The Valuations of the near \(2n\)-gon \(\mathbb{I}_n\)

Abstract

The maximal and next-to-maximal subspaces of a nonsingular parabolic quadric \(Q(2n,2)\), \(n \geq 2\), which are not contained in a given hyperbolic quadric \(Q_+(2n-1,q) \subset Q(2n,q)\) define a sub near polygon \(\mathbb{I}_n\) of the dual polar space \(DQ(2n,2)\). It is known that every valuation of \(DQ(2n,2)\) induces a valuation of \(\mathbb{I}_n\). In this paper, we show that also the converse is true: every valuation of \(\mathbb{I}_n\) is induced by a valuation of \(DQ(2n,2)\). We will also study the structure of the valuations of \(\mathbb{I}_n\).