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http://hdl.handle.net/11320/14706
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Pole DC | Wartość | Język |
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dc.contributor.author | Pąk, Karol | - |
dc.date.accessioned | 2023-02-27T11:25:04Z | - |
dc.date.available | 2023-02-27T11:25:04Z | - |
dc.date.issued | 2022 | - |
dc.identifier.citation | Formalized Mathematics, Volume 30, Issue 4, Pages 255-279 | pl |
dc.identifier.issn | 1426-2630 | - |
dc.identifier.uri | http://hdl.handle.net/11320/14706 | - |
dc.description.abstract | In this article we formalize in Mizar [1], [2] the final step of ourattempt to formally construct a prime representing polynomial with 10 variablesproposed by Yuri Matiyasevich in [4].The first part of the article includes many auxiliary lemmas related to multivariate polynomials. We start from the properties of monomials, among them their evaluation as well as the power function on polynomials to define the substitution for multivariate polynomials. For simplicity, we assume that a polynomialand substituted ones as i-th variable have the same number of variables. Then we study the number of variables that are used in given multivariate polynomials. By the used variable we mean a variable that is raised at least once to a non-zeropower. We consider both adding unused variables and eliminating them. The second part of the paper deals with the construction of the polynomialproposed by Yuri Matiyasevich. First, we introduce a diophantine polynomialover 4 variables that has roots in integers if and only if indicated variable is thesquare of a natural number, and another two is the square of an odd naturalnumber. We modify the polynomial by adding two variables in such a way thatthe root additionally requires the divisibility of these added variables. Then wemodify again the polynomial by adding two variables to also guarantee the nonnegativity condition of one of these variables. Finally, we combine the prime diophantine representation proved in [7] with the obtained polynomial constructinga prime representing polynomial with 10 variables. This work has been partiallypresented in [8] with the obtained polynomial constructing a prime representingpolynomial with 10 variables in Theorem (85). | pl |
dc.language.iso | en | pl |
dc.publisher | DeGruyter Open | pl |
dc.rights | Attribution-ShareAlike 3.0 Unported (CC BY-SA 3.0) | pl |
dc.rights.uri | https://creativecommons.org/licenses/by-sa/3.0/ | pl |
dc.subject | polynomial reduction | pl |
dc.subject | prime representing polynomial | pl |
dc.title | Prime Representing Polynomial with 10 Unknowns | pl |
dc.type | Article | pl |
dc.rights.holder | © 2022 The Author(s) | pl |
dc.rights.holder | CC BY-SA 3.0 license | pl |
dc.identifier.doi | 10.2478/forma-2022-0021 | - |
dc.description.Affiliation | Institute of Computer Science, University of Białystok, Poland | pl |
dc.description.references | Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pąk, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors,Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261–279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007/978-3-319-20615-8_17. | pl |
dc.description.references | Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pąk. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9–32,2018. doi:10.1007/s10817-017-9440-6. | pl |
dc.description.references | Artur Korniłowicz and Adam Naumowicz. Niven’s theorem. Formalized Mathematics, 24(4):301–308, 2016. doi:10.1515/forma-2016-0026. | pl |
dc.description.references | Yuri Matiyasevich. Primes are nonnegative values of a polynomial in 10 variables. Journalof Soviet Mathematics, 15:33–44, 1981. doi:10.1007/BF01404106. | pl |
dc.description.references | Karol Pąk. Diophantine sets. Preliminaries. Formalized Mathematics, 26(1):81–90, 2018.doi:10.2478/forma-2018-0007. | pl |
dc.description.references | Karol Pąk. Prime representing polynomial with 10 unknowns – Introduction. Formalized Mathematics, 30(3):169–198, 2022. doi:10.2478/forma-2022-0013. | pl |
dc.description.references | Karol Pąk. Prime representing polynomial with 10 unknowns – Introduction. Part II. Formalized Mathematics, 30(4):245–253, 2022. doi:10.2478/forma-2022-0020. | pl |
dc.description.references | Karol Pąk and Cezary Kaliszyk. Formalizing a diophantine representation of the set of prime numbers. In June Andronick and Leonardo de Moura, editors,13th International Conference on Interactive Theorem Proving, ITP 2022, August 7-10, 2022, Haifa, Israel,volume 237 of LIPIcs, pages 26:1–26:8. Schloss Dagstuhl - Leibniz-Zentrum für Informatik,2022. doi:10.4230/LIPIcs.ITP.2022.26. | pl |
dc.description.references | Christoph Schwarzweller and Artur Korniłowicz. Characteristic of rings. Prime fields. Formalized Mathematics, 23(4):333–349, 2015. doi:10.1515/forma-2015-0027. | pl |
dc.identifier.eissn | 1898-9934 | - |
dc.description.volume | 30 | pl |
dc.description.issue | 4 | pl |
dc.description.firstpage | 255 | pl |
dc.description.lastpage | 279 | pl |
dc.identifier.citation2 | Formalized Mathematics | pl |
Występuje w kolekcji(ach): | Artykuły naukowe (WInf) Formalized Mathematics, 2022, Volume 30, Issue 4 |
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