The complexity of root-finding in orders
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2021
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866916815811117056 |
|---|---|
| author | Spelier, Pim |
| author_facet | Spelier, Pim |
| contents | Given an order, a commutative ring whose additive group is free of finite rank, a natural computational question is whether a fixed univariate polynomial $f \in \mathbb{Z}[X]$ has a root in this ring. In this paper, we show that the computational difficulty of this depends strongly on the arithmetic properties of $f$. We show that with probability 1, determining whether $f$ has a root is NP-complete. For $\text{deg } f \leq 3$ we give a full classification of the computational complexity: some special $f$ admit a polynomial-time algorithm, and for all other $f$ the problem is NP-complete. Additionally, we prove the problem is undecidable for $f = (X^2+1)^2$, conditional on Hilberts Tenth Problem for $\mathbb{Q}(i)$. The key ingredients for proving NP-completeness are a new source of NP-complete group-theoretic problems developed in previous work, and a full classification of cubic polynomials with discriminant divisible only by $3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2101_06165 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | The complexity of root-finding in orders Spelier, Pim Rings and Algebras Number Theory 11R54, 68Q17 (Primary) 03D35 (Secondary) Given an order, a commutative ring whose additive group is free of finite rank, a natural computational question is whether a fixed univariate polynomial $f \in \mathbb{Z}[X]$ has a root in this ring. In this paper, we show that the computational difficulty of this depends strongly on the arithmetic properties of $f$. We show that with probability 1, determining whether $f$ has a root is NP-complete. For $\text{deg } f \leq 3$ we give a full classification of the computational complexity: some special $f$ admit a polynomial-time algorithm, and for all other $f$ the problem is NP-complete. Additionally, we prove the problem is undecidable for $f = (X^2+1)^2$, conditional on Hilberts Tenth Problem for $\mathbb{Q}(i)$. The key ingredients for proving NP-completeness are a new source of NP-complete group-theoretic problems developed in previous work, and a full classification of cubic polynomials with discriminant divisible only by $3$. |
| title | The complexity of root-finding in orders |
| topic | Rings and Algebras Number Theory 11R54, 68Q17 (Primary) 03D35 (Secondary) |
| url | https://arxiv.org/abs/2101.06165 |