Strength is bounded linearly by Birch rank
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866918133766291456 |
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| author | Baily, Benjamin Lampert, Amichai |
| author_facet | Baily, Benjamin Lampert, Amichai |
| contents | Let $f$ be a homogeneous polynomial over a field. For many fields, including number fields and function fields, we prove that the strength of $f$ is bounded above by a constant multiple of the Birch rank of $f.$ The constant depends only on the degree of $f$ and the absolute transcendence degree of the field. This is the first linear bound obtained for forms of degree greater than three, partially resolving a conjecture of Adiprasito, Kazhdan and Ziegler.
Our result has applications for the Hardy-Littlewood circle method. The circle method yields an asymptotic formula for counting integral zeros of (collections of) homogeneous polynomials, provided the Birch rank is sufficiently large -- a natural geometric condition. Our main theorem implies that these formulas hold even if we only assume a similar lower bound on the strength of the (collection of) homogeneous polynomials -- an arithmetic condition which is a priori weaker. This answers questions of Cook-Magyar and Skinner, and also yields a new proof of a seminal result of Schmidt as a consequence of Birch's earlier work.
Over finite fields we obtain a quasi-linear bound for partition rank of tensors in terms of analytic rank, improving Moshkovitz-Zhu's state of the art bound. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_00248 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Strength is bounded linearly by Birch rank Baily, Benjamin Lampert, Amichai Number Theory Algebraic Geometry Let $f$ be a homogeneous polynomial over a field. For many fields, including number fields and function fields, we prove that the strength of $f$ is bounded above by a constant multiple of the Birch rank of $f.$ The constant depends only on the degree of $f$ and the absolute transcendence degree of the field. This is the first linear bound obtained for forms of degree greater than three, partially resolving a conjecture of Adiprasito, Kazhdan and Ziegler. Our result has applications for the Hardy-Littlewood circle method. The circle method yields an asymptotic formula for counting integral zeros of (collections of) homogeneous polynomials, provided the Birch rank is sufficiently large -- a natural geometric condition. Our main theorem implies that these formulas hold even if we only assume a similar lower bound on the strength of the (collection of) homogeneous polynomials -- an arithmetic condition which is a priori weaker. This answers questions of Cook-Magyar and Skinner, and also yields a new proof of a seminal result of Schmidt as a consequence of Birch's earlier work. Over finite fields we obtain a quasi-linear bound for partition rank of tensors in terms of analytic rank, improving Moshkovitz-Zhu's state of the art bound. |
| title | Strength is bounded linearly by Birch rank |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2410.00248 |