325 research outputs found
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
TFT construction of RCFT correlators IV: Structure constants and correlation functions
We compute the fundamental correlation functions in two-dimensional rational
conformal field theory, from which all other correlators can be obtained by
sewing: the correlators of three bulk fields on the sphere, one bulk and one
boundary field on the disk, three boundary fields on the disk, and one bulk
field on the cross cap. We also consider conformal defects and calculate the
correlators of three defect fields on the sphere and of one defect field on the
cross cap.
Each of these correlators is presented as the product of a structure constant
and the appropriate conformal two- or three-point block. The structure
constants are expressed as invariants of ribbon graphs in three-manifolds.Comment: 98 pages, some figures; v2 (version published in NPB): typos
correcte
Connected Lie Groupoids are Internally Connected and Integral Complete in Synthetic Differential Geometry
We extend some fundamental definitions and constructions in the established
generalisation of Lie theory involving Lie groupoids by reformulating them in
terms of groupoids internal to a well-adapted model of synthetic differential
geometry. In particular we define internal counterparts of the definitions of
source path and source simply connected groupoid and the integration of
-paths. The main results of this paper show that if a classical Hausdorff
Lie groupoid satisfies one of the classical connectedness conditions it also
satisfies its internal counterpart.Comment: Statement of Theorem 4.7 and notation in Section 4.3 correcte
Computing Puiseux series : a fast divide and conquer algorithm
Let be a polynomial of total degree defined over
a perfect field of characteristic zero or greater than .
Assuming separable with respect to , we provide an algorithm that
computes the singular parts of all Puiseux series of above in less
than operations in , where
is the valuation of the resultant of and its partial derivative with
respect to . To this aim, we use a divide and conquer strategy and replace
univariate factorization by dynamic evaluation. As a first main corollary, we
compute the irreducible factors of in up to an
arbitrary precision with arithmetic
operations. As a second main corollary, we compute the genus of the plane curve
defined by with arithmetic operations and, if
, with bit operations
using a probabilistic algorithm, where is the logarithmic heigth of .Comment: 27 pages, 2 figure
Equivariant virtual Betti numbers
We define a generalised Euler characteristic for arc-symmetric sets endowed
with a group action. It coincides with equivariant homology for compact
nonsingular sets, but is different in general. We lay emphasis on the
particular case of , and give an application to the study of the
singularities of Nash function germs via an analog of the motivic zeta function
of Denef & Loeser.Comment: 20 pages, to appear in Ann. Inst. Fourie
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