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A Direttissimo Algorithm for Equidimensional Decomposition

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Paper breakdownThe authors' claims, conditions and measurements, as JSON Signed receiptProof this review has not been altered
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Christian Eder, Pierre Lairez, Rafael Mohr, Mohab Safey El Din

Symbolic Computation · Submitted 16 Feb 2023

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Practical experiments demonstrate its efficiency compared to state-of-the-art implementations.

how it is stated · asserted 1.00 read the record →

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arXiv ID2302.08174 (version 2)
Pith IDFUCYLV3E, Pith's permanent record of this paper and its review
CitationChristian Eder, Mohab Safey El Din, Pierre Lairez, Rafael Mohr. "A Direttissimo Algorithm for Equidimensional Decomposition." arXiv:2302.08174 (2023). https://pith.science/paper/FUCYLV3E
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The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,318 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.13 · human participants 0.01 · controlled comparison 0.34 · randomized assignment 0.01 · states quantitative result 0.07 · compares with prior results 0.89 · mentions code or data availability 0.03.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

We describe a recursive algorithm that decomposes an algebraic set into locally closed equidimensional sets, i.e. sets which each have irreducible components of the same dimension. At the core of this algorithm, we combine ideas from the theory of triangular sets, a.k.a. regular chains, with Gr\"obner bases to encode and work with locally closed algebraic sets. Equipped with this, our algorithm avoids projections of the algebraic sets that are decomposed and certain genericity assumptions frequently made when decomposing polynomial systems, such as assumptions about Noether position. This makes it produce fine decompositions on more structured systems where ensuring genericity assumptions often destroys the structure of the system at hand. Practical experiments demonstrate its efficiency compared to state-of-the-art implementations.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We describe a recursive algorithm that decomposes an algebraic set into locally closed equidimensional sets, i.e. sets which each have irreducible components of the same dimension. contribution 0.97 0.92 0.37 asserted 1.00
s1 At the core of this algorithm, we combine ideas from the theory of triangular sets, a.k.a. regular chains, with Gr\"obner bases to encode and work with locally closed algebraic sets. method 0.97 0.84 0.07 asserted 1.00
s2 Equipped with this, our algorithm avoids projections of the algebraic sets that are decomposed and certain genericity assumptions frequently made when decomposing polynomial systems, such as assumptions about Noether position. contribution 0.51 0.83 0.18 asserted 1.00
s3 This makes it produce fine decompositions on more structured systems where ensuring genericity assumptions often destroys the structure of the system at hand. interpretation 0.71 0.73 0.24 asserted 0.97
s4 Practical experiments demonstrate its efficiency compared to state-of-the-art implementations. result 0.99 0.94 0.21 asserted 1.00

What kind of paper the classifier reports this is: method or tool 1.00 · evidence from simulation or computation 0.48.

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  • T body sentences
  • C citations
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

We describe a recursive algorithm that decomposes an algebraic set into locally closed equidimensional sets, i.e. sets which each have irreducible components of the same dimension. At the core of this algorithm, we combine ideas from the theory of triangular sets, a.k.a. regular chains, with Gr\"obner bases to encode and work with locally closed algebraic sets. Equipped with this, our algorithm avoids projections of the algebraic sets that are decomposed and certain genericity assumptions frequently made when decomposing polynomial systems, such as assumptions about Noether position. This makes it produce fine decompositions on more structured systems where ensuring genericity assumptions often destroys the structure of the system at hand. Practical experiments demonstrate its efficiency compared to state-of-the-art implementations.

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