Higher Order Goh Conditions for Singular Extremals of Corank 1

We prove Goh conditions of order n >= 3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\geqq 3$$\end{document} for strictly singular length-minimizing curves of corank 1, under the assumption that the domain of the nth instrinsic differential is of finite codimension. This result relies upon the proof of an open mapping theorem for maps with a regular nth differential.