Research Article
Advancing Solutions to Higher-Degree Polynomials:
A Novel Recurrence Approach via EMS’s Theorem
Mourad Sultan Ezouidi*
,
Taoufik Gassoumi
Issue:
Volume 13, Issue 2, June 2026
Pages:
59-73
Received:
7 April 2026
Accepted:
16 April 2026
Published:
29 April 2026
Abstract: Solving higher-degree polynomial equations remains a fundamental challenge in both pure and applied mathematics. While quadratic, cubic, and quartic equations have known algebraic solutions, no general radical solution exists for degree five and higher (Abel–Ruffini theorem). This work introduces a novel recurrence-based methodology for deriving exact roots of polynomials of arbitrary degree, based on Ezouidi Mourad Sultan's Theorem (EMST). Unlike traditional algebraic techniques that are often restricted to degrees four or less or rely on numerical approximations, this framework allows for the explicit determination of roots, including irrational, complex, and multiple roots, across any polynomial degree. By systematically leveraging the structure of polynomial coefficients through recursive relationships, this approach extends the capabilities of classical methods and enhances their precision. The method is demonstrated through comprehensive examples involving irreducible and high-degree polynomials of degree 8, producing exact roots in closed form. Comparative analyses with established techniques such as Cardano's method, Newton's Method, and the Rational Root Theorem highlight the advantages of this recurrence formulation, including exactness, no reliance on initial guesses, and applicability to any degree. The EMST-based methodology offers a unified pathway toward exact solutions for longstanding algebraic problems.
Abstract: Solving higher-degree polynomial equations remains a fundamental challenge in both pure and applied mathematics. While quadratic, cubic, and quartic equations have known algebraic solutions, no general radical solution exists for degree five and higher (Abel–Ruffini theorem). This work introduces a novel recurrence-based methodology for deriving ex...
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Research Article
From Polynomial to Linear Equation to Matrix: The Ezouidi Duality and Second Theorem
Mourad Sultan Ezouidi*
Issue:
Volume 13, Issue 2, June 2026
Pages:
74-87
Received:
21 April 2026
Accepted:
30 April 2026
Published:
13 May 2026
DOI:
10.11648/j.ajaa.20261302.12
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Abstract: This paper introduces a new algebraic method based on the Ezouidi substitution and its inverse. The Ezouidi substitution expresses powers of the variable in terms of power sums of the roots and new auxiliary variables. A central result is the Ezouidi Second Theorem, an identity that relates power sums of the roots to the polynomial coefficients. The parameter q plays a fundamental role: when q equals zero, the polynomial has a single root repeated n times, giving the binomial form; when q equals one, the polynomial has n roots; when q equals two, the roots are the squares of the original roots; when q equals three, the roots are the cubes; and for higher q, the roots are raised to the corresponding power. Using the Ezouidi substitution together with the Second Theorem, the polynomial is transformed into a linear equation whose coefficients match the original polynomial. From this linear equation we construct the Ezouidi matrix. The matrix is singular, its rows satisfy the linear equation, and its columns sum to zero. The inverse substitution and the Second Theorem reverse the process, recovering the linear equation and the polynomial from the matrix. This establishes a complete triple duality: polynomial, linear equation, and Ezouidi matrix. A detailed example for degree six demonstrates the reverse direction, starting from the matrix to the linear equation and finally to the polynomial. Numerical examples for degrees two, three, and four further confirm the theoretical results. The method provides a new tool for connecting polynomial equations to linear systems and matrix theory.
Abstract: This paper introduces a new algebraic method based on the Ezouidi substitution and its inverse. The Ezouidi substitution expresses powers of the variable in terms of power sums of the roots and new auxiliary variables. A central result is the Ezouidi Second Theorem, an identity that relates power sums of the roots to the polynomial coefficients. Th...
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