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Generic polynomial
About: Generic polynomial is a research topic. Over the lifetime, 608 publications have been published within this topic receiving 6784 citations.
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TL;DR: In this article, a discrete version of the Riemann-hilbert problem is solved for a dessin d'enfants, which is defined as finding a Fuchsian differential equation satisfied by the local inverses of a Shabat polynomial.
Abstract: We state and solve a discrete version of the classical Riemann-Hilbert problem. In particular, we associate a Riemann-Hilbert problem to every dessin d'enfants. We show how to compute the solution for a dessin that is a tree. This amounts to finding a Fuchsian differential equation satisfied by the local inverses of a Shabat polynomial. We produce a uni- versal annihilating operator for the inverses of a generic polynomial. We classify those plane trees that have a representation by Mobius transformations and those that have a linear rep- resentation of dimension at most two. This yields an analogue for trees of Schwarz's classical list, that is, a list of those plane trees whose Riemann-Hilbert problem has a hypergeometric solution of order at most two.
6 citations
TL;DR: In this article, the authorship of a paper has been defined as "that author whose name is followed by \"(p)\" pre-s of the papers presented.".
Abstract: s of the papers presented follow. Those having the letter V after their numbers were read by title. Where a paper has more than one author, that author whose name is followed by \"(p)\" pre-
6 citations
01 Sep 2006
TL;DR: For analytic vector fields on the complex plane, it was shown in this article that a generic polynomial vector field of degree higher than 2 on the plane has countably many complex limit cycles that are homologically independent on the leaves.
Abstract: It is well known that a generic polynomial vector field of degree higher than 2 on the plane has countably many complex limit cycles that are homologically independent on the leaves. In the paper, a similar assertion is proved for analytic vector fields on the complex plane. The proof essentially uses the results of D.S. Volk and T.S. Firsova.
6 citations
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TL;DR: In this paper, the Wedderburn decomposition of a number field is explicitly computable and each component is in fact a matrix ring over a field, which leads to an algorithm that either gives an A-basis for X or determines that no such basis exists.
Abstract: Let E be a number field and G be a finite group. Let A be any O_E-order of full rank in the group algebra E[G] and X be a (left) A-lattice. We give a necessary and sufficient condition for X to be free of given rank d over A. In the case that the Wedderburn decomposition of E[G] is explicitly computable and each component is in fact a matrix ring over a field, this leads to an algorithm that either gives an A-basis for X or determines that no such basis exists.
Let L/K be a finite Galois extension of number fields with Galois group G such that E is a subfield of K and put d=[K:E]. The algorithm can be applied to certain Galois modules that arise naturally in this situation. For example, one can take X to be O_L, the ring of algebraic integers of L, and A to be the associated order A of O_L in E[G]. The application of the algorithm to this special situation is implemented in Magma under certain extra hypotheses when K=E=Q.
6 citations
01 Jan 2005
TL;DR: In this paper, the authors provide algorithms for dynamically obtaining both a possible representation of the splitting field and the Galois group of a given separable polynomial from its universal decomposition algebra.
Abstract: We provide some algorithms for dynamically obtaining both a possible representation of the splitting field and the Galois group of a given separable polynomial from its universal decomposition algebra.
6 citations