𝔖 Bobbio Scriptorium
✦   LIBER   ✦

Representations by Quadratic Forms in a Finite Field of Characteristic Two

✍ Scribed by John D. Fulton

Publisher
John Wiley and Sons
Year
1977
Tongue
English
Weight
392 KB
Volume
77
Category
Article
ISSN
0025-584X

No coin nor oath required. For personal study only.

✦ Synopsis

Let F be a finite field of cliaracteristic two and'let F'xm and FIXn denote vector spaces of m-tuples and n-tuples, respectively, over P. Let Q be a quadratic form of rank m defined on FIXm and let Q, be a quadratic form of rank n defined on F I X n . Then relative to given ordered bases for .FIXm and FIXn, the number of 1-1 linear transformations T : F'x'2+F'xtn is determined such that Q(xT)= =Q1(x) for all x in FIX". Equivalently, the number of n x m matrices X with elements in F is determined such that X A X T is congruent to B niodulo the subgroup of alternate matrices in the additive group of all i z ~n matrices overF, where A is the upper triangular matrix o€Q and B is the upper triangular niatrix of QI relative to the given bases of F t X m and F I X , , respectively.

πŸ“œ SIMILAR VOLUMES

Zeros of a Pair of Quadratic Forms Defin
Zeros of a Pair of Quadratic Forms Defined over a Finite Field
✍ David B. Leep; Laura Mann Schueller πŸ“‚ Article πŸ“… 1999 πŸ› Elsevier Science 🌐 English βš– 155 KB

Let N be the number of affine zeros of a pair of quadratic forms in n#1 variables defined over a finite field F O . We give upper and lower bounds for N and show that these bounds are optimal. One result states that if n#1510 and every quadratic form in the pencil has order at least three, then "N!q

Simultaneous Representation of Integers
Simultaneous Representation of Integers by a Pair of Ternary Quadratic Formsβ€”With an Application to Index Form Equations in Quartic Number Fields
✍ IstvΓ‘n GaΓ‘l; Attila Pethő; Michael Pohst πŸ“‚ Article πŸ“… 1996 πŸ› Elsevier Science 🌐 English βš– 411 KB

Let Q 1 , Q 2 # Z[X, Y, Z] be two ternary quadratic forms and u 1 , u 2 # Z. In this paper we consider the problem of solving the system of equations (1) Q 2 (x, y, z)=u 2 in x, y, z # Z with gcd(x, y, z)=1. According to Mordell [12] the coprime solutions of can be presented by finitely many expr

A Determinant Representation for the Dis
A Determinant Representation for the Distribution of Quadratic Forms in Complex Normal Vectors
✍ Hongsheng Gao; Peter J Smith πŸ“‚ Article πŸ“… 2000 πŸ› Elsevier Science 🌐 English βš– 161 KB

Let the column vectors of X: M\_N, M<N, be distributed as independent complex normal vectors with the same covariance matrix 7. Then the usual quadratic form in the complex normal vectors is denoted by Z=XLX H where L: N\_N is a positive definite hermitian matrix. This paper deals with a representat

A Determinant Representation for the Dis
A Determinant Representation for the Distribution of a Generalised Quadratic Form in Complex Normal Vectors
✍ Peter J Smith; Hongsheng Gao πŸ“‚ Article πŸ“… 2000 πŸ› Elsevier Science 🌐 English βš– 186 KB

Consider the quadratic form Z=Y H (XL X H ) &1 Y where Y is a p\_m complex Gaussian matrix, X is an independent p\_n complex Gaussian matrix, L is a Hermitian positive definite matrix, and m p n. The distribution of Z has been studied for over 30 years due to its importance in certain multivariate s