Some factorization results for bivariate polynomials
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866929736789262336 |
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| author | Bonciocat, Nicolae Ciprian Garg, Rishu Singh, Jitender |
| author_facet | Bonciocat, Nicolae Ciprian Garg, Rishu Singh, Jitender |
| contents | We provide upper bounds on the total number of irreducible factors, and in particular irreducibility criteria for some classes of bivariate polynomials $f(x,y)$ over an arbitrary field $\mathbb{K}$. Our results rely on information on the degrees of the coefficients of $f$, and on information on the factorization of the constant term and of the leading coefficient of $f$, viewed as a polynomial in $y$ with coefficients in $\mathbb{K}[x]$. In particular, we provide a generalization of the bivariate version of Perron's irreducibility criterion, and similar results for polynomials in an arbitrary number of indeterminates. The proofs use non-Archimedean absolute values, that are suitable for finding information on the location of the roots of $f$ in an algebraic closure of $\mathbb{K}(x)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_02324 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Some factorization results for bivariate polynomials Bonciocat, Nicolae Ciprian Garg, Rishu Singh, Jitender Number Theory Commutative Algebra Algebraic Geometry 11R09, 12J25, 12E05, 11C08 We provide upper bounds on the total number of irreducible factors, and in particular irreducibility criteria for some classes of bivariate polynomials $f(x,y)$ over an arbitrary field $\mathbb{K}$. Our results rely on information on the degrees of the coefficients of $f$, and on information on the factorization of the constant term and of the leading coefficient of $f$, viewed as a polynomial in $y$ with coefficients in $\mathbb{K}[x]$. In particular, we provide a generalization of the bivariate version of Perron's irreducibility criterion, and similar results for polynomials in an arbitrary number of indeterminates. The proofs use non-Archimedean absolute values, that are suitable for finding information on the location of the roots of $f$ in an algebraic closure of $\mathbb{K}(x)$. |
| title | Some factorization results for bivariate polynomials |
| topic | Number Theory Commutative Algebra Algebraic Geometry 11R09, 12J25, 12E05, 11C08 |
| url | https://arxiv.org/abs/2402.02324 |