Hermite character sums

For non-negative integers \(l_{1}, l_{2},\ldots , l_{n}\), we define character sums \(\varphi _{(l_{1}, l_{2},\ldots , l_{n})}\) and \(\psi _{(l_{1}, l_{2},\ldots , l_{n})}\) over a finite field which are generalizations of Jacobsthal and modified Jacobsthal sums, respectively. We express these character sums in terms of Greene’s finite field hypergeometric series. We then express the number of points on the hyperelliptic curves \(y^2=(x^m+a)(x^m+b)(x^m+c)\) and \(y^2=x(x^m+a)(x^m+b)(x^m+c)\) over a finite field in terms of the character sums \(\varphi _{(l_{1}, l_{2}, l_{3})}\) and \(\psi _{(l_{1}, l_{2}, l_{3})}\), and finally obtain expressions in terms of the finite field hypergeometric series.

Finite fields play important roles in many application areas such as coding theory, cryptography, Monte Carlo and quasi-Monte Carlo methods, pseudorandom number generation, quantum computing, and wireless communication. In this book we will focus on sequences, character sums, and polynomials over finite fields in view of the above mentioned application areas: Chapters 1 and 2 deal with sequences mainly constructed via characters and analyzed using bounds on character sums. Chapters 3, 5, and 6 deal with polynomials over finite fields. Chapters 4 and 9 consider problems related to coding theory studied via finite geometry and additive combinatorics, respectively. Chapter 7 deals with quasirandom points in view of applications to numerical integration using quasi-Monte Carlo methods and simulation. Chapter 8 studies aspects of iterations of rational functions from which pseudorandom numbers for Monte Carlo methods can be derived. The goal of this book is giving an overview of several recent research directions as well as stimulating research in sequences and polynomials under the unified framework of character theory.

Matrix valued Hermite polynomials, Burchnall formulas and non-abelian Toda lattice

There are two definitions for the discrete q-Hermite polynomials, one of them is defined for \(0 1\). This paper is devoted to extend these definitions to the discrete q-Hermite matrix polynomials by means of the generating matrix functions. Explicit expressions and Rodrigues-type formulas for the discrete q-Hermite matrix polynomials are obtained. Some recurrence relations for these matrix polynomials, in particular the three terms recurrence relations are given. Furthermore, some identities are proved.

Character Sums and Primitive Roots in Finite Fields

Let E be an elliptic curve over the finite field F p , and P P EpF p q be an F p -rational point.We obtain nontrivial estimates for multiplicative character sums associated with the division polynomials n pP q twisted by several multiplicative functions, where n pP q denotes the n-th division polynomial evaluated at P .

We prove analogs of the Burgess estimates for character sums over n-dimensional segments in the field \( \mathbb{F}_{p^n } \).

Linearly recurring solution sequences for equations over finite fields

In this note, we give estimates for a class of character sums that occur as eigenvalues of adjacency matrices of certain graphs constructed by F. R. K. Chung.Her situation is as follows.We are given a finite field F , an integer n ~ 1 , an extension field E of F of degree n, and an element x in E that generates E over F, i.e., an element x such that E is F(x) , Theorem 1.Let X be any nontrivial complex-valued multiplicative character of EX (extended by zero to all of E). and x in E any element that generates E over F.

Character Sums, Primitive Elements, and Powers in Finite Fields

There has recently been interest in relating properties of matrices drawn at random from the classical compact groups to statistical characteristics of number-theoretical L-functions. One example is the relationship conjectured to hold between the value distributions of the characteristic polynomials of such matrices and value distributions within families of L-functions. These connections are here extended to non-classical groups. We focus on an explicit example: the exceptional Lie group G_2. The value distributions for characteristic polynomials associated with the 7- and 14-dimensional representations of G_2, defined with respect to the uniform invariant (Haar) measure, are calculated using two of the Macdonald constant term identities. A one parameter family of L-functions over a finite field is described whose value distribution in the limit as the size of the finite field grows is related to that of the characteristic polynomials associated with the 7-dimensional representation of G_2. The random matrix calculations extend to all exceptional Lie groups

1. Symmetries of vector spaces: 1.1. What is a symmetry? 1.2. Distance is fundamental 1.3. Groups of symmetries 1.4. Bilinear forms and symmetries of spacetime 1.5. Putting the pieces together 1.6. A broader view: Lie groups 2. Complex numbers, quaternions and geometry: 2.1. Complex numbers 2.2. Quaternions 2.3. The geometry of rotations of R3 2.4. Putting the pieces together 2.5. A broader view: octonions 3. Linearization: 3.1. Tangent spaces 3.2. Group homomorphisms 3.3. Differentials 3.4. Putting the pieces together 3.5. A broader view: Hilbert's fifth problem 4. One-parameter subgroups and the exponential map: 4.1. One-parameter subgroups 4.2. The exponential map in dimension one 4.3. Calculating the matrix exponential 4.4. Properties of the matrix exponential 4.5. Using exp to determine L(G) 4.6. Differential equations 4.7. Putting the pieces together 4.8. A broader view: Lie and differential equations 4.9. Appendix on convergence 5. Lie algebras: 5.1. Lie algebras 5.2. Adjoint maps { big `A' and small `a' 5.3. Putting the pieces together 5.4. A broader view: Lie theory 6. Matrix groups over other fields: 6.1. What is a field? 6.2. The unitary group 6.3. Matrix groups over finite fields 6.4. Putting the pieces together 6.5. A broader view of finite groups of Lie type and simple groups Appendix I. Linear algebra facts Appendix II. Paper assignment used at Mount Holyoke College Appendix III. Opportunities for further study Solutions to selected problems Bibliography.

We estimate mixed character sums of polynomial values over elements of a finite field with sparse representations in a fixed ordered basis over the subfield . First we use a combination of the inclusion–exclusion principle with bounds on character sums over linear subspaces to get nontrivial bounds for large . Then we focus on the particular case , which is more intricate. The bounds depend on certain natural restrictions. We also provide families of examples for which the conditions of our bounds are fulfilled. In particular, we completely classify all monomials as argument of the additive character for which our bound is applicable. Moreover, we also show that it is applicable for a large family of rational functions, which includes all reciprocal monomials.

Matrix polynomials satisfying first order differential equations and three term recurrence relations

For classical polynomials orthogonal with respect to a positive measure supported on the real line, the moment matrix is Hankel and positive definite. The polynomials satisfy a three term recurrence relation. When the measure is supported on the complex unit circle, the moment matrix is positive definite and Toeplitz. Then they satisfy a coupled Szegő recurrence relation but also a three term recurrence relation. In this paper we study the generalization for formal polynomials orthogonal with respect to an arbitrary moment matrix and consider arbitrary Hankel and Toeplitz matrices as special cases. The relation with Padé approximation and with Krylov subspace iterative methods is also outlined.