Estimates on volumes of homogeneous polynomial spaces

In this paper we develop the "local part" of our local/global approach to globally valued fields (GVFs). The "global part", which relies on these results, is developed in a subsequent paper.We study virtual divisors on projective varieties defined over a valued field $K$, as well as sub-valuations on polynomial rings over $K$ (analogous to homogeneous polynomial ideals). We prove a Nullstellensatz-style duality between projective varieties equipped with virtual divisors (analogous to projective varieties over a plain field) and certain sub-valuations on polynomial rings over $K$ (analogous to homogeneous polynomial ideals). Our main result compares the \emph{volume} of a virtual divisor on a variety $W$, namely its $(\dim W + 1)$-fold self-intersection, with the asymptotic behaviour of the volume of the dual sub-valuation, restricted to the space of polynomial functions of degree $m$, as $m \rightarrow \infty$.