Invariants of plane curve singularities and Pl\"ucker formulas in positive characteristic

We study classical invariants for plane curve singularities $f\in K[[x,y]]$, $K$ an algebraically closed field of characteristic $p\geq 0$: Milnor number, delta invariant, kappa invariant and multiplicity. It is known, in characteristic zero, that $\mu(f)=2\delta(f)-r(f)+1$ and that $\kappa(f)=2\delta(f)-r(f)+\mathrm{mt}(f)$. For arbitrary characteristic, Deligne prove that there is always the inequality $\mu(f)\geq 2\delta(f)-r(f)+1$ by showing that $\mu(f)-\left( 2\delta(f)-r(f)+1\right)$ measures the wild vanishing cycles. By introducing new invariants $\gamma,\tilde{\gamma}$, we prove in this note that $\kappa(f)\geq \gamma(f)+\mathrm{mt}(f)-1\geq 2\delta(f)-r(f)+\mathrm{mt}(f)$ with equalities if and only if the characteristic $p$ does not divide the multiplicity of any branch of $f$. As an application we show that if $p$ is "big" for $f$ (in fact $p > \kappa(f)$), then $f$ 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 $f\in K[[x]]$, where $K$ is an algebraically closed field of characteristic $p>0$ by explicit normal forms. We show that the right determinacy of $f$ is completely determined by its support. Moreover we prove that the right modality of $f$ is equal to the integer part of $\mu/p$, where $\mu$ is the Milnor number of $f$. As a consequence we prove in this case that the modality is equal to the proper modality, which is the dimension of the $\mu$-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 $\mu$-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 $p>0$ of the ground field, the Milnor number of $f$ satisfies $\mu(f)\leq 4p$, if the right modality of $f$ 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 $\{f(x,y) = 0\}$ and introduce the new notion of weighted homogeneous Newton non-degeneracy (WHNND). It is known that the Milnor number $\mu$ resp. the delta-invariant $\delta$ can be computed by explicit formulas $\mu_N$ resp. $\delta_N$ from the Newton diagram of $f$ if $f$ is NND resp. WNND. It was however unknown whether the equalities $\mu=\mu_N$ resp. $\delta=\delta_N$ can be characterized by a certain non-degeneracy condition on $f$ and, if so, by which one. We show that $\mu=\mu_N$ resp. $\delta=\delta_N$ 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