From Polynomial to Linear Equation to Matrix: The Ezouidi Duality and Second Theorem

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.