Technical Summary: Morse Theory of Euclidean Distance Functions from Algebraic Hypersurfaces

1. Problem Statement

The paper addresses the construction of a systematic Morse theory framework for the Euclidean distance function restricted to a smooth manifold $X$ from a closed definable subset $Y \subseteq \mathbb{R}^n$. Specifically, the authors analyze the function:
$$\text{dist}_Y|_X : X \to \mathbb{R}, \quad \text{where } \text{dist}_Y(z) := \min_{y \in Y} \|y - z\|.$$

While distance functions are central to differential geometry and computational algebraic geometry (defining concepts like the Euclidean Distance Degree and bottlenecks), they are generally non-differentiable and non-convex. Classical Morse theory, which relies on smoothness and non-degenerate critical points, cannot be directly applied. The core problem is to define a robust notion of "critical points" and "indices" for these Lipschitz functions that allows for topological analysis (via changes in sublevel sets) and provides bounds on the number of such critical points, particularly when $X$ and $Y$ are generic algebraic hypersurfaces.

2. Methodology

The authors develop a framework based on Lipschitz geometry and the theory of continuous selections, rather than classical smooth analysis.

2.1. Critical Points via Subdifferentials

Following Clarke's theory for locally Lipschitz functions, the authors define a critical point $x \in X$ as a point where the Clarke subdifferential $\partial_x (\text{dist}_Y|_X)$ contains the zero vector.

• The subdifferential is constructed as the convex hull of limits of gradients at points of differentiability.
• For the distance function, the subdifferential at $x$ is explicitly described as the convex hull of normalized vectors pointing from $x$ to its closest points in $Y$.
• A point is critical if the zero vector lies in the projection of this convex hull onto the tangent space $T_x X$.

2.2. Continuous Selections and Nondegeneracy

To apply Morse-theoretic results, the authors restrict their attention to cases where $\text{dist}_Y|_X$ behaves as a continuous selection of finitely many $C^2$ functions.

• Continuous Selection: A function $f$ is a continuous selection of $(f_0, \dots, f_m)$ if $f(x) = f_i(x)$ for some $i$ at every $x$.
• Nondegenerate Critical Points: A critical point $x$ is nondegenerate if:
  1. The differentials of the active functions $f_i$ (where $f(x)=f_i(x)$) satisfy a linear independence condition.
  2. A specific convex combination of these functions, denoted $\lambda_f$, has a nondegenerate second differential on the intersection of their kernels.
• Dual Indices: Unlike classical Morse theory which has a single index, nondegenerate critical points in this framework possess two indices:
  • Piecewise Linear Index ($k$): Defined as $k = \#I(x) - 1$, where $I(x)$ is the set of indices of active functions. This relates to the number of closest points in $Y$ (geometric bottlenecks).
  • Quadratic Index ($\iota$): The number of negative eigenvalues of the Hessian of $\lambda_f$ restricted to the relevant subspace.

2.3. Genericity via Multijet Transversality

To ensure these definitions apply to the specific case of algebraic hypersurfaces, the authors employ a parametric multijet transversality theorem. They prove that for generic polynomials defining $X$ and $Y$:

• The distance function is a continuous selection near critical points.
• All critical points are nondegenerate.
• The number of closest points to any $x$ is bounded by $n+1$.

3. Key Contributions and Results

3.1. Unification of Bottlenecks and Nearest Point Problems

The framework unifies two major problems in computational algebraic geometry:

1. Bottlenecks: When $X = \mathbb{R}^n$, critical points correspond to geometric bottlenecks (points equidistant to multiple points on $Y$). The piecewise linear index $k$ corresponds to the order of the bottleneck.
2. Euclidean Distance Degree (EDD): When $Y$ is a single point, the critical points correspond to the standard EDD problem.
   The authors show that both are instances of critical points of $\text{dist}_Y|_X$, distinguished by their indices.

3.2. Topological Change and Morse Inequalities

The paper establishes a Deformation Lemma for distance functions: if an interval contains no critical values, the sublevel sets deformation retract.
For nondegenerate critical points, the change in homology when passing a critical value is controlled by the two indices $(k, \iota)$. Specifically, the relative homology $H_*(X_{c+\epsilon}, X_{c-\epsilon})$ is isomorphic to the reduced homology of a sphere $S^{k+\iota}$.
This leads to Strong Morse Inequalities relating the Betti numbers of $X$, $X \cap Y$, and the critical points:
$$\sum_{i=0}^\lambda (-1)^{i+\lambda} b_i(X) \leq \sum_{i=0}^\lambda (-1)^{i+\lambda} \left( b_i(X \cap Y) + \sum_{k+\iota=i} \#C_{k,\iota}(X, Y) \right)$$
where $C_{k,\iota}(X, Y)$ denotes the set of nondegenerate critical points with indices $k$ and $\iota$.

3.3. Duality Formula

A significant result is a duality formula for the Euler characteristics of $X$ and $Y$ (assuming compactness and nondegeneracy):
$$\chi(Y) + \sum_{k,\iota \geq 0} (-1)^{k+\iota} \#C_{k,\iota}(X, Y) = \chi(X) + \sum_{k,\iota \geq 0} (-1)^{k+\iota} \#C_{k,\iota}(Y, X)$$
This generalizes the classical relationship between the Euler characteristic and the alternating sum of critical points.

3.4. Bounds on Critical Points

The authors provide explicit bounds on the number of critical points for generic algebraic hypersurfaces $X = Z(p)$ and $Y = Z(q)$ with degrees $d_1$ and $d_2$.

• Finiteness: For $d_1, d_2 \geq 3$, the number of critical points is finite.
• Degree Bounds: The number of $k$-critical points (those with $k+1$ closest points) not in $Y$ is bounded by:
  $$\#C_{k,\ast}(\text{dist}_Y|_X) \leq c(k, n) \cdot d_1^n \cdot d_2^{n(k+1)}$$
  where $c(k, n)$ is a constant depending only on $n$ and $k$.
• This result generalizes previous bounds for Bottleneck Degrees (where $k=1$) and Euclidean Distance Degrees.

4. Significance and Claims

The paper claims to provide a general systematic framework that recovers and unifies various notions of critical points in distance geometry using the language of Lipschitz geometry.

• Theoretical Unification: It bridges the gap between classical differential geometry (smooth Morse theory) and computational algebraic geometry (EDD, bottlenecks) by treating distance functions as continuous selections.
• Topological Interpretation: It offers a topological interpretation of geometric phenomena (like bottlenecks) via the piecewise linear index, showing that topology changes depend on both the number of closest points ($k$) and the curvature properties ($\iota$).
• Technical Toolset: The paper provides a "technical toolset" (specifically the multijet parametric transversality theorem for polynomials) that guarantees the applicability of this theory to generic algebraic hypersurfaces. This allows for the derivation of degree bounds without relying on complex algebraic geometry techniques like polar classes, although the authors note their techniques extend to the complex setting.
• Modesty: The authors explicitly state that while they focus on smooth algebraic hypersurfaces, they expect results to extend to complete intersections, leaving this for future work. They also note that the definition of "Real Euclidean Distance Degree" relative to a set $Y$ is introduced, but the study of its complex counterpart is left for future investigation.

In summary, the paper establishes that for generic algebraic hypersurfaces, the distance function behaves sufficiently like a Morse function (in the sense of continuous selections) to allow for precise topological analysis and quantitative bounds on the number of critical points.