Rank r Solutions to the Matrix Equation XAXT = C, A Nonalternate, C Alternate, Over GF(2y).

1974 ◽  
Vol 26 (1) ◽  
pp. 78-90 ◽  
Author(s):  
Philip G. Buckhiester
Keyword(s):  
Finite Field ◽  
Matrix Equation ◽  
Symmetric Matrices ◽  
Number Of Solutions ◽  
The Matrix ◽  
Image Position

Let GF(q) denote a finite field of order q = py, p a prime. Let A and C be symmetric matrices of order n, rank m and order s, rank k, respectively, over GF(q). Carlitz [6] has determined the number N(A, C, n, s) of solutions X over GF(q), for p an odd prime, to the matrix equation1.1where n = m. Furthermore, Hodges [9] has determined the number N(A, C, n, s, r) of s × n matrices X of rank r over GF(q), p an odd prime, which satisfy (1.1). Perkin [10] has enumerated the s × n matrices of given rank r over GF(q), q = 2y, such that XXT = 0. Finally, the author [3] has determined the number of solutions to (1.1) in case C = 0, where q = 2y.

On a Normal Form of the Orthogonal Transformation III

1958 ◽  
Vol 1 (3) ◽  
pp. 183-191 ◽  
Author(s):  
Hans Zassenhaus
Keyword(s):  
Normal Form ◽  
Linear Space ◽  
Bilinear Form ◽  
Matrix Equation ◽  
Orthogonal Transformation ◽  
The Matrix ◽  
Image Position

Under the assumptions of case of theorem 1 we derive from (3.32) the matrix equationso that there corresponds the matrix B to the bilinear form4.1on the linear space4.2and fP,μ, is symmetric if ɛ = (-1)μ+1, anti-symmetric if ɛ = (-1)μ.The last statement remains true in the case a) if P is symmetric irreducible because in that case fP,μ is 0.

On General Polynomials

1967 ◽  
Vol 10 (4) ◽  
pp. 579-583 ◽  
Author(s):  
Kenneth S. Williams
Keyword(s):  
Finite Field ◽  
Number Of Solutions ◽  
Fixed Integer ◽  
Image Position

Let d denote a fixed integer > 1 and let GF(q) denote the finite field of q = pn elements. We consider q fixed ≥ A(d), where A(d) is a (large) constant depending only on d. Let1where each aiεGF(q). Let nr(r = 2, 3, …, d) denote the number of solutions in GF(q) offor which x1, x2, …, xr are all different.

The Meromorphisms of an Elliptic Function-Field

1936 ◽  
Vol 32 (2) ◽  
pp. 212-215 ◽  
Author(s):  
H. Davenport
Keyword(s):  
Finite Field ◽  
Elliptic Function ◽  
Function Field ◽  
Unit Element ◽  
Number Of Solutions ◽  
Complicated Case ◽  
Zero Divisors ◽  
The Subject ◽  
Image Position ◽  
The Given

1. Hasse's second proof of the truth of the analogue of Riemann's hypothesis for the congruence zeta-function of an elliptic function-field over a finite field is based on the consideration of the normalized meromorphisms of such a field. The meromorphisms form a ring of characteristic 0 with a unit element and no zero divisors, and have as a subring the natural multiplications n (n = 0, ± 1, …). Two questions concerning the nature of meromorphisms were left open, first whether they are commutative, and secondly whether every meromorphism μ satisfies an algebraic equation with rational integers n0, … not all zero. I have proved that except in the case (which is equivalent to |N−q|=2 √q, where N is the number of solutions of the Weierstrassian equation in the given finite field of q elements), both these results are true. This proof, of which I give an account in this paper, suggested to Hasse a simpler treatment of the subject, which throws still more light on the nature of meromorphisms. Consequently I only give my proof in full in the case in which the given finite field is the mod p field, and indicate briefly in § 4 how it generalizes to the more complicated case.

Note on the Number of Solutions of f(x1) = f(x2) = … =f(xr) over a Finite Field

1971 ◽  
Vol 14 (3) ◽  
pp. 429-432
Author(s):  
Kenneth S. Williams
Keyword(s):  
Finite Field ◽  
Nonnegative Integer ◽  
Number Of Solutions ◽  
Image Position

Let GF(q) denote the finite field with q = pn elements and let(1)where each ai ∊ GF(q) and 1 < d <p. For r=2, 3, …, d we let nr denote the number of solutions (x1, …, xr) over GF(q) of(2)for which x1, …, xr are all different. Birch and Swinnerton-Dyer [1] have shown that(3)where each vr is a nonnegative integer depending on f and q and the constant implied by the O-symbol depends here, and throughout the paper, only on d.

Compounds of Skew-Symmetric Matrices

1964 ◽  
Vol 16 ◽  
pp. 473-478 ◽  
Author(s):  
Marvin Marcus ◽  
Adil Yaqub
Keyword(s):  
Symmetric Matrices ◽  
The Real ◽  
Real Solutions ◽  
Interesting Paper ◽  
The Matrix ◽  
Image Position ◽  
Square Matrix

In a recent interesting paper (3) H. Schwerdtfeger answered a question of W. R. Utz (4) on the structure of the real solutions A of A* = B, where A is skew-symmetric. (Utz and Schwerdtfeger call A* the "adjugate" of A ; A* is the n-square matrix whose (i, j) entry is (—1)i+j times the determinant of the (n — 1)-square matrix obtained by deleting row i and column j of A. The word "adjugate," however, is more usually applied to the matrix (AT)*, where AT denotes the transposed matrix of A ; cf. (1, 2).)The object of the present paper is to find all real n-square skew-symmetric solutions A to the equation

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