Density of solutions for systems of forms
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918132407336960 |
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| author | Lampert, Amichai |
| author_facet | Lampert, Amichai |
| contents | Let $K$ be a field of characteristic zero over which every diagonal form in sufficiently many variables admits a nontrivial solution. For example, $K$ may be a totally imaginary number field or a finite extension of a $p$-adic field. Suppose $f_1,\ldots,f_s$ are forms of degree $d$ over $K.$ Bik, Draisma and Snowden recently proved that there exists a constant $B = B(d,s,K)$ such that the rational solutions to the system of equations $f_1=\ldots=f_s = 0$ are Zariski dense, as long as the Birch rank of $f_1,\ldots,f_s$ is greater than $B.$ We establish an effective bound for this constant, improving vastly on the astronomical bound coming from their proof. Our result has applications for surjectivity of polynomial maps and for the Hardy-Littlewood circle method. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_11514 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Density of solutions for systems of forms Lampert, Amichai Number Theory Algebraic Geometry 11D72, 11G25, 11G35, 11P55, 14G05 Let $K$ be a field of characteristic zero over which every diagonal form in sufficiently many variables admits a nontrivial solution. For example, $K$ may be a totally imaginary number field or a finite extension of a $p$-adic field. Suppose $f_1,\ldots,f_s$ are forms of degree $d$ over $K.$ Bik, Draisma and Snowden recently proved that there exists a constant $B = B(d,s,K)$ such that the rational solutions to the system of equations $f_1=\ldots=f_s = 0$ are Zariski dense, as long as the Birch rank of $f_1,\ldots,f_s$ is greater than $B.$ We establish an effective bound for this constant, improving vastly on the astronomical bound coming from their proof. Our result has applications for surjectivity of polynomial maps and for the Hardy-Littlewood circle method. |
| title | Density of solutions for systems of forms |
| topic | Number Theory Algebraic Geometry 11D72, 11G25, 11G35, 11P55, 14G05 |
| url | https://arxiv.org/abs/2507.11514 |