Parametric K-Formula: O(n) Closed-Form Solutions for Separable Polynomial Constraints

Separable polynomial equations of the form Pn i=1 aixpi i + a0 = 0 appear frequently in engineering applications including trilateration systems, regularized optimization, and geometric constraint satisfaction. Current methods for solving such systems rely primarily on iterative approaches like Newton-Raphson methods, which require O (n ³ ) operations per iteration due to Jacobian matrix computations and can experience convergence difficulties. However, the separable structure of these polynomials presents opportunities for more efficient solution methods that current approaches do not exploit. This study addresses this gap by developing the parametric K-formula (PK-formula), a novel noniterative method that generates closed-form parametric solutions for separable polynomial equations. The approach strategically introduces a tunable parameter k that decomposes the constraint into directly solvable components, enabling solution generation without iterative refinement. We provide complete mathematical derivation with formal proofs, characterize solution domains, and analyze computational complexity. The method’s performance is validated through three industrial applications: real-time positioning systems, regularized optimization, and robotic constraint satisfaction. The PK-formula achieves O(n) computational complexity compared to O (n ³ ) per iteration for Newton-Raphson methods. Experimental validation demonstrates computational speedups ranging from 50× to 120× across the three application domains, with deterministic timing characteristics suitable for real-time systems. The method generates solutions along a specific parametric curve on the constraint manifold where the last n−1 terms contribute equally, providing predictable computational performance. Numerical precision analysis confirms solution accuracy within machine precision limits across all tested scenarios. The results establish that structure-specific approaches can achieve substantial computational advantages for targeted polynomial classes while maintaining mathematical rigor. This work provides a foundation for developing specialized solvers for structured polynomial systems and demonstrates practical utility in applications requiring real-time constraint satisfaction with limited computational resources.