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arxiv: 2402.08639 · v4 · submitted 2024-02-13 · 🧮 math.AG · math.AT· math.MG

Morse theory of Euclidean distance functions from algebraic hypersurfaces

Andrea Guidolin , Antonio Lerario , Isaac Ren , Martina Scolamiero This is my paper

Pith reviewed 2026-05-24 03:12 UTC · model grok-4.3

classification 🧮 math.AG math.ATmath.MG
keywords Morse theoryEuclidean distancealgebraic hypersurfacesbottlenecksnearest point problemsLipschitz functionscontinuous selectionsdefinable sets
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The pith

A Morse theory for the Euclidean distance from one hypersurface restricted to another equips nondegenerate critical points with both a quadratic index and a piecewise linear index.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs a Morse theory for the restriction to a smooth manifold X of the Euclidean distance function to a closed definable set Y. It proceeds by defining critical points via the theory of Lipschitz functions and continuous selections of nearest-point maps. Nondegenerate critical points receive two indices: one quadratic, as in classical Morse theory, and one piecewise linear that tracks bottleneck structure. The resulting framework simultaneously handles bottlenecks and nearest-point problems on generic algebraic hypersurfaces, supplying bounds on the number of such critical points.

Core claim

We construct a version of Morse theory for the restriction to X of the Euclidean distance function from Y using the notion of critical points of Lipschitz functions and applying the theory of continuous selections. In this theory, nondegenerate critical points have two indices: a quadratic index (as in classical Morse theory), and a piecewise linear index (that relates to the notion of bottlenecks). This framework is flexible enough to simultaneously treat and unify the study of two cases of interest for computational algebraic geometry: bottlenecks and nearest point problems.

What carries the argument

Critical points of the restricted distance function equipped with a quadratic index and a piecewise linear index, obtained via continuous selections of the nearest-point map from Y.

If this is right

  • The number of critical points of the distance from Y restricted to X admits an upper bound when X and Y are generic algebraic hypersurfaces.
  • Bottlenecks between two hypersurfaces are captured by the piecewise linear index within the same Morse-theoretic setting.
  • Nearest-point problems on algebraic hypersurfaces become instances of the same indexed critical-point count.
  • The theory supplies a single technical toolset that applies to both problems without separate case-by-case arguments.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The dual-index structure could be used to design algorithms that enumerate bottlenecks by tracking only the piecewise-linear contribution.
  • Removing the genericity assumption would require a stratified version of the theory to handle degenerate loci.
  • The same construction may extend to distance functions between semi-algebraic sets of higher codimension.

Load-bearing premise

The construction requires continuous selections for the nearest-point map together with genericity of the hypersurfaces X and Y so that all critical points remain nondegenerate.

What would settle it

An explicit pair of generic algebraic hypersurfaces X and Y for which the nearest-point map admits no continuous selection or for which a critical point of the restricted distance function is degenerate.

Figures

Figures reproduced from arXiv: 2402.08639 by Andrea Guidolin, Antonio Lerario, Isaac Ren, Martina Scolamiero.

Figure 1
Figure 1. Figure 1: Example of the subdifferential of f = distY |X at a critical point x. The subdifferential ∂xf is the interval between the projections of the vectors x−y0 kx−y0k and x−y1 kx−y1k onto TxX. our functions of interest, if Y ⊆ R n is a closed set and x ∈ R n \ Y , then the subdifferential of distY at x can be easily described, see Proposition 2.1.2: (1) ∂x(distY ) = co  x − y kx − yk [PITH_FULL_IMAGE:figures/f… view at source ↗
Figure 2
Figure 2. Figure 2: Example in R 3 of the characterization of critical points using normal spaces. X is a sphere and Y = {y0, y1} consists of two points. The plane represents the medial axis MY ; the small arrows represent normal vectors from various points of X; Nx1X (the horizontal dashed line) is the normal space of X at x1. 2.3. The deformation lemma. Given f : X → R and t ∈ R, we denote by Xt := {x ∈ X | f(x) ≤ t} and in… view at source ↗
Figure 3
Figure 3. Figure 3: Examples of critical points of f = distY |X and their degeneracy. (a) and (b): Y is a finite set of coplanar points in R 3 and X is a line that intersects the plane of the paper at a single point x. (c): X and Y are lines. X y1 x y2 (a) index (1, 0) X x y1 (b) index (0, 0) X Y x y1 (c) index (0, 1) [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Different examples of distY |X with different piecewise linear and quadratic indices, where Y is a finite set of points or a circle and X is a line. Indices are written as (k(x), ι(x)). Suppose that X is given as the zero set of a regular equation p = 0, with p : N → R a smooth function (in Section 4 we are interested in the case N = R n and p is a polynomial). Then, for all j = 0, . . . , m, Dxfj = Dxgj |… view at source ↗
read the original abstract

Let $Y\subseteq \mathbb{R}^n$ be a closed definable subset and $X\subseteq \mathbb{R}^n$ be a smooth manifold. We construct a version of Morse theory for the restriction to $X$ of the Euclidean distance function from $Y$. This is done using the notion of critical points of Lipschitz functions and applying the theory of continuous selections. In this theory, nondegenerate critical points have two indices: a quadratic index (as in classical Morse theory), and a piecewise linear index (that relates to the notion of bottlenecks). This framework is flexible enough to simultaneously treat and unify the study of two cases of interest for computational algebraic geometry: bottlenecks and nearest point problems. We provide a technical toolset guaranteeing the applicability of the theory to the case where $X, Y$ are generic algebraic hypersurfaces and use it to bound the number of critical points of the distance from $Y$ restricted to $X$, among other applications.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper constructs a Morse theory for the restriction to a smooth manifold X of the Euclidean distance function from a closed definable subset Y in R^n. Using critical points of Lipschitz functions and the theory of continuous selections, nondegenerate critical points are assigned a quadratic index and a piecewise linear index. The framework unifies the study of bottlenecks and nearest-point problems, and supplies a technical toolset to apply the results to generic algebraic hypersurfaces X and Y, including bounds on the number of critical points.

Significance. If the genericity conditions and continuous-selection hypotheses are rigorously established, the two-index construction provides a coherent unification of classical Morse theory with distance geometry that is directly applicable to computational algebraic geometry problems. The explicit toolset for algebraic hypersurfaces is a concrete strength that could enable new bounds and algorithms.

minor comments (2)
  1. The abstract states that a technical toolset guarantees applicability to generic algebraic hypersurfaces, but the main text should include an explicit theorem listing the precise genericity hypotheses (e.g., on transversality or nondegeneracy) so that readers can verify the conditions without reconstructing them from the proofs.
  2. Notation for the two indices (quadratic and piecewise linear) should be introduced with a short comparison table or diagram early in the paper to clarify their relation to classical Morse index and to the notion of bottlenecks.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the recognition of its unification of Morse theory with distance geometry and the concrete toolset for algebraic hypersurfaces. The recommendation for minor revision is noted, and we will incorporate any necessary clarifications or editorial adjustments in the revised version.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from external theories

full rationale

The paper derives its Morse theory variant by applying the established theory of critical points for Lipschitz functions together with continuous selections to the Euclidean distance function restricted to X. The two-index definition (quadratic plus piecewise-linear) and the unification of bottlenecks with nearest-point problems are presented as formal consequences once the continuous-selection and nondegeneracy hypotheses are granted for generic algebraic hypersurfaces. No equations, fitted parameters, or self-citations appear in the abstract that would reduce any claimed result to its own inputs by construction. The framework is therefore independent of the target applications and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The construction rests on standard facts from Lipschitz analysis and o-minimal geometry that are not derived inside the paper; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Existence of continuous selections for the nearest-point set-valued map
    Invoked to define critical points of the distance function via continuous selections.
  • domain assumption Generic algebraic hypersurfaces yield nondegenerate critical points
    Used to guarantee that the two-index theory applies and to obtain finite bounds.

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