Non-Linear Extension of Interval Arithmetic and Exact Resolution of Interval Equations: Pseudo-Complex Numbers

This paper introduces a novel extension of interval arithmetic through the formulation of pseudo- complex numbers, a mathematical framework defined over the quotient of polynomials R[h] (h2−h) . By leveraging pseudo-complex numbers, we extend traditional interval arithmetic to enhance the resolution of interval equations in analytical and computational settings. The proposed method systematically addresses the challenges of non-linear interval functions and their singularities, offering new tools for solving equations with guaranteed inclusion of solutions. Key results include the isomorphism between pseudo-complex numbers and diagonal matrices, the completeness of the pseudo-complex space, and the formulation of a generalized resolution theorem for interval equations. Applications and examples illustrate the practicality of this approach in diverse scenarios, including error propagation and constraint satisfaction in interval computations.