On the Degree of the GCD of Random Polynomials over a Finite Field
In this paper, we focus on the degree of the greatest common divisor (gcd) of random polynomials over Fq. Here, Fq is the finite field with q elements. Firstly, we compute the probability distribution of the degree of the gcd of random and monic polynomials with fixed degree over Fq. Then, we consid...
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| Format: | Article |
| Language: | English |
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Wiley
2021-01-01
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| Series: | Journal of Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2021/3619347 |
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| author | Kui Liu Meijie Lu |
| author_facet | Kui Liu Meijie Lu |
| author_sort | Kui Liu |
| collection | DOAJ |
| description | In this paper, we focus on the degree of the greatest common divisor (gcd) of random polynomials over Fq. Here, Fq is the finite field with q elements. Firstly, we compute the probability distribution of the degree of the gcd of random and monic polynomials with fixed degree over Fq. Then, we consider the waiting time of the sequence of the degree of gcd functions. We compute its probability distribution, expectation, and variance. Finally, by considering the degree of a certain type gcd, we investigate the probability distribution of the number of rational (i.e., in Fq) roots (counted with multiplicity) of random and monic polynomials with fixed degree over Fq. |
| format | Article |
| id | doaj-art-b9534e05f0354da3a766b7ceca3b434b |
| institution | Kabale University |
| issn | 2314-4629 2314-4785 |
| language | English |
| publishDate | 2021-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Mathematics |
| spelling | doaj-art-b9534e05f0354da3a766b7ceca3b434b2025-02-03T07:23:54ZengWileyJournal of Mathematics2314-46292314-47852021-01-01202110.1155/2021/36193473619347On the Degree of the GCD of Random Polynomials over a Finite FieldKui Liu0Meijie Lu1School of Mathematics and Statistics, Qingdao University, 308 Ningxia Road, Shinan District, Qingdao 266000, Shandong, ChinaSchool of Mathematics and Statistics, Qingdao University, 308 Ningxia Road, Shinan District, Qingdao 266000, Shandong, ChinaIn this paper, we focus on the degree of the greatest common divisor (gcd) of random polynomials over Fq. Here, Fq is the finite field with q elements. Firstly, we compute the probability distribution of the degree of the gcd of random and monic polynomials with fixed degree over Fq. Then, we consider the waiting time of the sequence of the degree of gcd functions. We compute its probability distribution, expectation, and variance. Finally, by considering the degree of a certain type gcd, we investigate the probability distribution of the number of rational (i.e., in Fq) roots (counted with multiplicity) of random and monic polynomials with fixed degree over Fq.http://dx.doi.org/10.1155/2021/3619347 |
| spellingShingle | Kui Liu Meijie Lu On the Degree of the GCD of Random Polynomials over a Finite Field Journal of Mathematics |
| title | On the Degree of the GCD of Random Polynomials over a Finite Field |
| title_full | On the Degree of the GCD of Random Polynomials over a Finite Field |
| title_fullStr | On the Degree of the GCD of Random Polynomials over a Finite Field |
| title_full_unstemmed | On the Degree of the GCD of Random Polynomials over a Finite Field |
| title_short | On the Degree of the GCD of Random Polynomials over a Finite Field |
| title_sort | on the degree of the gcd of random polynomials over a finite field |
| url | http://dx.doi.org/10.1155/2021/3619347 |
| work_keys_str_mv | AT kuiliu onthedegreeofthegcdofrandompolynomialsoverafinitefield AT meijielu onthedegreeofthegcdofrandompolynomialsoverafinitefield |