On the cardinality of matrices with prescribed rank and partial trace over a finite field

被引：2

作者：

Balasubramanian, Kumar ^{[1
]}

Kaipa, Krishna ^{[2
]}

Khurana, Himanshi ^{[3
]}

机构：

[1] IISER Bhopal, Dept Math, Bhopal 462066, Madhya Pradesh, India

[2] IISER Pune, Dept Math, Pune 411008, Maharashtra, India

[3] Harish Chandra Res Inst, Dept Math, Prayagraj 211019, Uttar Pradesh, India

来源：

LINEAR ALGEBRA AND ITS APPLICATIONS | 2025年 / 704卷

关键词：

Rank; Trace; Cardinality; Generating function; DEGENERATE WHITTAKER MODELS; LINEAR-GROUPS; GAUSS SUMS;

D O I：

10.1016/j.laa.2024.10.011

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

Let F be the finite field of order q and M(n, n, r, F ) be the set of n x n matrices of rank r over the field F . For alpha E F and A E M(n, n, F ), let Z alpha A,r = {X X E M(n, n, r, F ) Tr(AX) AX ) = alpha } . In this article, we solve the problem of determining the cardinality of Z alpha A,r . We also solve the generalization of the problem to rectangular matrices. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

引用

页码：35 / 57

页数：23

共 15 条

[1]

Balasubramanian K, 2022, Arxiv, DOI arXiv:2206.03024

[2] Hyperplane sections of determinantal varieties over finite fields and linear codes [J].

Beelen, Peter ;

Ghorpade, Sudhir R. .

DISCRETE MATHEMATICS, 2020, 343 (09)

[3] NUMBER OF NXN MATRICES OF RANK R AND TRACE ALPHA OVER A FINITE FIELD [J].

BUCKHIES.PG .

DUKE MATHEMATICAL JOURNAL, 1972, 39 (04) :695-699

[4] BILINEAR FORMS OVER A FINITE-FIELD, WITH APPLICATIONS TO CODING THEORY [J].

DELSARTE, P .

JOURNAL OF COMBINATORIAL THEORY SERIES A, 1978, 25 (03) :226-241

[5] PROPERTIES AND APPLICATIONS OF RECURRENCE F(I+1, K+1, N+1) = QK+1F(I,K+1,N)-QKF(I,K,N) [J].

DELSARTE, P .

SIAM JOURNAL ON APPLIED MATHEMATICS, 1976, 31 (02) :262-270

[6]

Eichler M, 1938, J REINE ANGEW MATH, V179, P227

[7]

Fulman J, 2001, Arxiv, DOI arXiv:math/0105172

[8]

Gorodetsky O, 2019, MATH Z, V291, P609, DOI 10.1007/s00209-018-2097-y

[9] Gauss sums for general and special linear groups over a finite field [J].

Kim, DS .

ARCHIV DER MATHEMATIK, 1997, 69 (04) :297-304

[10]

Lamprecht E., 1957, J REINE ANGEW MATH, V197, P1, DOI DOI 10.1515/CRLL.1957.197.1

← 1 2 →