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| Author: | John [ Thu Jan 06, 2011 8:55 pm ] |
| Post subject: | gmspoly.lib |
the function "isTame" in gmspoly.lib is supposed to check some sort of tameness of a given polynomial. Could anyone comment what precisely is meant here? The source code is pretty short. Maybe someone could also comment on this too. |
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| Author: | seelisch [ Thu Mar 17, 2011 10:58 am ] |
| Post subject: | Re: gmspoly.lib |
Hello, sorry for answering only now. I will answer this post of yours on behalf of Martin Schulze who is the author of the respective SINGULAR-LIB. He added some documentation to his gmspoly.lib, and now the overview gives a hint where to find more about the different kinds of tameness. Hope that helps! Frank OVERVIEW: A library for computing the Gauss-Manin system of a cohomologically tame polynomial f. Schulze's algorithm [Sch05], based on Sabbah's theory [Sab98], is used to compute a good basis of (the Brieskorn lattice of) the Gauss-Manin system and the differential operation of f in terms of this basis. In addition, there is a test for tameness in the sense of Broughton. Tame polynomials can be considered as an affine algebraic analogue of local analytic isolated hypersurface singularities. They have only finitely many citical points, and those at infinity do not give rise to atypical values in a sense depending on the precise notion of tameness considered. Well-known notions of tameness like tameness, M-tameness, Malgrange-tameness, and cohomological tameness, and their relations, are reviewed in [Sab98,8]. For ordinary tameness, see Broughton [Bro88,3]. Sabbah [Sab98] showed that the Gauss-Manin system, the D-module direct image of the structure sheaf, of a cohomologically tame polynomial carries a similar structure as in the isolated singularity case, coming from a Mixed Hodge structure on the cohomology of the Milnor (typical) fibre (see gmssing.lib). The data computed by this library encodes the differential structure of the Gauss-Manin system, and the Mixed Hodge structure of the Milnor fibre over the complex numbers. As a consequence, it yields the Hodge numbers, spectral pairs, and monodromy at infinity. REFERENCES: [Bro88] S. Broughton: Milnor numbers and the topology of polynomial hypersurfaces. Inv. Math. 92 (1988) 217-241. [Sab98] C. Sabbah: Hypergeometric periods for a tame polynomial. arXiv.org math.AG/9805077. [Sch05] M. Schulze: Good bases for tame polynomials. J. Symb. Comp. 39,1 (2005), 103-126. |
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