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arxiv: 2606.06849 · v1 · pith:JZK5KONTnew · submitted 2026-06-05 · 🧮 math.AG

Parabolic second-order tangent sets of semialgebraic sets and applications to polynomial optimization

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keywords semialgebraic setsparabolic tangent setssecond-order conditionspolynomial optimizationquadratic growthalgebraic linearizationtangent cones
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The pith

For semialgebraic sets under rank stability and arc-realizability, parabolic second-order tangent sets coincide with their algebraic models from gradients and Hessians.

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

The paper aims to show that the true parabolic second-order tangent sets of basic closed semialgebraic sets match an algebraic second-order linearized set under directional rank stability and semialgebraic parabolic arc-realizability. This coincidence holds for the outer, inner, and arc-generated tangent sets. It provides exact formulas for several standard classes of semialgebraic sets and leads to algebraically checkable second-order conditions for quadratic growth in polynomial optimization. The work also illustrates how the theory identifies curvature, flatness, and branch dependence in optimization examples.

Core claim

Under directional rank stability and semialgebraic parabolic arc-realizability, the outer, inner, and arc-generated tangent sets coincide with the algebraic second-order linearized set determined by gradients and Hessians of the active constraints. Exact formulas are obtained for smooth hypersurfaces, regular complete intersections, smooth inequality systems, and stratified semialgebraic sets. These formulas yield algebraically checkable second-order necessary conditions and sufficient conditions for quadratic growth in polynomial optimization.

What carries the argument

The algebraic second-order linearized set, constructed from gradients and Hessians of active constraints, which equals the true parabolic tangent sets when the stability and realizability conditions are met.

Load-bearing premise

Directional rank stability and semialgebraic parabolic arc-realizability hold for the basic closed semialgebraic set under consideration.

What would settle it

A concrete basic closed semialgebraic set that satisfies directional rank stability and arc-realizability but where any of the parabolic tangent sets differs from the algebraic model would disprove the coincidence.

read the original abstract

We study parabolic second-order tangent sets of semialgebraic sets and their use in local polynomial optimization. For a basic closed semialgebraic feasible set, we compare the true parabolic tangent set with the algebraic second-order linearized set determined by gradients and Hessians of the active constraints. Under directional rank stability and semialgebraic parabolic arc-realizability, the outer, inner, and arc-generated tangent sets coincide with this algebraic model. Exact formulas are obtained for smooth hypersurfaces, regular complete intersections, smooth inequality systems, and stratified semialgebraic sets. These formulas yield algebraically checkable second-order necessary conditions and sufficient conditions for quadratic growth in polynomial optimization. Examples show how the theory detects curvature, flatness, branch dependence, and the failure of ordinary quadratic scaling.

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 / 3 minor

Summary. The paper studies parabolic second-order tangent sets of semialgebraic sets and their applications to local polynomial optimization. For a basic closed semialgebraic feasible set, it compares the true parabolic tangent set with the algebraic second-order linearized set determined by gradients and Hessians of the active constraints. Under the assumptions of directional rank stability and semialgebraic parabolic arc-realizability, the outer, inner, and arc-generated tangent sets are shown to coincide with this algebraic model. Exact formulas are derived for smooth hypersurfaces, regular complete intersections, smooth inequality systems, and stratified semialgebraic sets. These formulas are used to obtain algebraically checkable second-order necessary and sufficient conditions for quadratic growth, with examples illustrating detection of curvature, flatness, branch dependence, and failure of ordinary quadratic scaling.

Significance. If the directional rank stability and semialgebraic parabolic arc-realizability conditions hold as stated, the results provide a concrete algebraic bridge between geometric tangent-set constructions and computable second-order conditions in polynomial optimization over semialgebraic sets. The explicit formulas for standard classes (hypersurfaces, complete intersections, inequality systems) and the illustrative examples that detect curvature and flatness effects constitute a useful contribution to the literature on higher-order variational analysis.

minor comments (3)
  1. [Introduction] The abstract states that the main coincidence holds 'under directional rank stability and semialgebraic parabolic arc-realizability,' yet the introduction does not contain an early, self-contained paragraph listing the precise definitions or references for these two standing assumptions; adding such a paragraph would improve readability.
  2. [§2] Notation for the outer, inner, and arc-generated parabolic tangent sets is introduced without an explicit comparison table to the classical first-order tangent cone; a small table in §2 would clarify the distinction between first- and second-order objects.
  3. [Examples] The examples section is referenced in the abstract but the manuscript does not indicate how many examples are provided or whether they are numbered; consistent numbering and a brief summary table of what each example demonstrates would aid navigation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the clear summary of its contributions, and the recommendation for minor revision. No specific major comments appear in the report, so we have no point-by-point rebuttals to provide. We will prepare a revised version incorporating any minor editorial or typographical adjustments that may be identified during the revision process.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

This is a theoretical paper in semialgebraic geometry deriving comparisons between parabolic tangent sets and algebraic models under explicit assumptions (directional rank stability and semialgebraic parabolic arc-realizability). The exact formulas for hypersurfaces, complete intersections, and stratified sets are obtained conditionally from gradients/Hessians without any self-definitional loops, fitted parameters renamed as predictions, or load-bearing self-citations. All steps are algebraically checkable and rest on standard external concepts in the field, rendering the derivation chain self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only; no explicit free parameters, axioms, or invented entities are identifiable beyond standard background in real algebraic geometry and optimization theory.

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Reference graph

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