Relational Geometric Architecture of Polynomial Roots

RELATIONAL GEOMETRIC ARCHITECTURE OF POLYNOMIAL ROOTS A model of independent circular references, algebraic identity, configuration, transformation and dynamics of solutions The work presents a geometric and relational architecture for the individual representation, identification, organization, and monitoring of polynomial roots through independent circular references. Its central proposition establishes a correspondence between the degree of a polynomial equation and the number of circular units required to represent its roots, counted with their respective multiplicities over the complex field. Each circle functions as an independent geometric reference, preserving the mathematical identity of one occurrence of a root. Its information includes the root's numerical value, multiplicity, real or complex classification, and geometric position. This arrangement makes it possible to distinguish the individual identity of each solution from the global configuration formed by the complete set of roots. The architecture is developed through complementary dimensions: individual root identification, independent geometric positioning, classification of real and complex solutions, multiplicity, the relationship between roots and polynomial coefficients, and the transformation of configurations as the coefficients of an equation change. A fundamental principle of the model is the distinction between the algebraic value of a root and the geometric position of its corresponding circle. The geometric position serves as an independent reference and does not necessarily reproduce the numerical location of the root in the complex plane. Consequently, the circles can remain spatially separate while retaining the mathematical relationships that connect their respective solutions. The mathematical construction is illustrated through polynomial equations of different degrees, including fifth-degree equations, equations with complex conjugate roots, and equations containing repeated roots. Vieta's formulas establish the correspondence between the individually registered roots and the coefficients of the original polynomial. This relationship makes it possible to examine the same algebraic structure in two complementary directions: from coefficients to roots and from roots to coefficients. The study further introduces a dynamic dimension through a transformation parameter. When the coefficients of a polynomial vary, the values and classifications of its roots may also change, as may their multiplicities and relationships. Each circular reference can register these variations, allowing the evolution of the complete configuration to be followed across successive states of the equation. The resulting model is expressed as a relational architecture comprising the original polynomial equation, the set of independent circular references, the roots, their multiplicities and classifications, and a transformation operator. This structure is integrated with the Theory of Transduction, in which the passage from an algebraic expression to an individualized geometric configuration is understood as a transition between mathematical states. The proposed architecture provides a unified representational framework applicable to polynomial equations of arbitrary degree. Its central structural principle is expressed as follows: ONE EQUATION → n ROOTS → n CIRCLES → ONE ARCHITECTURE By combining geometric individualization, algebraic correspondence, configuration, transformation, and relational memory, the work establishes a framework for investigating the organization and evolution of polynomial solutions, with particular application to the structural study of fifth-degree equations.