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2013 New Quasi-Coincidence Point Polynomial Problems
Yi-Chou Chen, Hang-Chin Lai
J. Appl. Math. 2013(SI21): 1-8 (2013). DOI: 10.1155/2013/959464

Abstract

Let F : × be a real-valued polynomial function of the form F ( x , y ) = a s ( x ) y s + a s - 1 ( x ) y s - 1 + + a 0 ( x ) , where the degree s of y in F ( x , y ) is greater than or equal to 1 . For arbitrary polynomial function f ( x ) [ x ] , x , we will find a polynomial solution y ( x ) [ x ] to satisfy the following equation: ( * ): F ( x , y ( x ) ) = a f ( x ) , where a is a constant depending on the solution y ( x ) , namely, a quasi-coincidence (point) solution of ( * ), and a is called a quasi-coincidence value. In this paper, we prove that (i) the leading coefficient a s ( x ) must be a factor of f ( x ) , and (ii) each solution of ( * ) is of the form y ( x ) = - a s - 1 ( x ) / s a s ( x ) + λ p ( x ) , where λ is arbitrary and p ( x ) = c ( f ( x ) / a s ( x ) ) 1 / s is also a factor of f ( x ) , for some constant c , provided the equation ( * ) has infinitely many quasi-coincidence (point) solutions.

Citation

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Yi-Chou Chen. Hang-Chin Lai. "New Quasi-Coincidence Point Polynomial Problems." J. Appl. Math. 2013 (SI21) 1 - 8, 2013. https://doi.org/10.1155/2013/959464

Information

Published: 2013
First available in Project Euclid: 14 March 2014

zbMATH: 06950959
MathSciNet: MR3100826
Digital Object Identifier: 10.1155/2013/959464

Rights: Copyright © 2013 Hindawi

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