We investigate the problem of robust and real-time rendering of algebraic surfaces. We show that expressing the intersection of the ray and the algebraic surface as a single univariate polynomial is not robust in practice, comparing results between monomial, Bernstein, Lagrange, and Chebyshev basis fits. We show that fitting multiple polynomials over subintervals, such as a unit length subdivision of the ray extent within the region of interest, improves robustness at a negligible performance cost.