arxiv: 2605.00278 · v1 · submitted 2026-04-30 · 🧮 math.AC · cs.CV

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Elimination Templates in Macaulay2

Manav Batavia , Cheng Chen , Anna Natalie Chlopecki , Timothy Duff , William Huang , Aolong Li , Wanchun Shen

Authors on Pith no claims yet

Pith reviewed 2026-05-09 19:17 UTC · model grok-4.3

classification 🧮 math.AC cs.CV

keywords EliminationTemplatesMacaulay2elimination theoryzero-dimensional idealsradical idealsparametric polynomial systemscomputer visionautomatic solvers

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The pith

The EliminationTemplates package in Macaulay2 constructs automatic solvers for families of zero-dimensional radical ideals depending on algebraically independent parameters.

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

The paper introduces a software package called EliminationTemplates inside the Macaulay2 computer algebra system. The package supplies tools that let users build solvers capable of handling entire families of polynomial systems whose solution sets are finite and simple, with some variables treated as free parameters. It gives a self-contained account of how elimination templates are built for these families and how the templates behave when the parameters receive concrete numerical values. The authors also document the package's main datatypes and functions, and they show the tools working on concrete cases drawn from computer vision. Readers would care because the automation removes the need to derive each solver by hand, which is a recurring bottleneck when algebraic methods are applied to engineering problems.

Core claim

We introduce the package EliminationTemplates for the Macaulay2 computer algebra system, which provides tools for constructing automatic solvers for families of zero-dimensional radical ideals depending on algebraically independent parameters. This article provides a self-contained description of how elimination templates are constructed for such families and their specialization properties. Additionally, we describe the main functionality and datatypes provided by our package, and illustrate its usage on several examples, including applications from computer vision from which elimination templates originated.

What carries the argument

Elimination templates, precomputed algebraic objects that encode variable eliminations for a whole parametric family and support direct substitution of parameter values to obtain concrete solvers.

If this is right

Users can generate solvers for new parametric families without deriving the elimination relations manually.
The templates retain the radical and zero-dimensional character under generic specialization of the parameters.
Computer-vision problems that rely on solving polynomial systems can incorporate the package to produce solvers more quickly.
The supplied datatypes allow systematic storage, manipulation, and reuse of the constructed templates across different families.

Where Pith is reading between the lines

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

The same template-construction workflow could be adapted for ideals that are not radical if multiplicity information is tracked separately.
Similar packages could be written for other computer-algebra systems, lowering the effort needed to solve parametric systems in those environments.
The approach suggests a general pattern for turning manual elimination steps into reusable code that researchers in robotics or algebraic statistics could exploit.

Load-bearing premise

The ideals in each family must be zero-dimensional and radical while the parameters remain algebraically independent, so that specialization preserves the expected solution structure.

What would settle it

Take a concrete family that is not radical or not zero-dimensional, generate its template with the package, specialize to a numerical parameter value, and compare the output solutions against the true solutions of the specialized system; systematic mismatch would refute the claimed specialization properties.

read the original abstract

We introduce the package \texttt{EliminationTemplates} for the Macaulay2 computer algebra system, which provides tools for constructing automatic solvers for families of zero-dimensional radical ideals depending on algebraically independent parameters. This article provides a self-contained description of how elimination templates are constructed for such families and their specialization properties. Additionally, we describe the main functionality and datatypes provided by our package, and illustrate its usage on several examples, including applications from computer vision from which elimination templates originated.

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.

Desk Editor's Note private letter to a colleague

This paper ships a Macaulay2 package that turns elimination template methods into documented, reusable code for building solvers on zero-dimensional radical ideals with independent parameters.

read the letter

This paper ships a Macaulay2 package that turns elimination template methods into documented, reusable code for building solvers on zero-dimensional radical ideals with independent parameters. The main new element is the software itself plus the explicit account of how the templates are constructed and how they specialize when parameters are substituted. Earlier template work existed mostly in computer vision papers; this version packages the steps into datatypes and functions that other Macaulay2 users can call directly. The authors also supply a self-contained description of the construction process and the specialization rules that follow from the zero-dimensional and radical assumptions. That part is useful because it lets readers see the algebraic conditions without hunting through scattered references. The examples, especially the computer vision ones, show concrete usage and make the motivation clear. A reader who already works in Macaulay2 and needs to set up parametric solvers will save time on the manual elimination steps. The central limitation is the set of hypotheses required. Everything rests on the ideals being zero-dimensional and radical with algebraically independent parameters; if those fail, the specialization guarantees do not apply. The paper states the conditions up front, which is honest, but users still have to check them for each new problem. Implementation details such as numerical stability or handling of non-generic cases are not visible from the abstract, so a referee would want to inspect the actual package code and run the examples. This work is aimed at algebraic geometers and computer vision researchers who use Macaulay2 for polynomial solving. It is not a theoretical result but a practical tool that organizes existing ideas into accessible software. It deserves a serious referee because the package is new content and the documentation appears careful enough to warrant community testing and feedback. I would send it for peer review.

Referee Report

0 major / 3 minor

Summary. The manuscript introduces the EliminationTemplates package for Macaulay2, which supplies tools for constructing automatic solvers for families of zero-dimensional radical ideals depending on algebraically independent parameters. It provides a self-contained account of elimination-template construction and the associated specialization properties, describes the package's main functionality and datatypes, and demonstrates usage through examples drawn from computer vision.

Significance. If the described construction and specialization properties hold under the stated hypotheses (zero-dimensionality, radicalness, and algebraic independence of parameters), the work supplies a practical, reproducible implementation that can streamline the generation of solvers for parametric polynomial systems. The explicit provision of a Macaulay2 package together with a self-contained theoretical description constitutes a concrete strength for researchers who need to handle families of ideals in algebraic geometry and computer-vision applications.

minor comments (3)

Abstract: the phrase 'specialization properties' is used without a one-sentence indication of what those properties are; adding a brief qualifier would improve immediate readability for readers outside the immediate subfield.
Section describing the datatypes: the relationship between the new EliminationTemplate datatype and existing Macaulay2 ideal and ring objects should be stated explicitly (e.g., whether the template stores a Gröbner basis or a resultant matrix) to avoid ambiguity for users.
Computer-vision examples: the manuscript should include a short table or paragraph comparing the size of the templates produced by the package with those obtained by hand or by other solvers, so that the practical gain is quantified.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our manuscript, as well as for recognizing the significance of the EliminationTemplates package and its potential utility in algebraic geometry and computer-vision applications. The recommendation for minor revision is noted. However, the report contains no specific major comments or requests for clarification, so we have no points to address point-by-point.

Circularity Check

0 steps flagged

No significant circularity; self-contained package description

full rationale

The paper introduces the EliminationTemplates package and supplies a self-contained description of template construction for zero-dimensional radical ideals with algebraically independent parameters, together with specialization properties and usage examples. All claims are conditioned on the stated hypotheses of zero-dimensionality, radicalness, and algebraic independence; the account relies on standard algebraic geometry without any self-definitional loops, fitted inputs renamed as predictions, load-bearing self-citations, or imported uniqueness results. No derivation step reduces to its own inputs by construction, and the central contribution is an implementation tool rather than a closed mathematical claim.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard properties of zero-dimensional radical ideals and parameter specialization in algebraic geometry; no free parameters or invented entities are introduced in the abstract.

axioms (2)

standard math Zero-dimensional radical ideals over algebraically closed fields have finitely many solutions
Invoked implicitly when describing automatic solvers for such ideals
domain assumption Specialization preserves the elimination template structure for algebraically independent parameters
Central to the described specialization properties

pith-pipeline@v0.9.0 · 5382 in / 1200 out tokens · 27832 ms · 2026-05-09T19:17:38.083004+00:00 · methodology

discussion (0)

Reference graph

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