Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 $m \rightarrow \infty$.

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The categories of this paper are: atom-ph
10
The categories of this paper are: q-bio
14
The categories of this paper are: math-ph
11
The categories of this paper are: hep-lat
23
The categories of this paper are: solv-int
8
The categories of this paper are: hep-ex
25
The categories of this paper are: patt-sol
13
The categories of this paper are: quant-ph
9
The categories of this paper are: nlin
19
The categories of this paper are: eess
16
The categories of this paper are: stat
35
The categories of this paper are: mtrl-th
27
The categories of this paper are: comp-gas
3
The categories of this paper are: physics
6
The categories of this paper are: astro-ph
4
The categories of this paper are: cond-mat