The number of roots of a lacunary bivariate polynomial on a line
We prove that a polynomial f ∈ R [x, y] with t non-zero terms, restricted to a real line y = a x + b, either has at most 6 t - 4 zeros or vanishes over the whole line. As a consequence, we derive an alternative algorithm for deciding whether a linear polynomial y - a x - b ∈ K [x, y] divides a lacun...
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2009
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| Acceso en línea: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_07477171_v44_n9_p1280_Avendano http://hdl.handle.net/20.500.12110/paper_07477171_v44_n9_p1280_Avendano |
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paper:paper_07477171_v44_n9_p1280_Avendano2023-06-08T15:45:09Z The number of roots of a lacunary bivariate polynomial on a line Avendaño, Martín Descartes' rule of signs Factorization of polynomials Fewnomials We prove that a polynomial f ∈ R [x, y] with t non-zero terms, restricted to a real line y = a x + b, either has at most 6 t - 4 zeros or vanishes over the whole line. As a consequence, we derive an alternative algorithm for deciding whether a linear polynomial y - a x - b ∈ K [x, y] divides a lacunary polynomial f ∈ K [x, y], where K is a real number field. The number of bit operations performed by the algorithm is polynomial in the number of non-zero terms of f, in the logarithm of the degree of f, in the degree of the extension K / Q and in the logarithmic height of a, b and f. © 2009 Elsevier Ltd. All rights reserved. Fil:Avendaño, M. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. 2009 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_07477171_v44_n9_p1280_Avendano http://hdl.handle.net/20.500.12110/paper_07477171_v44_n9_p1280_Avendano |
| institution |
Universidad de Buenos Aires |
| institution_str |
I-28 |
| repository_str |
R-134 |
| collection |
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) |
| topic |
Descartes' rule of signs Factorization of polynomials Fewnomials |
| spellingShingle |
Descartes' rule of signs Factorization of polynomials Fewnomials Avendaño, Martín The number of roots of a lacunary bivariate polynomial on a line |
| topic_facet |
Descartes' rule of signs Factorization of polynomials Fewnomials |
| description |
We prove that a polynomial f ∈ R [x, y] with t non-zero terms, restricted to a real line y = a x + b, either has at most 6 t - 4 zeros or vanishes over the whole line. As a consequence, we derive an alternative algorithm for deciding whether a linear polynomial y - a x - b ∈ K [x, y] divides a lacunary polynomial f ∈ K [x, y], where K is a real number field. The number of bit operations performed by the algorithm is polynomial in the number of non-zero terms of f, in the logarithm of the degree of f, in the degree of the extension K / Q and in the logarithmic height of a, b and f. © 2009 Elsevier Ltd. All rights reserved. |
| author |
Avendaño, Martín |
| author_facet |
Avendaño, Martín |
| author_sort |
Avendaño, Martín |
| title |
The number of roots of a lacunary bivariate polynomial on a line |
| title_short |
The number of roots of a lacunary bivariate polynomial on a line |
| title_full |
The number of roots of a lacunary bivariate polynomial on a line |
| title_fullStr |
The number of roots of a lacunary bivariate polynomial on a line |
| title_full_unstemmed |
The number of roots of a lacunary bivariate polynomial on a line |
| title_sort |
number of roots of a lacunary bivariate polynomial on a line |
| publishDate |
2009 |
| url |
https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_07477171_v44_n9_p1280_Avendano http://hdl.handle.net/20.500.12110/paper_07477171_v44_n9_p1280_Avendano |
| work_keys_str_mv |
AT avendanomartin thenumberofrootsofalacunarybivariatepolynomialonaline AT avendanomartin numberofrootsofalacunarybivariatepolynomialonaline |
| _version_ |
1768544690757435392 |