40 research outputs found
Factorisation theorems for generalised power series
Fields of generalised power series (or Hahn fields), with coefficients in a
field and exponents in a divisible ordered abelian group, are a fundamental
tool in the study of valued and ordered fields and asymptotic expansions. The
subring of the series with non-positive exponents appear naturally when
discussing exponentiation, as done in transseries, or integer parts. A notable
example is the ring of omnific integers inside the field of Conway's surreal
numbers.
In general, the elements of such subrings do not have factorisations into
irreducibles. In the context of omnific integers, Conway conjectured in 1976
that certain series are irreducible (proved by Berarducci in 2000), and that
any two factorisations of a given series share a common refinement.
Here we prove a factorisation theorem for the ring of series with
non-positive real exponents: every series is shown to be a product of
irreducible series with infinite support and a factor with finite support which
is unique up to constants. From this, we shall deduce a general factorisation
theorem for series with exponents in an arbitrary divisible ordered abelian
group, including omnific integers as a special case. We also obtain new
irreducibility and primality criteria.
To obtain the result, we prove that a new ordinal-valued function, which we
call degree, is a valuation on the ring of generalised power series with real
exponents, and we formulate some structure results on the associated RV monoid.Comment: 40 page
Factorisation of germ-like series
A classical tool in the study of real closed fields are the fields
of generalised power series (i.e., formal sums with well-ordered support) with
coefficients in a field of characteristic 0 and exponents in an ordered
abelian group . A fundamental result of Berarducci ensures the existence of
irreducible series in the subring of consisting of
the generalised power series with non-positive exponents.
It is an open question whether the factorisations of a series in such subring
have common refinements, and whether the factorisation becomes unique after
taking the quotient by the ideal generated by the non-constant monomials. In
this paper, we provide a new class of irreducibles and prove some further cases
of uniqueness of the factorisation.Comment: 11 pages; minor corrections and numbering changes; to appear in J.
Log. Ana
Decidability of the theory of modules over Pr\"ufer domains with infinite residue fields
We provide algebraic conditions ensuring the decidability of the theory of
modules over effectively given Pr\"ufer (in particular B\'ezout) domains with
infinite residue fields in terms of a suitable generalization of the prime
radical relation. For B\'{e}zout domains these conditions are also necessary.Comment: Updated so that the title and abstract matches the published version.
Other minor corrections and changes mad