Pairs of Bilinear Equations in a Finite Field

1966 ◽  
Vol 18 ◽  
pp. 561-565
Author(s):  
A. Duane Porter
Keyword(s):  
Finite Field ◽  
Quadratic Forms ◽  
Number Of Solutions ◽  
Image Position ◽  
Bilinear Equations ◽  
Special Case ◽  
Bilinear Equation

Let F = GF(g) be the finite field of q = pr elements, p arbitrary. We wish to consider the system of bilinear equations1.1where all coefficients are from F. The number of solutions in F of a single bilinear equation may be obtained from a theorem of John H. Hodges (3, Theorem 3) by properly defining the matrices U, V, A, B. In 1954, L. Carlitz (1) obtained, as a special case of his work on quadratic forms, the number of simultaneous solutions in F of (1.1) when all aj = 1 and p is odd. Carlitz considered the case p = 2 separately.

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.

General description on extended homoclinic orbit solutions of the KdV-type bilinear equations

2020 ◽  
pp. 2150092
Author(s):  
Shu-Zhi Liu ◽  
Da-Jun Zhang
Keyword(s):  
Soliton Solution ◽  
Homoclinic Orbit ◽  
Soliton Solutions ◽  
General Description ◽  
De Vries ◽  
Bilinear Derivatives ◽  
Bilinear Equations ◽  
Special Case ◽  
Bilinear Equation

The Korteweg–de Vries (KdV)-type bilinear equations always allow 2-soliton solutions. In this paper, for a general KdV-type bilinear equation, we interpret how the so-called extended homoclinic orbit solutions arise from a special case of its 2-soliton solution. Two properties of bilinear derivatives are developed to deal with bilinear equation deformations. A non-integrable (3+1)-dimensional bilinear equation is employed as an example.

scholarly journals ON THE SOLVABILITY OF BILINEAR EQUATIONS IN FINITE FIELDS

2008 ◽  
Vol 50 (3) ◽  
pp. 523-529 ◽  
Author(s):  
IGOR E. SHPARLINSKI
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Absolute Constant ◽  
Character Sums ◽  
Multiplicative Character ◽  
Image Position ◽  
Multiplicative Character Sums ◽  
Bilinear Equations

AbstractWe consider the equation over a finite field q of q elements, with variables from arbitrary sets $\cA,\cB, \cC, \cD \subseteq \F_q$. The question of solvability of such and more general equations has recently been considered by Hart and Iosevich, who, in particular, prove that if for some absolute constant C > 0, then above equation has a solution for any λ ∈ q*. Here we show that using bounds of multiplicative character sums allows us to extend the class of sets which satisfy this property.

On the Number of Solutions of Some General Types of Equations in a Finite Field

1952 ◽  
Vol 4 ◽  
pp. 343-351 ◽  
Author(s):  
Olin B. Faircloth
Keyword(s):  
Number Theory ◽  
Finite Field ◽  
Riemann Hypothesis ◽  
Quadratic Forms ◽  
Function Fields ◽  
Commutative Rings ◽  
Number Of Solutions ◽  
Fermat's Last Theorem ◽  
Conditional Equation ◽  
And Algebra

The conditional equation f(x1, … , xs) = 0, where f is a polynomial in the x´s with coefficients in a finite field F(pn), is connected with many well-known developments in number theory and algebra, such as: Waring's problem, the arithmetical theory of quadratic forms, the Riemann hypothesis for function fields, Fermat's Last Theorem, cyclotomy, and the theory of congruences in commutative rings.

Inequalities Associated with the Triangle

1965 ◽  
Vol 8 (5) ◽  
pp. 615-626 ◽  
Author(s):  
W. J. Blundon
Keyword(s):  
Equilateral Triangle ◽  
Quadratic Forms ◽  
Real Functions ◽  
Image Position ◽  
Special Case ◽  
Real Quadratic

Let R, r, s represent respectively the circumradius, the inradius and the semiperimeter of a triangle with sides a, b, c. Let f(R, r) and F(R, r) be homogeneous real functions. Let q(R, r) and Q(R, r) be real quadratic forms. The latter functions are thus a special case of the former. Our main result is to derive the strongest possible inequalities of the form1with equality only for the equilateral triangle.

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.

XIV.—The Minimum System of Two Quadratic Forms in n Variables

1926 ◽  
Vol 45 (2) ◽  
pp. 149-165 ◽  
Author(s):  
H. W. Turnbull ◽  
J. Williamson
Keyword(s):  
Quadratic Forms ◽  
Projective Group ◽  
Canonical Forms ◽  
Linear Transformations ◽  
Irreducible System ◽  
Simultaneous System ◽  
Image Position ◽  
Minimum System ◽  
Special Case ◽  
Rational Integral

The following pages deal with the simultaneous system of two general quadratic forms in n homogeneous variables. It is a special case of Gordan's Theorem which proves such systems to be finite, for the general projective group of linear transformations. While several works have dealt with the cases when n = 2, 3, or 4, nothing seems to have been written on the general case except a memoir in the year 1908. We continue, and simplify, the results there obtained, and now establish that(1) All rational integral concomitants of two quadratic forms in n variables and any number of sets of linear variables may be expressed as rational integral functions of (3n + 1) concomitants, and forms derivable by polarisation.(2) These (3n + 1) forms, called the H system, constitute a strictly irreducible system.This system is exhibited in §7 astogether with polars.The work is divided into three chapters: I §§ 1–4 is introductory notation, II §§ 5–17 provides a proof of these theorems, while III §§18–21 gives the non-symbolic and canonical forms of the results.

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.

Shelah's work on non-semi-proper iterations, II

2001 ◽  
Vol 66 (4) ◽  
pp. 1865-1883 ◽  
Author(s):  
Chaz Schlindwein
Keyword(s):  
Closed Subset ◽  
Stationary Subset ◽  
Finite Support ◽  
Countable Support ◽  
Final Model ◽  
Axiom A ◽  
Mahlo Cardinal ◽  
The One ◽  
Image Position ◽  
Special Case

One of the main goals in the theory of forcing iteration is to formulate preservation theorems for not collapsing ω1 which are as general as possible. This line leads from c.c.c. forcings using finite support iterations to Axiom A forcings and proper forcings using countable support iterations to semi-proper forcings using revised countable support iterations, and more recently, in work of Shelah, to yet more general classes of posets. In this paper we concentrate on a special case of the very general iteration theorem of Shelah from [5, chapter XV]. The class of posets handled by this theorem includes all semi-proper posets and also includes, among others, Namba forcing.In [5, chapter XV] Shelah shows that, roughly, revised countable support forcing iterations in which the constituent posets are either semi-proper or Namba forcing or P[W] (the forcing for collapsing a stationary co-stationary subset ofwith countable conditions) do not collapse ℵ1. The iteration must contain sufficiently many cardinal collapses, for example, Levy collapses. The most easily quotable combinatorial application is the consistency (relative to a Mahlo cardinal) of ZFC + CH fails + whenever A ∪ B = ω2 then one of A or B contains an uncountable sequentially closed subset. The iteration Shelah uses to construct this model is built using P[W] to “attack” potential counterexamples, Levy collapses to ensure that the cardinals collapsed by the various P[W]'s are sufficiently well separated, and Cohen forcings to ensure the failure of CH in the final model.In this paper we give details of the iteration theorem, but we do not address the combinatorial applications such as the one quoted above.These theorems from [5, chapter XV] are closely related to earlier work of Shelah [5, chapter XI], which dealt with iterated Namba and P[W] without allowing arbitrary semi-proper forcings to be included in the iteration. By allowing the inclusion of semi-proper forcings, [5, chapter XV] generalizes the conjunction of [5, Theorem XI.3.6] with [5, Conclusion XI.6.7].

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