scholarly journals The Motivic Igusa Zeta Series of Some Hypersurfaces Non-Degenerated with Respect to their Newton Polyhedron

2016 ◽  
Vol 30 (1) ◽  
pp. 143-179
Author(s):  
Hans Schoutens
Keyword(s):  
Newton Polyhedron ◽  
Resolution Of Singularities

AbstractWe describe some algorithms, without using resolution of singularities, that establish the rationality of the motivic Igusa zeta series of certain hypersurfaces that are non-degenerated with respect to their Newton polyhedron. This includes, in any characteristic, the motivic rationality for polydiagonal hypersurfaces, vertex singularities, binomial hypersurfaces, and Du Val singularities.

A STRENGTHENING OF RESOLUTION OF SINGULARITIES IN CHARACTERISTIC ZERO

2003 ◽  
Vol 86 (2) ◽  
pp. 327-357 ◽  
Author(s):  
A. BRAVO ◽  
O. VILLAMAYOR U.
Keyword(s):  
Characteristic Zero ◽  
Resolution Of Singularities ◽  
Closed Subscheme ◽  
Subject Classification ◽  
Invertible Sheaf ◽  
Simple Manner ◽  
Birational Morphism ◽  
Exceptional Locus

Let $X$ be a closed subscheme embedded in a scheme $W$, smooth over a field ${\bf k}$ of characteristic zero, and let ${\mathcal I} (X)$ be the sheaf of ideals defining $X$. Assume that the set of regular points of $X$ is dense in $X$. We prove that there exists a proper, birational morphism, $\pi : W_r \longrightarrow W$, obtained as a composition of monoidal transformations, so that if $X_r \subset W_r$ denotes the strict transform of $X \subset W$ then:(1) the morphism $\pi : W_r \longrightarrow W$ is an embedded desingularization of $X$ (as in Hironaka's Theorem);(2) the total transform of ${\mathcal I} (X)$ in ${\mathcal O}_{W_r}$ factors as a product of an invertible sheaf of ideals ${\mathcal L}$ supported on the exceptional locus, and the sheaf of ideals defining the strict transform of $X$ (that is, ${\mathcal I}(X){\mathcal O}_{W_r} = {\mathcal L} \cdot {\mathcal I}(X_r)$).Thus (2) asserts that we can obtain, in a simple manner, the equations defining the desingularization of $X$.2000 Mathematical Subject Classification: 14E15.

Resolution of Singularities of Embedded Algebraic Surfaces

1968 ◽  
Vol 52 (379) ◽  
pp. 88
Author(s):  
J. A. Todd ◽  
S. S. Abhyankar
Keyword(s):  
Resolution Of Singularities ◽  
Algebraic Surfaces

Algorithmic Resolution of Singularities

2009 ◽  
pp. 157-184 ◽  
Author(s):  
Anne Frühbis-Krüger ◽  
Gerhard Pfister
Keyword(s):  
Resolution Of Singularities

scholarly journals LOCAL VANISHING AND HODGE FILTRATION FOR RATIONAL SINGULARITIES

2018 ◽  
Vol 19 (3) ◽  
pp. 801-819
Author(s):  
Mircea Mustaţă ◽  
Sebastián Olano ◽  
Mihnea Popa
Keyword(s):  
Toric Variety ◽  
Exceptional Divisor ◽  
Resolution Of Singularities ◽  
Isolated Singularities ◽  
Rational Singularities ◽  
Smooth Complex ◽  
Hodge Filtration ◽  
Generation Level ◽  
Image Position

Given an $n$-dimensional variety $Z$ with rational singularities, we conjecture that if $f:Y\rightarrow Z$ is a resolution of singularities whose reduced exceptional divisor $E$ has simple normal crossings, then $$\begin{eqnarray}\displaystyle R^{n-1}f_{\ast }\unicode[STIX]{x1D6FA}_{Y}(\log E)=0. & & \displaystyle \nonumber\end{eqnarray}$$ We prove this when $Z$ has isolated singularities and when it is a toric variety. We deduce that for a divisor $D$ with isolated rational singularities on a smooth complex $n$-dimensional variety $X$, the generation level of Saito’s Hodge filtration on the localization $\mathscr{O}_{X}(\ast D)$ is at most $n-3$.

Fractional Order Dynamics in the Trajectory Planning of Redundant and Hyper-Redundant Manipulators

2003 ◽  
Author(s):  
Fernando B. M. Duarte ◽  
J. A. Tenreiro Machado
Keyword(s):  
Fractional Calculus ◽  
Fractional Order ◽  
Trajectory Planning ◽  
Free Space ◽  
Trajectory Optimization ◽  
Generalized Inverse ◽  
Resolution Of Singularities ◽  
Control Algorithms ◽  
Redundant Manipulators ◽  
Kinematic Control

Redundant manipulators have some advantages when compared with classical arms because they allow the trajectory optimization, both on the free space and on the presence of obstacles, and the resolution of singularities. For this type of arms the proposed kinematic control algorithms adopt generalized inverse matrices but, in general, the corresponding trajectory planning schemes show important limitations. Motivated by these problems this paper studies the pseudoinverse-based trajectory planning algorithms, using the theory of fractional calculus.

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