Fourier restriction to polynomial curves I: a geometric inequality

We prove a Fourier restriction result for general polynomial curves in ${\Bbb R}^d$. Measuring the Fourier restriction with respect to the affine arclength measure of the curve, we obtain a universal estimate for the class of all polynomial curves of bounded degree. Our method relies on establishing a geometric inequality for general polynomial curves which is of interest in its own right. Applications of this geometric inequality to other problems in euclidean harmonic analysis have recently been established.