Totally Real Integral Points on a Plane Algebraic Curve
نویسندگان
چکیده
Michel LAURENT Abstract. Let F (X,Y ) = ∑m i=0 ∑n j=0 ai,jX iY j be an absolutely irreducible polynomial in Z[X,Y ]. Suppose that m ≥ 1, n ≥ 2 and that the polynomial ∑n j=0 am,jY j is reducible in Q[Y ], has n simple roots and an unique real root. Let L be a totally real number field and let (ξ, ζ) ∈ OL ×L be such that F (ξ, ζ) = 0. We give an upper bound for the absolute height H(ξ) which depends only upon the polynomial F . Our result may be viewed as a natural extension of Runge’s method to arbitrary totally real number fields.
منابع مشابه
Visualization of Points and Segments of Real Algebraic Plane Curves
This thesis presents an exact and complete approach for visualization of segments and points of real plane algebraic curves given in implicit form f(x, y) = 0. A curve segment is a distinct curve branch consisting of regular points only. Visualization of algebraic curves having self-intersection and isolated points constitutes the main challenge. Visualization of curve segments involves even mo...
متن کاملAn Improved Upper Complexity Bound for the Topology Computation of a Real Algebraic Plane Curve
The computation of the topological shape of a real algebraic plane curve is usually driven by the study of the behavior of the curve around its critical points (which includes also the singular points). In this paper we present a new algorithm computing the topological shape of a real algebraic plane curve whose complexity is better than the best algorithms known. This is due to the avoiding, t...
متن کاملComputing real inflection points of cubic algebraic curves
Shape modeling using planar cubic algebraic curves calls for computing the real inflection points of these curves since inflection points represents important shape feature. A real inflection point is also required for transforming projectively a planar cubic algebraic curve to the normal form, in order to facilitate further analysis of the curve. However, the naive method for computing the inf...
متن کاملReal Plane Algebraic Curves
We study real algebraic plane curves, at an elementary level, using as little algebra as possible. Both cases, affine and projective, are addressed. A real curve is infinite, finite or empty according to the fact that a minimal polynomial for the curve is indefinite, semi–definite nondefinite or definite. We present a discussion about isolated points. By means of the operator, these points can ...
متن کاملPeriods of Hilbert Modular Forms and Rational Points on Elliptic Curves
Let E be a modular elliptic curve over a totally real field. In [7, Chapter 8] the first author formulates a conjecture allowing the construction of canonical algebraic points on E by suitably integrating the associated Hilbert modular form. The main goal of the present paper is to obtain numerical evidence for this conjecture in the first case where it asserts something nontrivial, namely, whe...
متن کامل