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Combinatorial study of morsifications of real univariate singularities

Arnaud Bodin, Evelia Rosa García Barroso, Patrick Popescu-Pampu, Miruna-Stefana Sorea

TL;DR

This work analyzes morsifications of real univariate singularities by translating the combinatorial type of the resulting Morse functions into planar contact-tree data derived from real and complex Newton-Puiseux roots. Central to the framework are the real contact tree $T_{\mathbb{R}}(f)$, the integrated contact tree $T_{\mathop{\mathrm{int}}}(f)$, and the area-series valuations that yield the integrated exponent function $\sigma$, enabling an explicit description of the signs and orderings of critical values. The injectivity condition ensures a robust correspondence between the real and complex root data and the two planar structures, culminating in Theorem A, which characterizes morsifications via a bi-ordered leaf set $(\mathcal{R}_{\mathbb{R}}(f),<_{{\mathbb{R}}},<_{\mathop{\mathrm{int}}})$. Theorem B strengthens this connection by identifying the integrated contact tree with the real contact tree of the apparent contour in the target, linking the critical-value combinatorics to the geometry of the discriminant. An explicit three-cusp example demonstrates how the combinatorial type and resulting snakes depend on parameter values, illustrating the practical utility of the theory for understanding real morsifications and their geometric shadows.

Abstract

We study a broad class of morsifications of germs of univariate real analytic functions. We characterize the combinatorial types of the resulting Morse functions, via planar contact trees constructed from Newton-Puiseux roots of the polar curves of the morsifications.

Combinatorial study of morsifications of real univariate singularities

TL;DR

This work analyzes morsifications of real univariate singularities by translating the combinatorial type of the resulting Morse functions into planar contact-tree data derived from real and complex Newton-Puiseux roots. Central to the framework are the real contact tree , the integrated contact tree , and the area-series valuations that yield the integrated exponent function , enabling an explicit description of the signs and orderings of critical values. The injectivity condition ensures a robust correspondence between the real and complex root data and the two planar structures, culminating in Theorem A, which characterizes morsifications via a bi-ordered leaf set . Theorem B strengthens this connection by identifying the integrated contact tree with the real contact tree of the apparent contour in the target, linking the critical-value combinatorics to the geometry of the discriminant. An explicit three-cusp example demonstrates how the combinatorial type and resulting snakes depend on parameter values, illustrating the practical utility of the theory for understanding real morsifications and their geometric shadows.

Abstract

We study a broad class of morsifications of germs of univariate real analytic functions. We characterize the combinatorial types of the resulting Morse functions, via planar contact trees constructed from Newton-Puiseux roots of the polar curves of the morsifications.
Paper Structure (33 sections, 7 theorems, 48 equations, 15 figures)

This paper contains 33 sections, 7 theorems, 48 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Morsifications
  3. Bi-ordered critical sets as measures of the combinatorial types of morsifications
  4. The meaning of the injectivity condition
  5. Structure of the paper
  6. Related works
  7. An explanatory picture
  8. Real Newton-Puiseux series, right semi-branches and primitives
  9. Newton-Puiseux series
  10. Right semi-branches
  11. Primitives
  12. Morsifications
  13. Morse functions
  14. The canonical bi-order on the critical graph of a Morse function
  15. Right-reduced functions, morsifications and their Morse rectangles
  16. ...and 18 more sections

Key Result

Proposition 3.8

Let $f\in \mathbb{R}\{x, y\}$ be a right-reduced series and $F_x(y)$ be a primitive of $f$. Assume that the series $F_x(\xi_i) \in \mathbb{R}\{ x^{\frac{1}{\mathbb{N}}} \}$ are pairwise distinct when $\xi_i$ varies among the real Newton-Puiseux roots of $f$. Then a Morse rectangle $[0, \varepsilon]

Figures (15)

  • Figure 1: The graph of a morsification $(x,y) \mapsto F_x(y)$
  • Figure 2: A Morse function and the two total orders on its critical set.
  • Figure 3: A Morse rectangle of $f$.
  • Figure 4: A set of outgoing edges and the greatest lower bound of two vertices.
  • Figure 5: Canonical total order on the leaves of a planar tree.
  • ...and 10 more figures

Theorems & Definitions (41)

  • Definition 2.1
  • Definition 3.1
  • Definition 3.2
  • Remark 3.3
  • Example 3.4
  • Definition 3.5
  • Definition 3.6
  • Example 3.7
  • Proposition 3.8
  • proof
  • ...and 31 more