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On Polynomial Ideals And Overconvergence In Tate Algebras

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Xavier Caruso (LFANT, IMB), Tristan Vaccon (XLIM), Thibaut Verron (JKU)

Symbolic Computation · Submitted 15 Feb 2022

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

The authors' main claim Finally, we prove the existence of a universal analytic Gr{\"o}bner basis for polynomial ideals in Tate algebras, compatible with all convergence radii.

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arXiv ID2202.07509 (version 1)
Pith IDBB6DMJN5, Pith's permanent record of this paper and its review
CitationIMB), Thibaut Verron (JKU), Tristan Vaccon (XLIM), Xavier Caruso (LFANT. "On Polynomial Ideals And Overconvergence In Tate Algebras." arXiv:2202.07509 (2022). https://pith.science/paper/BB6DMJN5
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,998 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.24 · human participants 0.01 · controlled comparison 0.05 · randomized assignment 0.01 · states quantitative result 0.86 · compares with prior results 0.84 · mentions code or data availability 0.03.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

In this paper, we study ideals spanned by polynomials or overconvergent series in a Tate algebra. With state-of-the-art algorithms for computing Tate Gr{\"o}bner bases, even if the input is polynomials, the size of the output grows with the required precision, both in terms of the size of the coefficients and the size of the support of the series. We prove that ideals which are spanned by polynomials admit a Tate Gr{\"o}bner basis made of polynomials, and we propose an algorithm, leveraging Mora's weak normal form algorithm, for computing it. As a result, the size of the output of this algorithm grows linearly with the precision. Following the same ideas, we propose an algorithm which computes an overconvergent basis for an ideal spanned by overconvergent series. Finally, we prove the existence of a universal analytic Gr{\"o}bner basis for polynomial ideals in Tate algebras, compatible with all convergence radii.

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 In this paper, we study ideals spanned by polynomials or overconvergent series in a Tate algebra. contribution 0.60 0.75 0.04 asserted 0.95
s1 With state-of-the-art algorithms for computing Tate Gr{\"o}bner bases, even if the input is polynomials, the size of the output grows with the required precision, both in terms of the size of the coefficients and the size of the support of the series. background 0.88 0.17 0.04 asserted 0.98
s2 We prove that ideals which are spanned by polynomials admit a Tate Gr{\"o}bner basis made of polynomials, and we propose an algorithm, leveraging Mora's weak normal form algorithm, for computing it. contribution 0.93 0.96 0.44 asserted 1.00
s3 As a result, the size of the output of this algorithm grows linearly with the precision. result 0.95 0.90 0.23 asserted 1.00
s4 Following the same ideas, we propose an algorithm which computes an overconvergent basis for an ideal spanned by overconvergent series. contribution 0.78 0.94 0.07 asserted 0.95
s5 Finally, we prove the existence of a universal analytic Gr{\"o}bner basis for polynomial ideals in Tate algebras, compatible with all convergence radii. result 0.89 0.91 0.35 asserted 1.00

What kind of paper the classifier reports this is: method or tool 0.55 · evidence from proof or derivation 1.00.

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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

In this paper, we study ideals spanned by polynomials or overconvergent series in a Tate algebra. With state-of-the-art algorithms for computing Tate Gr{\"o}bner bases, even if the input is polynomials, the size of the output grows with the required precision, both in terms of the size of the coefficients and the size of the support of the series. We prove that ideals which are spanned by polynomials admit a Tate Gr{\"o}bner basis made of polynomials, and we propose an algorithm, leveraging Mora's weak normal form algorithm, for computing it. As a result, the size of the output of this algorithm grows linearly with the precision. Following the same ideas, we propose an algorithm which computes an overconvergent basis for an ideal spanned by overconvergent series. Finally, we prove the existence of a universal analytic Gr{\"o}bner basis for polynomial ideals in Tate algebras, compatible with all convergence radii.

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