On computing the Weierstrass points of a plane algebraic curve

Abstract

Weierstrass points are an important geometric feature and birational invariant of an algebraic curve. A place P on a curve of genus g is a Weierstrass point if the corresponding point on the canonical embedding of the curve is an inflection point where the tangent plane is tangent to order at least g. This condition is equivalent to there being a holomorphic differential with a g -fold zero at P. Knowing the Weierstrass points of a curve gives you information about its automorphisms, since an automorphism must carry a Weierstrass point to a Weierstrass point of the same type. Also there is ongoing research into determining the gonality of a curve by studying its Weierstrass points.In this poster, we present an algorithm to compute the Weierstrass points of a curve and show technical improvements over the classical method. We partially estimate the complexity of this algorithm at O ( g 9 d 2 ) operations for certain key steps.The algorithm follows a classical construction that determines Weierstrass points by the vanishing of the Wronskian determinant. Our improvements relate to two aspects:1. avoiding the direct computation of the full Wronskian determinant as a rational function in favor of an interpolation strategy where we evaluate the determinant at closed points, and2. computing a partial resultant of the Wronskian numerator polynomial with the curve equation to obtain the factors corresponding to Weierstrass points, thus avoiding the contribution of singularities.A manuscript describing this work in detail can be found at the website listed above.