منابع مشابه
Geometry of Cubic Polynomials
Imagine a sphere with its equator inscribed in an equilateral triangle. This Saturn-like figure will help us understand from where Cardano’s formula for finding the roots of a cubic polynomial p(z) comes. It will also help us find a new proof of Marden’s theorem, the surprising result that the roots of the derivative p(z) are the foci of the ellipse inscribed in and tangent to the midpoints of ...
متن کاملReducible cubic CNS polynomials
The concept of a canonical number system can be regarded as a natural generalization of decimal representations of rational integers to elements of residue class rings of polynomial rings. Generators of canonical number systems are CNS polynomials which are known in the linear and quadratic cases, but whose complete description is still open. In the present note reducible CNS polynomials are tr...
متن کاملThe Geometry of Cubic Polynomials
Given a complex cubic polynomial p(z) = (z − 1)(z − r1)(z − r2) with |r1| = 1 = |r2|, where are the critical points? Marden’s Theorem tells us that the critical points are the foci of the Steiner ellipse of 41r1r2. In this paper we further explore the structure of these critical points. If we let Tγ be the circle of diameter γ passing through 1 and 1 − γ, then there are α, β ∈ [0, 2] such that ...
متن کاملTriangles, Ellipses, and Cubic Polynomials
1. INTRODUCTION. Discussions that led to this paper began during an electronic version of the Secondary School Teachers Program, a part of the Park City Mathematics Institute (PCMI), offered at the University of Cincinnati in the summer of 2006 for local high school mathematics teachers. The authors were working with the teachers on daily problem sets provided by PCMI; complex numbers and polyn...
متن کاملCharacter Sums with Division Polynomials
We obtain nontrivial estimates of quadratic character sums of division polynomials Ψn(P ), n = 1, 2, . . ., evaluated at a given point P on an elliptic curve over a finite field of q elements. Our bounds are nontrivial if the order of P is at least q for some fixed ε > 0. This work is motivated by an open question about statistical indistinguishability of some cryptographically relevant sequenc...
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ژورنال
عنوان ژورنال: Proceedings of the American Mathematical Society
سال: 1987
ISSN: 0002-9939
DOI: 10.2307/2046420