Soft Computing

, Volume 23, Issue 3, pp 837–845 | Cite as

Resolution of single-variable fuzzy polynomial equations and an upper bound on the number of solutions

  • Hamed Farahani
  • Mahmoud ParipourEmail author
  • Saeid Abbasbandy
Methodologies and Application


In this paper, the single-variable fuzzy polynomial equations are studied. We firstly define two solution types for the equations, called solution and r-cut solution. Then, sufficient and necessary conditions are proposed for existence of the solution and r-cut solution of the equations, respectively. Also, a new algorithm is designed to find all the solutions and r-cut solutions of the equations using algebraic computations. Based on Descartes’ rule of signs, we express and prove a fuzzy version of fundamental theorem of algebra to obtain the number of real roots of a single-variable fuzzy polynomial. Moreover, we present an upper bound on the number of solutions of the equations and show that each single-variable fuzzy polynomial equation has at most two distinct solutions.


Fuzzy numbers Fuzzy polynomials Real solution r-Cut solution Fundamental theorem 


Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  • Hamed Farahani
    • 1
  • Mahmoud Paripour
    • 2
    Email author
  • Saeid Abbasbandy
    • 3
  1. 1.Department of Basic SciencesChabahar Maritime UniversityChabaharIran
  2. 2.Department of Computer Engineering and Information TechnologyHamedan University of TechnologyHamedanIran
  3. 3.Department of MathematicsImam Khomeini International UniversityQazvinIran

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