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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: In this article, the simple group PSL_2(F_p) occurs as the Galois group of an extension of the rationals for all primes p>3.
Abstract: We show that the simple group PSL_2(F_p) occurs as the Galois group of an extension of the rationals for all primes p>3. We obtain our Galois extensions by studying the Galois action on the second etale cohomology groups of a specific elliptic surface.

12 citations

Journal ArticleDOI
01 Nov 2007
TL;DR: In this article, it was shown that for p > 5, the number of Hopf Galois structures on L/K afforded by K-Hopf algebras with associated group G is greater than p s, where s = (m-1) 2 3-m.
Abstract: Let p be an odd prime, G =Z m p , the elementary abelian p-group of rank m, and let Γ be the group of principal units of the ring Fp [x]/(x m+1 ). If L/K is a Galois extension with Galois group Γ, then we show that for p > 5, the number of Hopf Galois structures on L/K afforded by K-Hopf algebras with associated group G is greater than p s , where s = (m-1) 2 3- m.

12 citations

Journal ArticleDOI
TL;DR: The aim is to find sequences of functions which are the product of a polynomial of degree < n and an easy computable second factor and such that (f n ) n converges essentially faster to f on K than the sequence (P n * ) n of best approximating polynomials of degree ≤ n.
Abstract: Let f be an entire function of positive order and finite type. The subject of this note is the convergence acceleration of polynomial approximants of f by incorporating information about the growth of f(z) for z → ∞. We consider near polynomial approximation on a compact plane set K, which should be thought of as a circle or a real interval. Our aim is to find sequences (f n ) n of functions which are the product of a polynomial of degree < n and an easy computable second factor and such that (f n ) n converges essentially faster to f on K than the sequence (P n * ) n of best approximating polynomials of degree ≤ n. The resulting method, which we call Reduced Growth method (RG-method) is introduced in Section 2. In Section 5, numerical examples of the RG-method applied to the complex error function and to Bessel functions are given.

12 citations

01 Jan 2001
TL;DR: In this paper, the authors considered the distribution of the roots of the generalized Fibonacci polynomial and showed that the root θ is irreducible, and that all the other roots are in the open unit disk.
Abstract: be the generalized Fibonacci polynomial. There are a few papers in the literature in which the distribution of the roots of fn is considered. For example, from [5] and [6] we know that there exists a unique positive root θ of fn, which is larger than 1, and all the other roots of fn have modulus less than 1. Moreover, if n is odd, then fn has only one real root, while if n is even, fn has exactly two real roots (the root θ > 1 and one other root in the interval (−1, 0)). For an analysis of the real roots of f ′ n and f ′′ n see [3]. Since fn has a unique positive root θ which is larger than 1 and all the other roots are in the open unit disk, it follows that θ is a Pisot number and fn is a Pisot polynomial. In particular, fn is irreducible. This observation is due to Boyd (see [9]). Thus, we may denote the roots of fn by θ (i) for i = 1, 2, . . . , n, with the usual convention that θ = θ. We also let n = r1 + 2r2, where r1 is the number of real roots of fn and 2r2 is the number of complex non-real roots of fn. From the above comments we know that r2 = ⌊

12 citations

Journal ArticleDOI
TL;DR: In this article, the authors investigated the problem of second-order linear recurrence over finite fields with prime order and the characteristic polynomial of the relation is irreducible over F. They showed that a finite extension of F and a subgroup M of the multiplicative group of L such that the elements of M may be written, without repetition, so as to form a cyclically closed sequence which obeys the recurrence.

12 citations

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