We study the statistics of the number of connected components and the volume of a random real algebraic hypersurface in RPn defined by a homogeneous polynomial f of degree d in the real Fubini-Study ensemble. We prove that for the expectation of the number of connected components, Eb(0)(Z(RPn)(f)) = Theta(d(n)), (1) the asymptotic being in d for n fixed. We do not restrict ourselves to the random homogeneous case, and we consider more generally random polynomials belonging to a window of eigenspaces of the Laplacian on the sphere S-n, proving that the same asymptotic holds. As for the volume properties, we provide an exact formula: EVol(Z(RPn)(f))= delta(1/2)Vol(Sn-1), (2) where delta (which we specify exactly) is asymptotically a constant times d(2). Both Equations (1) and (2) exhibit expectation of maximal order in light of Milnor's bound b(0)(Z(RPn)(f))= O(d(n)) and the bound Vol(Z(RPn)(f))= O(d).

Statistics on Hilbert’s Sixteenth Problem / Lerario, Antonio; Lundberg, E.. - In: INTERNATIONAL MATHEMATICS RESEARCH NOTICES. - ISSN 1073-7928. - 2015:12(2015), pp. 4293-4321. [10.1093/imrn/rnu069]

Statistics on Hilbert’s Sixteenth Problem

Lerario, Antonio;
2015-01-01

Abstract

We study the statistics of the number of connected components and the volume of a random real algebraic hypersurface in RPn defined by a homogeneous polynomial f of degree d in the real Fubini-Study ensemble. We prove that for the expectation of the number of connected components, Eb(0)(Z(RPn)(f)) = Theta(d(n)), (1) the asymptotic being in d for n fixed. We do not restrict ourselves to the random homogeneous case, and we consider more generally random polynomials belonging to a window of eigenspaces of the Laplacian on the sphere S-n, proving that the same asymptotic holds. As for the volume properties, we provide an exact formula: EVol(Z(RPn)(f))= delta(1/2)Vol(Sn-1), (2) where delta (which we specify exactly) is asymptotically a constant times d(2). Both Equations (1) and (2) exhibit expectation of maximal order in light of Milnor's bound b(0)(Z(RPn)(f))= O(d(n)) and the bound Vol(Z(RPn)(f))= O(d).
2015
2015
12
4293
4321
https://arxiv.org/abs/1212.3823
Lerario, Antonio; Lundberg, E.
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/32534
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 31
  • ???jsp.display-item.citation.isi??? 25
social impact