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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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Journal ArticleDOI
TL;DR: The first explicitly known polynomials in Z(x) with nonsolvable Galois group and field discriminant of the form ±p A for p 7 a prime were presented in this paper.
Abstract: We present the first explicitly known polynomials in Z(x) with nonsolvable Galois group and field discriminant of the form ±p A for p 7 a prime. Our main polyno- mial has degree 25, Galois group of the form PSL2(5) 5 .10, and field discriminant 5 69 . A closely related polynomial has degree 120, Galois group of the form SL2(5) 5 .20, and field discriminant 5 311 . We completely describe 5-adic behavior, finding in particular that the root discriminant of both splitting fields is 125·5 1/12500 124.984 and the class number of the latter field is divisible by 5 4 .

6 citations

Journal ArticleDOI
TL;DR: For a wildly ramified p-extension E/F of algebraic function fields of one variable in an algebraically closed field k of characteristic p with Galois Group G, Nakajima obtained two exact sequences which determined implicitly the structure of the holomorphic semisimple differentials as k[G]-module as mentioned in this paper.
Abstract: For a wildly ramified p-extension E/F of algebraic function fields of one variable in an algebraically closed field k of characteristic p with Galois Group G, Nakajima obtained two exact sequences which determined implicitly the structure of the holomorphic semisimple differentials as k[G]-module. In this paper, in many cases, e.g., if there is a fully ramified prime, the structure is determined explicitly. Analogous results are obtained for p-extensions of ℤp-fields of CM-type. In the latter situation, if E/F is unramified, the structure of the minus part of the p-class group of E is determined as ℤp[G]-module.

6 citations

Posted Content
TL;DR: In this article, a discrete version of the Riemann-hilbert problem is solved for a dessin d'enfants, where the objective is to find 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 universal 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 representation 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

Journal ArticleDOI
TL;DR: In this paper, Binomial and Maclaurin series expansions are used for expressing typical interatomic potential functions in a generic polynomial function, with the coefficients presented in a tabular format.
Abstract: The use of polynomial functionals for describing two-body interactions in computational chemistry softwares has been surveyed and found to be prevalent. In this paper, Binomial and Maclaurin series expansions are used for expressing typical interatomic potential functions – such as Lennard-Jones, Morse, Rydberg and Buckingham potential – in a generic polynomial function, with the coefficients presented in a tabular format. Theoretical plots of these potential functions and their corresponding polynomial forms show increasing correlation with the order of polynomial, thereby validating the obtained polynomial’s coefficients. Conversely, a polynomial functional obtained by curve-fitting of experimental data can be converted into Morse, Rydberg and Buckingham potentials by using the generated table.

6 citations

Book ChapterDOI
Yūichi Rikuna1
01 Jan 2004
TL;DR: For a finite group G and a field k, a G-Galois extension over k by G/k-extension has been shown in this article, which is the first version of inverse Galois problem.
Abstract: For a finite group G and a field k,we call a G-Galois extension over k by G/k-extension. Whether a G/k-extension exists or not is the first version of inverse Galois problem. Especially the case when k = Q the rational number field, plays an important role in the study of the absolute Galois Group of Q. By many mathematicians, the existence of G/Q-extensions has been shown for a lot of finite groups G by now (cf. Malle-Matzat [14], Serre [19], etc.)

6 citations

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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20234
20227
20216
202010
20196
20186