A New Insight on Ronkin Functions or Currents

摘要

We propose a new approach (through Fourier decomposition of currents on the complex torus) to the notion of Ronkin function Rf of a Laurent polynomial with complex coefficients and extend it to that of Ronkin current Rf, where f is more generally a Laurent polynomial mapping (f1,...,fm):TnCm. The concept of Ronkin function was introduced by Ronkin (On zeroes of almost periodic functions generated by holomorphic functions in a multicircular domain, Complex analysis in modern mathematics (in Russian), Fazis, Moscow, pp 243-256, 2000); it appears to be closely related to that of archimedean amOEba introduced by Gelfand et al. (Discriminants, resultants and multidimensional determinants, mathematics: theory and applications, Birkhauser Boston, Inc., Boston, 1994); the interest of such notions in pure or applied mathematics lie in the fact they connect complex geometry with max-plus (so called tropical) geometry. Inspired precisely by image processing methods in the particular case n=2, we use the Laplace differential operator (acting on distributions) in order to visualize the contour of the corresponding archimedean amOEba Af. We also introduce a notion of refined spine in accordance with the persistence of the geometric genus under complex versus tropical deformation of Af. Numerical illustrations conducted under Sage and Matlab (with codes provided) are detailed and commented as illustrations. Ronkin function as well as Ronkin current are also interpreted from the pluripotential point of view as Green currents with respect to the affine or toric Lelong-Poincare equation, the setting (here toric) being inspired by that of the (d,d)-differential calculus on Rn (based on the introduction of a ghost copy of Rn) such as introduced by Lagerberg (Math Z 270(3-4):1011-1050, 2012). Related potential applications to diophantine geometry, following a recent result by Gualdi (Heights of hypersurfaces in toric varieties, preprint, arXiv:1711.00710, 2017), are finally discussed.