Summary: We first recall the main features of fractional calculus. In the expression of fractional derivatives of a real polynomial \(f(x)\), we view the order of differentiation \(q\) as a new indeterminate; then we define a new bivariate polynomial \(P_{f}(x,q)\). For \(0\leq q\leq 1, P_{f}(x,q)\) defines a homotopy between the polynomials \(f(x)\) and \(xf^{\prime}(x)\). Iterating this construction, we associate to \(f(x)\) a plane spline curve, called the stem of \(f\). Stems of classic random polynomials exhibits intriguing patterns; moreover in the complex plane \(P_{f}(x,q)\) creates an unexpected correspondence between the complex roots and the critical points of \(f(x)\). We propose 3 conjectures to describe and explain these phenomena. Illustrations are provided relying on the computer algebra system Maple.