Soft Computing

, Volume 23, Issue 3, pp 837–845

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

• Hamed Farahani
• Mahmoud Paripour
• Saeid Abbasbandy
Methodologies and Application

## Abstract

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.

## Keywords

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

## Notes

### Conflict of interest

The authors declare that they have no conflict of interest

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© 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