Self-intersection is a fundamental feasibility constraint in shape optimization: a self-crossing boundary leaves its interior, normal field, and finite-element mesh ill-defined, yet most existing barriers rely on heuristic geometric-proximity measures rather than certifying self-intersection directly. We propose a self-intersection barrier grounded in algebraic detection. We derived two bivariate polynomials from a curve’s Fourier coefficients whose common roots correspond exactly to its self-intersections, and solved the resulting system using the existing colleague-pencil linearization of the Bézout resultant in a Chebyshev basis, evaluating resultant coefficients via the Fast Fourier Transform. We proved that the resulting barrier diverges if and only if the curve self-intersects, remains continuous everywhere on its domain, and reduces constrained search to unconstrained optimization over the Fourier coefficients. We identified and corrected three failure modes in the underlying polynomial system, with numerical reliability ensured by forward-error certification, Newton polishing, and a numerical safety net. We validated the detection method on six analytically solvable benchmark curves, matching MATLAB’s built-in solver while running ~35× faster. In shape-optimization tests against a heuristic proximity barrier, our approach converged using ~17× fewer function evaluations, ~37× less runtime, and reached a 24× lower final cost, while tracking target geometries more closely — enforcing simplicity by construction rather than rejecting it after the fact.