A proof of van der Waerden's Conjecture on random Galois groups of polynomials
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912078201094144 |
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| author | Bhargava, Manjul |
| author_facet | Bhargava, Manjul |
| contents | Of the $(2H+1)^n$ monic integer polynomials $f(x)=x^n+a_1 x^{n-1}+\cdots+a_n$ with $\max\{|a_1|,\ldots,|a_n|\}\leq H$, how many have associated Galois group that is not the full symmetric group $S_n$? There are clearly $\gg H^{n-1}$ such polynomials, as may be obtained by setting $a_n=0$. In 1936, van der Waerden conjectured that $O(H^{n-1})$ should in fact also be the correct upper bound for the count of such polynomials. The conjecture has been known previously for degrees $n\leq 4$, due to work of van der Waerden and Chow and Dietmann.
In this expository article, we outline a proof of van der Waerden's Conjecture for all degrees $n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_03792 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A proof of van der Waerden's Conjecture on random Galois groups of polynomials Bhargava, Manjul Number Theory 11R09, 11R32, 11R45, 11C08, 11N35 Of the $(2H+1)^n$ monic integer polynomials $f(x)=x^n+a_1 x^{n-1}+\cdots+a_n$ with $\max\{|a_1|,\ldots,|a_n|\}\leq H$, how many have associated Galois group that is not the full symmetric group $S_n$? There are clearly $\gg H^{n-1}$ such polynomials, as may be obtained by setting $a_n=0$. In 1936, van der Waerden conjectured that $O(H^{n-1})$ should in fact also be the correct upper bound for the count of such polynomials. The conjecture has been known previously for degrees $n\leq 4$, due to work of van der Waerden and Chow and Dietmann. In this expository article, we outline a proof of van der Waerden's Conjecture for all degrees $n$. |
| title | A proof of van der Waerden's Conjecture on random Galois groups of polynomials |
| topic | Number Theory 11R09, 11R32, 11R45, 11C08, 11N35 |
| url | https://arxiv.org/abs/2410.03792 |