Quartic Rings Associated to Binary Quartic Forms
نویسندگان
چکیده
منابع مشابه
The practical computation of areas associated with binary quartic forms
We derive formulas for practically computing the area of the region |F (x, y)| ≤ 1 defined by a binary quartic form F (X, Y ) ∈ R[X,Y ]. These formulas, which involve a particular hypergeometric function, are useful when estimating the number of lattice points in certain regions of the type |F (x, y)| ≤ h and will likely find application in many contexts. We also show that for forms F of arbitr...
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Let p ≡ 1 (mod 4) be a prime, m ∈ Z and p m. In this paper we obtain a general criterion for m to be a quartic residue (mod p) in terms of appropriate binary quadratic forms. Let d > 1 be a squarefree integer such that ( d p ) = 1, where ( d p ) is the Legendre symbol, and let εd be the fundamental unit of the quadratic field Q( √ d). Since 1942 many mathematicians tried to characterize those p...
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A reduction theory is developed for binary forms (homogeneous polynomials) of degrees three and four with integer coefficients. The resulting coefficient bounds simplify and improve on those in the literature, particularly in the case of negative discriminant. Applications include systematic enumeration of cubic number fields, and 2-descent on elliptic curves defined over Q. Remarks are given c...
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Historically, much of the theory and practice in nonlinear optimization has revolved around the quadratic models. Though quadratic functions are nonlinear polynomials, they are well structured and easy to deal with. Limitations of the quadratics, however, become increasingly binding as higher degree nonlinearity is imperative in modern applications of optimization. In the recent years, one obse...
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A reduction theory is developed for binary forms (homogeneous polynomials) of degrees three and four with integer coef-cients. The resulting coeecient bounds simplify and improve on those in the literature, particularly in the case of negative discriminant. Applications include systematic enumeration of cubic number elds, and 2-descent on elliptic curves deened over Q. Remarks are given concern...
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ژورنال
عنوان ژورنال: International Mathematics Research Notices
سال: 2011
ISSN: 1687-0247,1073-7928
DOI: 10.1093/imrn/rnr070