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Preprint Number 1372

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1372. Itaï Ben Yaacov (ICJ)
Estimates on volumes of homogeneous polynomial spaces
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Submission date: 22 January 2018

Abstract:

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 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 → ∞.

Mathematics Subject Classification:

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Full text arXiv 1801.06994: pdf, ps.


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