Enumeration of algebraic and tropical singular hypersurfaces

TL;DR

This paper extends Mikhalkin's lattice path algorithm to enumerate singular tropical hypersurfaces of arbitrary degree and dimension, establishing a correspondence theorem and constructing real hypersurfaces with many nodes.

Contribution

It develops a generalized lattice path algorithm for higher-dimensional hypersurfaces and proves a correspondence theorem linking tropical and algebraic singular hypersurfaces.

Findings

Constructs a δ-dimensional linear space of real hypersurfaces with many nodes.

Provides asymptotic counts of real hypersurfaces with δ nodes comparable to complex counts.

Improves the leading term estimate for the case δ=1.

Abstract

We develop a version of Mikhalkin's lattice path algorithm for projective hypersurfaces of arbitrary degree and dimension, which enumerates singular tropical hypersurfaces passing through appropriate configuration of points. By proving a correspondence theorem combined with the lattice path algorithm, we construct a $\delta$ dimensional linear space of degree $ d $ real hypersurfaces containing $\frac{1}{\delta!} (\gamma_nd^n)^\delta + O(d^{n\delta-1})$ hypersurfaces with $ \delta $ real nodes, where $ \gamma_n $ are positive and given by a recursive formula. This is asymptotically comparable to the number $ \frac{1}{\delta!} \left( (n+1)(d-1)^n \right)^{\delta}+O\left(d^{n(\delta-1)} \right) $ of complex hypersurfaces having $ \delta $ nodes in a $ \delta $ dimensional linear space. In the case $ \delta=1 $ we give a slightly better leading term.