scholarly journals The distribution of certain special values of the cubic Legendre symbol

1985 ◽  
Vol 27 ◽  
pp. 165-184 ◽  
Author(s):  
S. J. Patterson
Keyword(s):  
Cube Root ◽  
Legendre Symbol ◽  
Special Values ◽  
Root Of Unity ◽  
Residue Symbol ◽  
Image Position

Let ω be a primitive cube root of unity. We define the cubic residue symbol (Legendre symbol) on ℤ[ω] as follows. Let πεℤ[ω] be a prime, (3, π)=1. For α ε ℤ[ω] such that (α, π)=1 we let be that third root of unity so that

scholarly journals On the asymptotic expansion of Airy's Integral

1963 ◽  
Vol 6 (2) ◽  
pp. 113-115 ◽  
Author(s):  
E. T. Copson
Keyword(s):  
Differential Equation ◽  
Asymptotic Expansion ◽  
Integral Function ◽  
Cube Root ◽  
Three Solutions ◽  
Root Of Unity ◽  
Complex Cube ◽  
Image Position

The integral functionis known as Airy's Integral since, when z is real, it is equal to the integralwhich first arose in Airy's researches on optics. It is readily seen that w= Ai(z) satisfies the differential equation d2w/dz2 = zw, an equation which also has solutions Ai(ωz), Ai(ω2z), where ω is the complex cube root of unity, exp 2/3πi. The three solutions are connected by the relation.

7.—Elliptic Singular Perturbations of First-order Operators with Critical Points

1976 ◽  
Vol 74 ◽  
pp. 91-113 ◽  
Author(s):  
P. P. N. de Groen
Keyword(s):  
Boundary Value Problem ◽  
Critical Points ◽  
Elliptic Boundary ◽  
Elliptic Boundary Value Problem ◽  
Internal Layers ◽  
Order Operator ◽  
Elliptic Boundary Value ◽  
First Order ◽  
Special Values ◽  
Image Position

SynopsisWe study the asymptotic behaviour for ɛ→+0 of the solution Φ of the elliptic boundary value problemis a bounded domain in ℝ2, 2 is asecond-order uniformly elliptic operator, 1 is a first-order operator, which has critical points in the interior of , i.e. points at which the coefficients of the first derivatives vanish, ɛ and μ are real parameters and h is a smooth function on . We construct firstorder approximations to Φ for all types of nondegenerate critical points of 1 and prove their validity under some restriction on the range of μ.In a number of cases we get internal layers of nonuniformity (which extend to the boundary in the saddle-point case) near the critical points; this depends on the position of the characteristics of 1 and their direction. At special values of the parameter μ outside the range in which we could prove validity we observe ‘resonance’, a sudden displacement of boundary layers; these points are connected with the spectrum of the operator ɛ2 + 1 subject to boundary conditions of Dirichlet type.

Polynomials With {0, +1, -1} Coefficients and a Root Close to a Given Point

1997 ◽  
Vol 49 (5) ◽  
pp. 887-915 ◽  
Author(s):  
Peter Borwein ◽  
Christopher Pinner
Keyword(s):  
Algebraic Number ◽  
Real Root ◽  
Small Degree ◽  
Pisot Number ◽  
Roots Of Unity ◽  
Best Approximations ◽  
Rate Of Approximation ◽  
Root Of Unity ◽  
Extremal Polynomials ◽  
Image Position

AbstractFor a fixed algebraic number α we discuss how closely α can be approximated by a root of a {0, +1, -1} polynomial of given degree. We show that the worst rate of approximation tends to occur for roots of unity, particularly those of small degree. For roots of unity these bounds depend on the order of vanishing, k, of the polynomial at α.In particular we obtain the following. Let BN denote the set of roots of all {0, +1, -1} polynomials of degree at most N and BN(α k) the roots of those polynomials that have a root of order at most k at α. For a Pisot number α in (1, 2] we show thatand for a root of unity α thatWe study in detail the case of α = 1, where, by far, the best approximations are real. We give fairly precise bounds on the closest real root to 1. When k = 0 or 1 we can describe the extremal polynomials explicitly.

Special values of L-functions and height two formal groups

1989 ◽  
Vol 105 (1) ◽  
pp. 13-24
Author(s):  
Rodney I. Yager
Keyword(s):  
Elliptic Curve ◽  
Quadratic Field ◽  
Complex Multiplication ◽  
Maximal Order ◽  
Imaginary Quadratic Field ◽  
Fundamental Period ◽  
Formal Groups ◽  
Special Values ◽  
Image Position

Let ψ be the Grossencharacter attached to an elliptic curve E defined over an imaginary quadratic field K ⊂ of discriminant −dK, and having complex multiplication by the maximal order of K. We denote the conductor of ψ by and fix a Weierstrass model for E with coefficients in ,whose discriminant is divisible only by primes dividing 6. Let Kab be the abelian closure of K in and choose a fundamental period Ω ∈ for the above model of the curve.

Mixing properties of (α,β)-expansions

2009 ◽  
Vol 29 (4) ◽  
pp. 1119-1140 ◽  
Author(s):  
KARMA DAJANI ◽  
YUSUF HARTONO ◽  
COR KRAAIKAMP
Keyword(s):  
Dynamical System ◽  
Invariant Measure ◽  
Lebesgue Measure ◽  
Natural Extension ◽  
Mixing Properties ◽  
Special Values ◽  
Image Position ◽  
Relationship Of ◽  
The Relationship

AbstractLet 0<α<1 andβ>1. We show that everyx∈[0,1] has an expansion of the formwherehi=hi(x)∈{0,α/β}, andpi=pi(x)∈{0,1}. We study the dynamical system underlying this expansion and give the density of the invariant measure that is equivalent to the Lebesgue measure. We prove that the system is weakly Bernoulli, and we give a version of the natural extension. For special values ofα, we give the relationship of this expansion with the greedyβ-expansion.

The Maximal p-Extension of a Local Field

1971 ◽  
Vol 23 (3) ◽  
pp. 398-402 ◽  
Author(s):  
Murray A. Marshall
Keyword(s):  
Galois Group ◽  
Local Field ◽  
Residue Class ◽  
Prime Integer ◽  
Prime Field ◽  
Root Of Unity ◽  
Valued Field ◽  
Residue Class Field ◽  
Image Position ◽  
Relation Rank

1. Let k denote a local field, that is, a complete discrete-valued field with perfect residue class field . Let G denote the Galois group of the maximal separable algebraic extension M of k, and let g denote the corresponding object over . For a given prime integer p, let G(p) denote the Galois group of the maximal p-extension of k. The dimensions of the cohomology groupsconsidered as vector spaces over the prime field Z/pZ, are equal, respectively, to the rank and the relation rank of the pro-p-group G(p); see [4; 9]. These dimensions are well known in many cases, especially when k is finite [6; 3; (Hoechsmann) 2, pp. 297-304], but also when k has characteristic p, or when k contains a primitive pth root of unity [4, p. 205].

Special values of the hypergeometric series

1991 ◽  
Vol 109 (2) ◽  
pp. 257-261 ◽  
Author(s):  
G. S. Joyce ◽  
I. J. Zucker
Keyword(s):  
Gamma Function ◽  
Hypergeometric Series ◽  
Special Values ◽  
Image Position

Recently, several authors [1, 3, 9] have investigated the algebraic and transcendental values of the hypergeometric seriesfor rational parameters a, b, c and algebraic arguments z. This work has led to some interesting new identities such asand where Γ(x) denotes the gamma function.

On Primitive Prime Factors of an-bn

1962 ◽  
Vol 58 (4) ◽  
pp. 556-562 ◽  
Author(s):  
A. Schinzel
Keyword(s):  
Cyclotomic Polynomial ◽  
Root Of Unity ◽  
Prime Factors ◽  
Image Position

Let a, b be relatively prime integers with |a| > |b| > 0. For any integer n > 0, let π n denote the nth cyclotomic polynomial, denned bywhere ζn is a primitive nth root of unity.

scholarly journals A note on secure multiparty computation via higher residue symbols

2021 ◽  
Vol 15 (1) ◽  
pp. 284-297
Author(s):  
Ignacio Cascudo ◽  
Reto Schnyder
Keyword(s):  
Prime Number ◽  
Small Difference ◽  
Secure Multiparty Computation ◽  
Small Subset ◽  
Multiparty Computation ◽  
Legendre Symbol ◽  
Sign Function ◽  
Residue Symbol

Abstract We generalize a protocol by Yu for comparing two integers with relatively small difference in a secure multiparty computation setting. Yu's protocol is based on the Legendre symbol. A prime number p is found for which the Legendre symbol (· | p) agrees with the sign function for integers in a certain range {−N, . . . , N} ⊂ ℤ. This can then be computed efficiently. We generalize this idea to higher residue symbols in cyclotomic rings ℤ[ζr ] for r a small odd prime. We present a way to determine a prime number p such that the r-th residue symbol (· | p) r agrees with a desired function f : A → { ζ r 0 , … , ζ r r − 1 } f:A \to \left\{ {\zeta _r^0, \ldots ,\zeta _r^{r - 1}} \right\} on a given small subset A ⊂ ℤ[ζr ], when this is possible. We also explain how to efficiently compute the r-th residue symbol in a secret shared setting.

Periods and special values of the hypergeometric series

1989 ◽  
Vol 106 (3) ◽  
pp. 389-401 ◽  
Author(s):  
Matthias Flach
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Modular Forms ◽  
Hypergeometric Series ◽  
Joint Paper ◽  
Special Values ◽  
Precise Statement ◽  
Image Position

The aim of this paper is to complement results by Wolfart [14] about algebraic values of the classical hypergeometric seriesfor rational parameters a, b, c and algebraic arguments z. Wolfart essentially determines the set of a, b, c ∈ ℚ,z ∈ ℚ for which F(a, b, c; z) ∈ ℚ and indicates, in a joint paper with F. Beukers[1], that some of these values can be expressed in terms of special values of modular forms. This method yields a few strikingly explicit identities likebut it does not give general statements about the nature of the algebraic values in question. In this paper we identify F(a, b, c; z) as a generator of a Kummer extension of a certain number field depending on z, which in particular bounds its degree as an algebraic number in terms of the degree of z. Our theorem in §2 seems to be the most precise statement one can make in general but sometimes improvements are possible as we point out at the end of §2.

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