1,054 research outputs found
Invariants of plane curve singularities and Pl\"ucker formulas in positive characteristic
We study classical invariants for plane curve singularities ,
an algebraically closed field of characteristic : Milnor number,
delta invariant, kappa invariant and multiplicity. It is known, in
characteristic zero, that and that
. For arbitrary characteristic,
Deligne prove that there is always the inequality by showing that
measures the wild vanishing cycles. By introducing new invariants
, we prove in this note that with equalities
if and only if the characteristic does not divide the multiplicity of any
branch of . As an application we show that if is "big" for (in fact
), then has no wild vanishing cycle. Moreover we obtain some
Pl\"ucker formulas for projective plane curves in positive characteristic.Comment: 15 pages; final version; to appear in the Annales de l'Institut
Fourie
The right classification of univariate power series in positive characteristic
While the classification of univariate power series up to coordinate change
is trivial in characteristic 0, this classification is very different in
positive characteristic. In this note we give a complete classification of
univariate power series , where is an algebraically closed
field of characteristic by explicit normal forms. We show that the right
determinacy of is completely determined by its support. Moreover we prove
that the right modality of is equal to the integer part of , where
is the Milnor number of . As a consequence we prove in this case that
the modality is equal to the proper modality, which is the dimension of the
-constant stratum in an algebraic representative of the semiuniversal
deformation with trivial section.Comment: 17 pages, final versio
Right unimodal and bimodal singularities in positive characteristic
The problem of classification of real and complex singularities was initiated
by Arnol'd in the sixties who classified simple, unimodal and bimodal w.r.t.
right equivalence. The classification of right simple singularities in positive
characteristic was achieved by Greuel and the author in 2014. In the present
paper we classify right unimodal and bimodal singularities in positive
characteristic by giving explicit normal forms. Moreover we completely
determine all possible adjacencies of simple, unimodal and bimodal
singularities. As an application we prove that, for singularities of right
modality at most 2, the -constant stratum is smooth and its dimension is
equal to the right modality. In contrast to the complex analytic case, there
are, for any positive characteristic, only finitely many 1-dimensional (resp.
2-dimensional) families of right class of unimodal (resp. bimodal)
singularities. We show that for fixed characteristic of the ground field,
the Milnor number of satisfies , if the right modality of
is at most 2.Comment: 19 page
Euler reflexion formulas for motivic multiple zeta functions
We introduce a new notion of \boxast-product of two integrable series with
coefficients in distinct Grothendieck rings of algebraic varieties, preserving
the integrability and commuting with the limit of rational series. In the same
context, we define a motivic multiple zeta function with respect to an ordered
family of regular functions, which is integrable and connects closely to
Denef-Loeser's motivic zeta functions. We also show that the \boxast-product
is associative in the class of motivic multiple zeta functions.
Furthermore, a version of the Euler reflexion formula for motivic zeta
functions is nicely formulated to deal with the \boxast-product and motivic
multiple zeta functions, and it is proved using the theory of arc spaces. As an
application, taking the limit for the motivic Euler reflexion formula we
recover the well known motivic Thom-Sebastiani theorem.Comment: To appear in Journal of Algebraic Geometr
Some remarks on the planar Kouchnirenko's Theorem
We consider different notions of non-degeneracy, as introduced by
Kouchnirenko (NND), Wall (INND) and Beelen-Pellikaan (WNND) for plane curve
singularities and introduce the new notion of weighted
homogeneous Newton non-degeneracy (WHNND). It is known that the Milnor number
resp. the delta-invariant can be computed by explicit formulas
resp. from the Newton diagram of if is NND resp.
WNND. It was however unknown whether the equalities resp.
can be characterized by a certain non-degeneracy condition on
and, if so, by which one. We show that resp.
is equivalent to INND resp. WHNND and give some applications and interesting
examples related to the existence of "wild vanishing cycles". Although the
results are new in any characteristic, the main difficulties arise in positive
characteristic.Comment: 23 pages, 2 figures. Final versio
Does exchange rate policy matter for economic growth? Vietnam evidence from a co-integration approach
Both economic growth and exchange rate theories suggest that the exchange rate regime could have consequences for the medium-term growth of a country, directly, through its effects on the adjustment to shocks, and indirectly, through its impact on the important determinants of growth. It is, however, surprising that there was little empirical work investigating the indirect relationship between the exchange rate policy and economics growth in the case of a specific country. In a co-integration framework, our research attempts to fill the gap by econometrically investigating the possible impacts of exchange rate regime on economic growth through two main channels - Foreign direct investment (FDI) and Exports - in the case of Vietnam - a successful example of a transitional economy.Exports, Exchange Rate, FDI, Growth, Co-integration
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