热带方案理论中的射影超曲面I: Macaulay理想。

IF 1.2 3区数学 Q1 MATHEMATICS

Research in the Mathematical Sciences Pub Date : 2025-01-01 Epub Date: 2025-04-25 DOI:10.1007/s40687-025-00517-7

Alex Fink, Jeffrey Giansiracusa, Noah Giansiracusa, Joshua Mundinger

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引用次数: 0

摘要

“热带理想”是热带多项式的幂等半环中的理想，它也是一个热带线性空间。我们介绍了一个基于横拟阵的构造，它将任何主理想扩展到热带理想。我们称之为麦考利热带理想。它有一个普适性：将给定的主理想扩展到带期望希尔伯特函数的热带理想是麦考利热带理想的弱像。对于每个n≥2和d≥1，我们的构造在P n中得到一个不可实现的d次超曲面格式。Maclagan-Rincón为每个n生成了pn中不可实现的行，对于(d, n) =(1,2)，这两个结构一致。Mundinger的附录将Macaulay结构与另一种将理想扩展到热带理想的方法进行了比较。

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Projective hypersurfaces in tropical scheme theory I: the Macaulay ideal.

A "tropical ideal" is an ideal in the idempotent semiring of tropical polynomials that is also, degree by degree, a tropical linear space. We introduce a construction based on transversal matroids that canonically extends any principal ideal to a tropical ideal. We call this the Macaulay tropical ideal. It has a universal property: any other extension of the given principal ideal to a tropical ideal with the expected Hilbert function is a weak image of the Macaulay tropical ideal. For each $n \geq 2$ and $d \geq 1$ , our construction yields a non-realizable degree d hypersurface scheme in $P^{n}$ . Maclagan-Rincón produced a non-realizable line in $P^{n}$ for each n, and for $(d, n) = (1, 2)$ the two constructions agree. An appendix by Mundinger compares the Macaulay construction with another method for canonically extending ideals to tropical ideals.

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来源期刊

Research in the Mathematical Sciences Mathematics-Computational Mathematics

CiteScore

2.00

自引率

8.30%

发文量

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