Motivation
Main Results
References
New Quasi-Coincidence Point
Polynomial Problems
Yi-Chou Chen
National Army Academy, Taiwan
(joint with Hang-Chin Lai)
The 2013 Annual Meeting of the Taiwan Mathematical Society
Department of Applied Mathematics, Kaohsiung, Taiwan, Dec 6 – Dec 8, 2013
Yi-Chou Chen, (page: 1/33)
New Quasi-Coincidence Point Polynomial Problems
Motivation
Main Results
1
Motivation
2
Main Results
3
References
Yi-Chou Chen, (page: 2/33)
References
New Quasi-Coincidence Point Polynomial Problems
Motivation
Main Results
References
Motivation
1. F : R × R → R with
F(x, y) = as (x)ys + as−1 (x)ys−1 + · · · + a1 (x)y + a0 (x),
for some as (x), as−1 (x), . . . , a1 (x), a0 (x) ∈ R[x].
2. y = y(x) ∈ R[x].
Yi-Chou Chen, (page: 3/33)
New Quasi-Coincidence Point Polynomial Problems
Motivation
Main Results
References
If we let T : R[x] −→ R[x] with
T(y) = F(x, y).
Consider Problem
T(y(x)) = y(x).
(∗)
To solve the solutions y = y(x) is a fixed point problem.
Yi-Chou Chen, (page: 4/33)
New Quasi-Coincidence Point Polynomial Problems
Motivation
Main Results
References
Problem (∗) implies
as (x)ys (x) + as−1 (x)ys−1 (x) + · · · + a1 (x)y(x) + a0 (x) = y(x),
and then
as (x)ys (x) + as−1 (x)ys−1 (x) + · · · + (a1 (x) − 1)y(x) + a0 (x) = 0.
Yi-Chou Chen, (page: 5/33)
New Quasi-Coincidence Point Polynomial Problems
Motivation
Main Results
References
Let
F(x, y) = as (x)ys (x) + as−1 (x)ys−1 (x) + · · · + (a1 (x) − 1)y(x)
+ a0 (x) ∈ R[x],
then Problem (1) become to F(x, y) = 0.
Yi-Chou Chen, (page: 6/33)
New Quasi-Coincidence Point Polynomial Problems
Motivation
Main Results
References
This means that the fixed point problem
F(x, y(x)) = y(x)
is equivalent to the polynomial equation
F(x, y(x)) = 0.
Yi-Chou Chen, (page: 7/33)
New Quasi-Coincidence Point Polynomial Problems
Motivation
Main Results
References
(2). Consider the fixed coincidence problem, solving
y = y(x) ∈ R[x] to satisfy
F(x, y(x)) = f (x).
(2)
This implies
as (x)ys (x) + as−1 (x)ys−1 (x) + · · · + a1 (x)y(x) + a0 (x) = f (x),
and then
as (x)ys (x)+as−1 (x)ys−1 (x)+· · ·+a1 (x)y(x)+(a0 (x) − f (x)) = 0.
Yi-Chou Chen, (page: 8/33)
New Quasi-Coincidence Point Polynomial Problems
Motivation
Main Results
References
Let
G(x, y) = as (x)ys (x) + as−1 (x)ys−1 (x) + · · · + a1 (x)y(x)
+ (a0 (x) − f (x)) ∈ R[x].
Then Problem (2) also becomes the polynomial equation
G(x, y) = 0.
Yi-Chou Chen, (page: 9/33)
New Quasi-Coincidence Point Polynomial Problems
Motivation
Main Results
References
This means that the fixed coincidence problem
F(x, y(x)) = f (x)
is also equivalent to the problem
G(x, y) = 0.
Yi-Chou Chen, (page: 10/33)
New Quasi-Coincidence Point Polynomial Problems
Motivation
Main Results
References
(3). Consider the problem, solving y = y(x) ∈ R[x] to
satisfy
F(x, y(x)) = cy(x),
(3)
for some c ∈ R. This implies
as (x)ys (x) + as−1 (x)ys−1 (x) + · · · + a1 (x)y(x) + a0 (x) = cy(x)
Yi-Chou Chen, (page: 11/33)
New Quasi-Coincidence Point Polynomial Problems
Motivation
Main Results
References
and
as (x)ys (x) + as−1 (x)ys−1 (x) + · · · + (a1 (x) − c)y(x) + a0 (x) = 0.
Then
y(x) as (x)ys−1 (x) + as−1 (x)ys−2 (x) + · · · + (a1 (x) − c) = −a0 (x),
that is, y(x) is a factor of a0 (x).
Yi-Chou Chen, (page: 12/33)
New Quasi-Coincidence Point Polynomial Problems
Motivation
Main Results
References
Although we can obtain y(x) by solving all
factors of a0 (x) but this method can not be solved
by a polynomial time method. In fact, this
problem is still a NP-complete problem now.
Yi-Chou Chen, (page: 13/33)
New Quasi-Coincidence Point Polynomial Problems
Motivation
Main Results
References
Open Problem (1)
F : Q × Q −→ Q be a polynomial function.
Consider the problem
F(x, y) = cy(x)
(4)
for some c ∈ Q. Is there an co-NP complete algorithm to
solve all solutions y = y(x) of (4) ?
Yi-Chou Chen, (page: 14/33)
New Quasi-Coincidence Point Polynomial Problems
Motivation
Main Results
References
Consider the problem,
F(x, y(x)) = c f (x),
(5)
for some c ∈ R. To solve y = y(x) ∈ R[x] is a
quasi-coincidence problem.
Yi-Chou Chen, (page: 15/33)
New Quasi-Coincidence Point Polynomial Problems
Motivation
Main Results
References
This implies
as (x)ys (x) + as−1 (x)ys−1 (x) + · · · + a1 (x)y(x) + a0 (x) = c f (x)
Problem (5) is called a “Quasi-coincidence” Problem.
(which is defined by H.C. Lai in [6]).
Yi-Chou Chen, (page: 16/33)
New Quasi-Coincidence Point Polynomial Problems
Motivation
Main Results
References
Example 1:
Usually, many engineering systems are stable at infinity
like
lim F(x, yn ) = c,
n−→∞
this system will be stable at infinity and
F(x, lim yn ) = c.
n−→∞
Yi-Chou Chen, (page: 17/33)
New Quasi-Coincidence Point Polynomial Problems
Motivation
Main Results
References
Then to solve the solutions y = y(x) such that the system
F(x, y(x)) ∈ R
(6)
stable is a important work.
Yi-Chou Chen, (page: 18/33)
New Quasi-Coincidence Point Polynomial Problems
Motivation
Main Results
References
Open Problem (2)
F : Q × Q −→ Q be a polynomial function.
Consider the problem
F(x, y) = c f (x)
(5)
for some c ∈ Q. Is there an co-NP complete algorithm to
solve all solutions y = y(x) of (5) ?
Yi-Chou Chen, (page: 19/33)
New Quasi-Coincidence Point Polynomial Problems
Motivation
Main Results
References
In the following paper, we provide a easy
method to solve Problem (5) if the number of all
solutions is infinitely many.
Yi-Chou Chen, (page: 20/33)
New Quasi-Coincidence Point Polynomial Problems
Motivation
Main Results
References
Main Results
Let F : R × R −→ R with
F(x, y) = as (x)ys + as−1 (x)ys−1 + K + a1 (x)y + a0 (x)
for some as (x), as−1 (x), . . . , a1 (x),a0 (x) ∈ R[x].
Consider the problem, solving y = y(x) ∈ R[x] to satisfy
F(x, y(x)) = c f (x),
for some c ∈ R.
Yi-Chou Chen, (page: 21/33)
New Quasi-Coincidence Point Polynomial Problems
Motivation
Main Results
References
We depart this Problem into two parts: One is “Infinitely
many solutions, another is “finitely many solutions.
In this paper, we solve that (1) if the problem (4) has
infinitely many solutions. Any solutions can be
represented as
−
as−1 (x)
+ λp(x).
sas (x)
for any λ ∈ R[x], this p(x) = ( f (x)/as (x))1/s .
Yi-Chou Chen, (page: 22/33)
New Quasi-Coincidence Point Polynomial Problems
Motivation
Main Results
References
(1) Finitely many solutions: This problem can be solved
by a NP complete algorithm now. Recently, we try to find
it by a co-NP complete algorithm to solve it and haven’t
succeed.
(2) Infinitely many solutions: In the following theorems,
we give some results if the number of all solutions is
“infinitely many”.
Yi-Chou Chen, (page: 23/33)
New Quasi-Coincidence Point Polynomial Problems
Motivation
Main Results
References
Theorem 1. (Chen and Lai, 2013)
Suppose that the quasi-coincidence polynomial equation
(5) has infinitely many solutions, the polynomial function
F(x, y) must be represented as
s
X
i=0
ci
f (x)
pi (x)
y − y(x) i
for some ci ∈ R and some factor p(x) of f (x).
Yi-Chou Chen, (page: 24/33)
New Quasi-Coincidence Point Polynomial Problems
Motivation
Main Results
References
Theorem 2. (Chen and Lai, 2013)
Suppose that the quasi-coincidence polynomial equation
(5) has infinitely many solutions, any solutions can be
represented as
−
as−1 (x)
+ λp(x)
sas (x)
(7)
for some λ ∈ R, this p(x) = ( f (x)/as (x))1/s . Exactly, the
solution set of equation (5) is
{−as−1 (x)/sas (x) + λp(x) | λ ∈ R}.
Yi-Chou Chen, (page: 25/33)
New Quasi-Coincidence Point Polynomial Problems
Motivation
Main Results
References
Example 2.
Let x ∈ R, f (x) = x4 (x − 1)4 and
F(x, y) = a3 (x)y3 + a2 (x)y2 + a1 (x)y + a0 (x)
= x(x − 1)y3 + 0y2 + x3 (x − 1)3 y + 0.
We will solve all quasi-coincidence solutions of
F(x, y) = ax4 (x − 1)4 .
This polynomial function has at least 6(≥ s + 3, since s = 3)
quasi-coincidence solutions as follows:
Yi-Chou Chen, (page: 26/33)
New Quasi-Coincidence Point Polynomial Problems
Motivation
Main Results
References
F(x1 , x2 , x2 − x ) = 2 x4 (x − 1)4 ,
F(x1 , x2 , 2x2 − 2x ) = 10 x4 (x − 1)4 ,
F(x1 , x2 , −x2 + x ) = −2 x4 (x − 1)4 ,
F(x1 , x2 , −2x2 + 2x ) = −10 x4 (x − 1)4 ,
5
F(x1 , x2 , x2 /2 − x/2 ) = x4 (x − 1)4
8
5
F(x1 , x2 , −x2 /2 + x/2 ) = − x4 (x − 1)4 .
8
Yi-Chou Chen, (page: 27/33)
New Quasi-Coincidence Point Polynomial Problems
Motivation
Main Results
References
In fact, we have |QcsF |= ∞ and compare with formula
(7), we obtain
f (x)
p(x) =
as (x)
! 1s
x4 (x − 1)4
=
x(x − 1)
! 13
= x(x − 1).
Yi-Chou Chen, (page: 28/33)
New Quasi-Coincidence Point Polynomial Problems
Motivation
Main Results
References
By Theorem 2, any quasi-coincidence solution is written
as:
−
a1 (x)
0
+ λp(x) = + λcx(x − 1)
sa2 (x)
2
= µx(x − 1)
where µ = λc ∈ R is arbitrary. This shows the
quasi-coincidence (point) solutions have cardinal
|QcsF |= ∞.
Yi-Chou Chen, (page: 29/33)
New Quasi-Coincidence Point Polynomial Problems
Motivation
Main Results
References
References
1. Y.C. Chen and H.C. Lai,
Quasi-fixed point solutions of real-valued polynomial
equation,
Proceedings of the 9th International Conference on Fixed
Point Theory and Applications, on July 16-22, 2009 held at
Taiwan.
2. Joachim von Gahen and Jürgen Gerhard,
Mordern Computer Algebra,
second edition in 2003, printed in the United Kingdom at
the University Press, Cambridge.
Yi-Chou Chen, (page: 30/33)
New Quasi-Coincidence Point Polynomial Problems
Motivation
Main Results
References
3. A.K. Lenstra,
Factoring multivariate polynomials over algebraic
number fields,
SIAM J. Computing 16 (1987), 591-598.
4. H.C. Lai and Y.C. Chen,
A Quasi-Fixed Polynomial Problem for a Polynomial
Function,
JNCA, Volume 11, Number 1, (2010), 101-114.
Yi-Chou Chen, (page: 31/33)
New Quasi-Coincidence Point Polynomial Problems
Motivation
Main Results
References
5. H.C. Lai and Y.C. Chen,
Quasi-fixed Point for vector-valued Polynomial
Functions on Rn × R,
Fixed Point Theory, 12(2011), Number 2, 391-400.
6. Y.C. Chen and H.C. Lai,
New Quasi-Coincidence Point Polynomial Problems,
Journal of Applied Mathematics. Volume 2013, Article ID
959464,8 pages http://dx.doi.org/10.1155/2013/959464.
Yi-Chou Chen, (page: 32/33)
New Quasi-Coincidence Point Polynomial Problems
Motivation
Main Results
References
Thanks For Your Attention !!
Yi-Chou Chen, (page: 33/33)
New Quasi-Coincidence Point Polynomial Problems