On birational monomial transformations of plane
We study birational monomial transformations of the form φ(x:y:z)=(ϵ1xα1yβ1zγ1:ϵ2xα2yβ2zγ2:xα3yβ3zγ3), where ϵ1,ϵ2∈{−1,1}. These transformations form a group. We describe this group in terms of generators and relations and, for every such transformation φ, we prove a formula, which represents the tr...
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| Format: | Article |
| Language: | English |
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Wiley
2004-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/S0161171204306514 |
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| _version_ | 1850167484559130624 |
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| author | Anatoly B. Korchagin |
| author_facet | Anatoly B. Korchagin |
| author_sort | Anatoly B. Korchagin |
| collection | DOAJ |
| description | We study birational monomial transformations of the form φ(x:y:z)=(ϵ1xα1yβ1zγ1:ϵ2xα2yβ2zγ2:xα3yβ3zγ3), where ϵ1,ϵ2∈{−1,1}. These transformations form a group. We describe this group in terms of generators and relations and, for every such transformation φ, we prove a formula, which represents the transformation φ as a product of generators of the group. To prove this formula, we use birationally equivalent polynomials Ax+By+C and Axp+Byq+Cxrys. If φ is the transformation which carries one polynomial onto another, then the integral powers of generators in the product, which represents the transformation φ, can be calculated by the expansion of p/q in the continued fraction. |
| format | Article |
| id | doaj-art-d8f271232b6c4cd5a7a8864eda7cd5d5 |
| institution | OA Journals |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2004-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-d8f271232b6c4cd5a7a8864eda7cd5d52025-08-20T02:21:11ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252004-01-012004321671167710.1155/S0161171204306514On birational monomial transformations of planeAnatoly B. Korchagin0Department of Mathematics and Statistics, Texas Tech University, Lubbock 79409-1042, TX, USAWe study birational monomial transformations of the form φ(x:y:z)=(ϵ1xα1yβ1zγ1:ϵ2xα2yβ2zγ2:xα3yβ3zγ3), where ϵ1,ϵ2∈{−1,1}. These transformations form a group. We describe this group in terms of generators and relations and, for every such transformation φ, we prove a formula, which represents the transformation φ as a product of generators of the group. To prove this formula, we use birationally equivalent polynomials Ax+By+C and Axp+Byq+Cxrys. If φ is the transformation which carries one polynomial onto another, then the integral powers of generators in the product, which represents the transformation φ, can be calculated by the expansion of p/q in the continued fraction.http://dx.doi.org/10.1155/S0161171204306514 |
| spellingShingle | Anatoly B. Korchagin On birational monomial transformations of plane International Journal of Mathematics and Mathematical Sciences |
| title | On birational monomial transformations of plane |
| title_full | On birational monomial transformations of plane |
| title_fullStr | On birational monomial transformations of plane |
| title_full_unstemmed | On birational monomial transformations of plane |
| title_short | On birational monomial transformations of plane |
| title_sort | on birational monomial transformations of plane |
| url | http://dx.doi.org/10.1155/S0161171204306514 |
| work_keys_str_mv | AT anatolybkorchagin onbirationalmonomialtransformationsofplane |