On binary cubic and quartic forms
Journal de Théorie des Nombres de Bordeaux, Tome 31 (2019) no. 2, pp. 323-341.

Dans cet article, nous décrivons le groupe d’automorphismes rationnels d’une forme binaire cubique ou quartique à coefficients entiers et à discriminant non nul en termes de certains covariants quadratiques des formes cubiques et quartiques. Cela nous permet d’étendre les travaux de Hooley et de donner des formules asymptotiques précises pour le nombre d’entiers appartenant à un intervalle et représentables par une forme cubique ou quartique donnée. En outre, nous déterminons le corps de définition des droites contenues dans certaines surfaces cubiques et quartiques associées à des formes cubiques et quartiques binaires.

In this paper we determine the group of rational automorphisms of binary cubic and quartic forms with integer coefficients and non-zero discriminant in terms of certain quadratic covariants of cubic and quartic forms. This allows one to give precise asymptotic formulae for the number of integers in an interval representable by a binary cubic or quartic form and extends work of Hooley. Further, we give the field of definition of lines contained in certain cubic and quartic surfaces related to binary cubic and quartic forms.

Reçu le : 2018-06-11
Accepté le : 2019-06-14
Publié le : 2019-10-29
DOI : https://doi.org/10.5802/jtnb.1083
Classification : 11D45,  11E76,  11D25
Mots clés: Binary forms, cubic and quartic surfaces
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     author = {Stanley Yao Xiao},
     title = {On binary cubic and quartic forms},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {31},
     number = {2},
     year = {2019},
     pages = {323-341},
     doi = {10.5802/jtnb.1083},
     language = {en},
     url = {jtnb.centre-mersenne.org/item/JTNB_2019__31_2_323_0/}
}
Stanley Yao Xiao. On binary cubic and quartic forms. Journal de Théorie des Nombres de Bordeaux, Tome 31 (2019) no. 2, pp. 323-341. doi : 10.5802/jtnb.1083. https://jtnb.centre-mersenne.org/item/JTNB_2019__31_2_323_0/

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