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Mathieu-Zhao spaces over field of positive characteristic

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Fengli Liu, Dan Yan

Commutative Algebra · Submitted 20 Oct 2020

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

The authors' main claim Finally, we classify some nilpotent derivations of $K[x_1]$ and give a sufficient and necessary condition for $D(I)$ to be a Mathieu-Zhao space of $K[x_1]$ for any ideal $I$ of $K[x_1]$.

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arXiv ID2010.10219 (version 2)
Pith IDAR3OEGKW, Pith's permanent record of this paper and its review
CitationDan Yan, Fengli Liu. "Mathieu-Zhao spaces over field of positive characteristic." arXiv:2010.10219 (2020). https://pith.science/paper/AR3OEGKW
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,835 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.10 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.10 · compares with prior results 0.06 · mentions code or data availability 0.01.

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

Let $K$ be a field of characteristic $p$, $\delta$ a nonzero $\mathcal{E}$-derivation and $D=f(x_1)\partial_1$. We first prove that $\operatorname{Im}D$ is not a Mathieu-Zhao space of $K[x_1]$ if and only if $f(x_1)=x_1^rf_1(x_1^p)$ and $r\neq 1$. Then we prove that $\operatorname{Im}\delta$ is a Mathieu-Zhao space of $K[x_1]$ if and only if $\delta$ is not locally nilpotent. Finally, we classify some nilpotent derivations of $K[x_1]$ and give a sufficient and necessary condition for $D(I)$ to be a Mathieu-Zhao space of $K[x_1]$ for any ideal $I$ of $K[x_1]$.

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 Let $K$ be a field of characteristic $p$, $\delta$ a nonzero $\mathcal{E}$-derivation and $D=f(x_1)\partial_1$. background 0.99 0.09 0.02 unclear 0.74
s1 We first prove that $\operatorname{Im}D$ is not a Mathieu-Zhao space of $K[x_1]$ if and only if $f(x_1)=x_1^rf_1(x_1^p)$ and $r\neq 1$. result 0.99 0.93 0.27 asserted 1.00
s2 Then we prove that $\operatorname{Im}\delta$ is a Mathieu-Zhao space of $K[x_1]$ if and only if $\delta$ is not locally nilpotent. result 0.99 0.94 0.28 asserted 1.00
s3 Finally, we classify some nilpotent derivations of $K[x_1]$ and give a sufficient and necessary condition for $D(I)$ to be a Mathieu-Zhao space of $K[x_1]$ for any ideal $I$ of $K[x_1]$. contribution 0.57 0.94 0.27 asserted 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

Let $K$ be a field of characteristic $p$, $\delta$ a nonzero $\mathcal{E}$-derivation and $D=f(x_1)\partial_1$. We first prove that $\operatorname{Im}D$ is not a Mathieu-Zhao space of $K[x_1]$ if and only if $f(x_1)=x_1^rf_1(x_1^p)$ and $r\neq 1$. Then we prove that $\operatorname{Im}\delta$ is a Mathieu-Zhao space of $K[x_1]$ if and only if $\delta$ is not locally nilpotent. Finally, we classify some nilpotent derivations of $K[x_1]$ and give a sufficient and necessary condition for $D(I)$ to be a Mathieu-Zhao space of $K[x_1]$ for any ideal $I$ of $K[x_1]$.

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