studying the extent to which fixed-point logic with counting can
express problems in linear algebra over finite commutative rings, generalising
known results on the logical definability of linear-algebraic problems over
finite fields
Uniform definability of henselian valuation rings in the Macintyre language
We discuss definability of henselian valuation rings in the Macintyre
language , the language of rings expanded by n-th power
predicates. In particular, we show that henselian valuation rings with finite
or Hilbertian residue field are uniformly --definable in
, and henselian valuation rings with value group
are uniformly --definable in the ring
language, but not uniformly --definable in
. We apply these results to local fields
and , as well as to higher dimensional local fields
Generic Automorphisms and Green Fields
We show that the generic automorphism is axiomatisable in the green field of
Poizat (once Morleyised) as well as in the bad fields which are obtained by
collapsing this green field to finite Morley rank. As a corollary, we obtain
"bad pseudofinite fields" in characteristic 0. In both cases, we give geometric
axioms. In fact, a general framework is presented allowing this kind of
axiomatisation. We deduce from various constructibility results for algebraic
varieties in characteristic 0 that the green and bad fields fall into this
framework. Finally, we give similar results for other theories obtained by
Hrushovski amalgamation, e.g. the free fusion of two strongly minimal theories
having the definable multiplicity property. We also close a gap in the
construction of the bad field, showing that the codes may be chosen to be
families of strongly minimal sets.Comment: Some minor changes; new: a result of the paper (Cor 4.8) closes a gap
in the construction of the bad fiel
An existential 0-definition of F_q[[t]] in F_q((t))
We show that the valuation ring F_q[[t]] in the local field F_q((t)) is
existentially definable in the language of rings with no parameters. The method
is to use the definition of the henselian topology following the work of
Prestel-Ziegler to give an existential-F_q-definable bounded neighbouhood of 0.
Then we `tweak' this set by subtracting, taking roots, and applying Hensel's
Lemma in order to find an existential-F_q-definable subset of F_q[[t]] which
contains tF_q[[t]]. Finally, we use the fact that F_q is defined by the formula
x^q-x=0 to extend the definition to the whole of F_q[[t]] and to rid the
definition of parameters.
Several extensions of the theorem are obtained, notably an existential
0-definition of the valuation ring of a non-trivial valuation with divisible
value group.Comment: 9 page
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