On the number of equivalence classes of binary forms of given degree and given discriminant

Bérczes, Attila; Evertse, Jan–Hendrik; Győry, Kálmán

doi:10.4064/aa113-4-6

Cited by 7 publications

(8 citation statements)

References 11 publications

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Order By: Relevance

“…Theorem G (Bérczes, Evertse and Győry [3]). Let O be an order with quotient field of degree n ≥ 4 over Q.…”

Section: Bounds For the Number Of Equivalence Classes Letmentioning

confidence: 99%

“…If O is an order of a number field of degree n ≥ 4, then (7) implies an explicit upper bound for the number of strong equivalence classes of monic irreducible binary forms F ∈ Z[x, y] with O F = O. Recently, the following has been established in [3] for not necessarily monic binary forms.…”

Section: Bounds For the Number Of Equivalence Classes Letmentioning

confidence: 99%

“…Theorem G was used in [3] to derive upper bounds for the number of equivalence classes of binary forms with given discriminant. For given integers a, k ≥ 1 denote by d k (a) the number of tuples of positive integers…”

Section: Bounds For the Number Of Equivalence Classes Letmentioning

confidence: 99%

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Polynomials and binary forms with given discriminant

Győry¹

2006

Publ. Math. Debrecen

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There is an extensive literature of monic polynomials and binary forms with given discriminant. The first part of our paper gives a brief survey of the most important results over Z on such polynomials and binary forms as well as on their various applications. In the second part we improve some earlier general effective and quantitative results over number fields. As an application we obtain some new information about the arithmetical properties of discriminants of polynomials and binary forms.

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“…Theorem G (Bérczes, Evertse and Győry [3]). Let O be an order with quotient field of degree n ≥ 4 over Q.…”

Section: Bounds For the Number Of Equivalence Classes Letmentioning

confidence: 99%

Section: Bounds For the Number Of Equivalence Classes Letmentioning

confidence: 99%

See 1 more Smart Citation

Polynomials and binary forms with given discriminant

Győry¹

2006

Publ. Math. Debrecen

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“…In the case n = 3, the parametrization of cubic orders due to Levi and Delone-Faddeev implies that every order O in a cubic field is the invariant order of a unique integral binary cubic form up to equivalence. For degrees n ≥ 4, Bérczes, Evertse, and Győry proved in [7] that an order O in a number field K of degree n can be the invariant order of at most 2 24n 3 classes of integral binary n-ic forms. This upper bound was subsequently improved to 2 5n 2 by Evertse and Győry [17,Theorem 17.1.1].…”

Section: Introductionmentioning

confidence: 99%

On the number of monogenizations of a quartic order

Bhargava¹

2021

Preprint

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We show that an order in a quartic field has fewer than 3000 essentially different generators as a Z-algebra (and fewer than 200 if the discriminant of the order is sufficiently large). This significantly improves the previously best known bound of 2 72 .Analogously, we show that an order in a quartic field is isomorphic to the invariant order of at most 10 classes of integral binary quartic forms (and at most 7 if the discriminant is sufficiently large). This significantly improves the previously best known bound of 2 80 .

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“…The latter is a consequence of the quantitative subspace theorem. In our arguments we use ideas from [8], [7] and [2]. Theorem 2.3 is proved in Section 9.…”

mentioning

confidence: 99%