scholarly journals The number of solutions of a system of equations in a finite field

1967 ◽  
Vol 12 (4) ◽  
pp. 421-424
Author(s):  
Charles Wells
Keyword(s):  
Finite Field ◽  
System Of Equations ◽  
Number Of Solutions

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.

scholarly journals Inductive analysis of the deflection of a beam truss allowing kinematic variation

2018 ◽  
Vol 239 ◽  
pp. 01012
Author(s):  
Mikhail Kirsanov ◽  
Evgeny Komerzan ◽  
Olesya Sviridenko
Keyword(s):  
Optimal Design ◽  
External Load ◽  
Analytical Calculation ◽  
System Of Equations ◽  
Number Of Solutions ◽  
Recurrence Equations ◽  
General Terms ◽  
Cutting Out ◽  
Selection Of ◽  
Inductive Analysis

A scheme of a statically determinate planar truss is proposed and an analytical calculation of its deflection and displacement of the mobile support are obtained. The forces in the rods from the external load, uniformly distributed over the nodes of the lower or upper belt, are determined by the method of cutting out nodes using the computer mathematic system Maple. In the generalization of a number of solutions of trusses with a different number of panels to the general case, the general terms of the sequence of coefficients in the formulas are found from solutions of linear homogeneous recurrence equations. To compose and solve these equations, Maple operators were used. In the process of calculation it was revealed that for even numbers of panels in half the span, the determinant of the system of equations degenerates. This corresponds to the kinematic degeneracy of the structure. The corresponding scheme of possible speeds of the truss is given. The displacement was determined by the Maxwell-Mohr’s formula. The graphs of the obtained dependences have appreciable jumps, which in principle can be used in the selection of optimal design sizes.

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.

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