scholarly journals Ranked solutions of the matric equationA1X1=A2X2

1980 ◽  
Vol 3 (2) ◽  
pp. 293-304 ◽  
Author(s):  
A. Duane Porter ◽  
Nick Mousouris
Keyword(s):  
Finite Field ◽  
Number Of Solutions

LetGF(pz)denote the finite field ofpzelements. LetA1bes×mof rankr1andA2bes×nof rankr2with elements fromGF(pz). In this paper, formulas are given for finding the number ofX1,X2overGF(pz)which satisfy the matric equationA1X1=A2X2, whereX1ism×tof rankk1, andX2isn×tof rankk2. These results are then used to find the number of solutionsX1,…,Xn,Y1,…,Ym,m,n>1, of the matric equationA1X1…Xn=A2Y1…Ym.

Quantum computing and polynomial equations over Z_2

2005 ◽  
Vol 5 (2) ◽  
pp. 102-112
Author(s):  
C.M. Dawson ◽  
H.L. Haselgrove ◽  
A.P. Hines ◽  
D. Mortimer ◽  
M.A. Nielsen ◽  
...  
Keyword(s):  
Quantum Computing ◽  
Finite Field ◽  
Quantum Computation ◽  
Quantum Computer ◽  
Complexity Classes ◽  
Polynomial Equations ◽  
Computational Power ◽  
Number Of Solutions

What is the computational power of a quantum computer? We show that determining the output of a quantum computation is equivalent to counting the number of solutions to an easily computed set of polynomials defined over the finite field Z_2. This connection allows simple proofs to be given for two known relationships between quantum and classical complexity classes, namely BQP/P/\#P and BQP/PP.

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.

ON THE NUMBER OF SOLUTIONS TO THE EQUATION (x1 + ⋯ + xn)2 = ax1 ⋯ xn IN A FINITE FIELD

2008 ◽  
Vol 04 (05) ◽  
pp. 797-817 ◽  
Author(s):  
IOULIA BAOULINA
Keyword(s):  
Finite Field ◽  
Number Of Solutions

Let Nq be the number of solutions to the equation [Formula: see text] over the finite field 𝔽q = 𝔽ps. L. Carlitz found formulas for Nq when n = 3 or 4. In an earlier paper, we found formulas for Nq when d = gcd (n - 2, q - 1) = 1 or 2 or 3 or 4; and when there exists an ℓ such that pℓ ≡ -1 (mod d). In our other paper, the cases d = 7 or 14, p ≡ 2 or 4 (mod 7) were considered. Recently, we obtained formulas for Nq when d = 8. In this paper, we find formulas for Nq when d = 2t, t ≥ 4, p ≡ 3 or 5 (mod 8) or p ≡ 9 (mod 16).

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.

On the Number of Zeros Over a Finite Field of Certain Symmetric Polynomials

1980 ◽  
Vol 23 (3) ◽  
pp. 327-332
Author(s):  
P. V. Ceccherini ◽  
J. W. P. Hirschfeld
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Polynomial Equations ◽  
Symmetric Polynomials ◽  
Number Of Solutions ◽  
Number Of Zeros

A variety of applications depend on the number of solutions of polynomial equations over finite fields. Here the usual situation is reversed and we show how to use geometrical methods to estimate the number of solutions of a non-homogeneous symmetric equation in three variables.

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