Variations on the Arkhipov-Karačuba Type Counterexamples to Artin's Conjecture
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arXiv
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| Format: | Preprint |
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2018
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| _version_ | 1866916440938905600 |
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| author | Han, Zhaobo Tom |
| author_facet | Han, Zhaobo Tom |
| contents | It was conjectured by Emil Artin in the 1930's that every $d$-form $F(x_1, x_2, $\ldots$, x_n)$ over the $p$-adic field in more than $d^2$ variables has a solution that is not $(0, 0, \cdots, 0)$ (non-trivial solution) over the $p$-adic field. This is true for $d=2$ and $d=3$. However, many counterexamples for $d \geq 4$ were later discovered. The major types of counterexamples are Terjanian Type and Arkhipov-Karačuba Type. The degrees of all known counterexamples, however, are divisible by $p-1$, which means that they are even for all odd primes. In this article we apply modifications to the known Arkhipov-Karačuba Type counterexamples to construct counterexamples with odd degrees that are divisible by $\frac{p-1}2$ for all primes greater than $3$ and congruent to $3$ modulo $4$ and then propose some ideas about increasing the number of variables in the counterexamples. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1809_01175 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Variations on the Arkhipov-Karačuba Type Counterexamples to Artin's Conjecture Han, Zhaobo Tom Number Theory It was conjectured by Emil Artin in the 1930's that every $d$-form $F(x_1, x_2, $\ldots$, x_n)$ over the $p$-adic field in more than $d^2$ variables has a solution that is not $(0, 0, \cdots, 0)$ (non-trivial solution) over the $p$-adic field. This is true for $d=2$ and $d=3$. However, many counterexamples for $d \geq 4$ were later discovered. The major types of counterexamples are Terjanian Type and Arkhipov-Karačuba Type. The degrees of all known counterexamples, however, are divisible by $p-1$, which means that they are even for all odd primes. In this article we apply modifications to the known Arkhipov-Karačuba Type counterexamples to construct counterexamples with odd degrees that are divisible by $\frac{p-1}2$ for all primes greater than $3$ and congruent to $3$ modulo $4$ and then propose some ideas about increasing the number of variables in the counterexamples. |
| title | Variations on the Arkhipov-Karačuba Type Counterexamples to Artin's Conjecture |
| topic | Number Theory |
| url | https://arxiv.org/abs/1809.01175 |