Refined Irreducibility Certificates for Multivariate Polynomials
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| Format: | Recurso digital |
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2025
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| _version_ | 1866901403800174592 |
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| author | SÉRGIO DE ANDRADE, PAULO |
| author_facet | SÉRGIO DE ANDRADE, PAULO |
| contents | This paper explores refined methods for generating irreducibility certificates for multivariate polynomials over a field. Determining whether a polynomial can be factored into non-constant polynomials is a fundamental problem in computer algebra and algebraic geometry. While complete factorization algorithms exist, they are often computationally intensive. An irreducibility certificate, in contrast, is a piece of data that allows for a quick and simple verification of a polynomial's irreducibility. This work synthesizes and extends classical criteria with geometric approaches to produce more powerful and broadly applicable certificates. The methodology focuses on two primary refinements: the strategic use of invertible linear transformations of variables to augment the effectiveness of Newton polytope-based criteria, and the combination of modular arithmetic with geometric analysis. We formalize the concept of a 'transformational certificate', where the certificate consists of a transformation matrix and a proof of indecomposability for the resulting Newton polytope. The results demonstrate that many polynomials whose irreducibility is not provable by direct application of standard criteria can be certified using these refined methods. For instance, a polynomial whose Newton polytope is decomposable may be transformed into one with an indecomposable polytope, providing a concise certificate. The discussion evaluates the trade-offs between the computational cost of finding such certificates and the simplicity of their verification. We conclude that these refined certificates offer a practical intermediate step between simple tests and full factorization, enhancing the efficiency of irreducibility testing in computer algebra systems. |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17543074 |
| institution | Zenodo |
| language | |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Refined Irreducibility Certificates for Multivariate Polynomials SÉRGIO DE ANDRADE, PAULO This paper explores refined methods for generating irreducibility certificates for multivariate polynomials over a field. Determining whether a polynomial can be factored into non-constant polynomials is a fundamental problem in computer algebra and algebraic geometry. While complete factorization algorithms exist, they are often computationally intensive. An irreducibility certificate, in contrast, is a piece of data that allows for a quick and simple verification of a polynomial's irreducibility. This work synthesizes and extends classical criteria with geometric approaches to produce more powerful and broadly applicable certificates. The methodology focuses on two primary refinements: the strategic use of invertible linear transformations of variables to augment the effectiveness of Newton polytope-based criteria, and the combination of modular arithmetic with geometric analysis. We formalize the concept of a 'transformational certificate', where the certificate consists of a transformation matrix and a proof of indecomposability for the resulting Newton polytope. The results demonstrate that many polynomials whose irreducibility is not provable by direct application of standard criteria can be certified using these refined methods. For instance, a polynomial whose Newton polytope is decomposable may be transformed into one with an indecomposable polytope, providing a concise certificate. The discussion evaluates the trade-offs between the computational cost of finding such certificates and the simplicity of their verification. We conclude that these refined certificates offer a practical intermediate step between simple tests and full factorization, enhancing the efficiency of irreducibility testing in computer algebra systems. |
| title | Refined Irreducibility Certificates for Multivariate Polynomials |
| url | https://doi.org/10.5281/zenodo.17543074 |