Uniform boundedness in operators parametrized by polynomial curves

Multiple results in harmonic analysis involving integrals of functions over curves (such as restriction theorems, convolution estimates, maximal function estimates or decoupling estimates) depend strongly on the non-vanishing of the torsion of the associated curve. Over the past years there has been considerable interest in extending these results to a degenerate case where the torsion vanishes at a finite number of points by using the affine arc-length as an alternative integration measure. As a model case, multiple results have been proven in which the coordinate functions of the curve are polynomials. In this case one expects the bounds of the operators to depend only on the degree of the polynomial. In this talk I will introduce and motivate the concept of affine arclength measure, provide new decomposition theorems for polynomial curves over characteristic zero local fields, and provide some applications to uniformity results in harmonic analysis.