scholarly journals Integer points on algebraic curves with exceptional units

1997 ◽  
Vol 63 (2) ◽  
pp. 145-164 ◽  
Author(s):  
Dimitrios Poulakis
Keyword(s):  
Elliptic Equations ◽  
Algebraic Curves ◽  
Irreducible Polynomial ◽  
Algebraic Number Field ◽  
Genus Zero ◽  
Integer Points ◽  
Regular Functions ◽  
Affine Curves ◽  
Absolutely Irreducible Polynomial ◽  
Integral Solutions

AbstractLet F(X, Y) be an absolutely irreducible polynomial with coefficients in an algebraic number field K. Denote by C the algebraic curve defined by the equation F(X, Y) = 0 and by K[C] the ring of regular functions on Cover K. Assume that there is a unit ϕ in K[C] − K such that 1 − ϕ is also a unit. Then we establish an explicit upper bound for the size of integral solutions of the equation F(X, Y) = 0, defined over K. Using this result we establish improved explicit upper bounds on the size of integral solutions to the equations defining non-singular affine curves of genus zero, with at least three points at ‘infinity’, the elliptic equations and a class of equations containing the Thue curves.

HEIGHTS AND LOGARITHMIC gcd ON ALGEBRAIC CURVES

2008 ◽  
Vol 04 (02) ◽  
pp. 177-197 ◽  
Author(s):  
MOURAD ABOUZAID
Keyword(s):  
Singular Point ◽  
Algebraic Curves ◽  
Irreducible Polynomial ◽  
Plane Curve ◽  
Asymptotic Equality ◽  
Algebraic Solution ◽  
Greatest Common Divisor ◽  
The Absolute ◽  
Logarithmic Height ◽  
Integral Solutions

Let F(x,y) be an irreducible polynomial over ℚ, satisfying F(0,0) = 0. Skolem proved that the integral solutions of F(x,y) = 0 with fixed gcd are bounded [13] and Walsh gave an explicit bound in terms of d = gcd (x,y) and F [16]. Assuming that (0,0) is a non-singular point of the plane curve F(x,y) = 0, we extend this result to algebraic solution, and obtain an asymptotic equality instead of inequality. We show that for any algebraic solution (α,β), the quotient h(α)/ log d is approximatively equal to degyF and the quotient h(β)/ log d to deg x F; here h(·) is the absolute logarithmic height and d is the (properly defined) "greatest common divisor" of α and β.

scholarly journals Constructing irreducible polynomials with prescribed level curves over finite fields

2001 ◽  
Vol 27 (4) ◽  
pp. 197-200
Author(s):  
Mihai Caragiu
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Irreducible Polynomial ◽  
Irreducible Polynomials ◽  
Level Curves ◽  
Curves Over Finite Fields ◽  
Irreducibility Criterion ◽  
Absolutely Irreducible Polynomial

We use Eisenstein's irreducibility criterion to prove that there exists an absolutely irreducible polynomialP(X,Y)∈GF(q)[X,Y]with coefficients in the finite fieldGF(q)withqelements, with prescribed level curvesXc:={(x,y)∈GF(q)2|P(x,y)=c}.

Non-existence of solutions for some nonlinear elliptic equations involving measures

2000 ◽  
Vol 130 (1) ◽  
pp. 167-187 ◽  
Author(s):  
L. Orsina ◽  
A. Prignet
Keyword(s):  
Elliptic Equations ◽  
Dirichlet Boundary ◽  
Existence Of Solutions ◽  
Model Problem ◽  
Nonlinear Elliptic Equations ◽  
Dirichlet Boundary Conditions ◽  
Regular Functions ◽  
Nonlinear Elliptic ◽  
Bounded Open Subset ◽  
Image Position

In this paper, we study the non-existence of solutions for the following (model) problem in a bounded open subset Ω of RN: with Dirichlet boundary conditions, where p > 1, q > 1 and μ is a bounded Radon measure. We prove that if λ is a measure which is concentrated on a set of zero r capacity (p < r ≤ N), and if q > r (p − 1)/(r − p), then there is no solution to the above problem, in the sense that if one approximates the measure λ with a sequence of regular functions fn, and if un is the sequence of solutions of the corresponding problems, then un converges to zero.We also study the non-existence of solutions for the bilateral obstacle problem with datum a measure λ concentrated on a set of zero p capacity, with u in for every υ in K, finding again that the only solution obtained by approximation is u = 0.

Uniform bounds for the number of solutions to Yn = f(X)

1986 ◽  
Vol 100 (2) ◽  
pp. 237-248 ◽  
Author(s):  
J.-H. Evertse ◽  
J. H. Silverman
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Solutions To Equations ◽  
Number Of Solutions ◽  
Explicit Result ◽  
Uniform Bounds ◽  
The Past ◽  
Image Position ◽  
Integral Solutions

Let K be an algebraic number field and f(X) ∈ K[X]. The Diophantine problem of describing the solutions to equations of the formhas attracted considerable interest over the past 60 years. Siegel [12], [13] was the first to show that under suitable non-degeneracy conditions, the equation (+) has only finitely many integral solutions in K. LeVeque[7] proved the following, more explicit, result. Letwhere a ∈ K* and αl,…,αk are distinct and algebraic over K. Then (+) has only finitely many integral solutions unless (nl,…,nk) is a permutation of one of the n-tuples

scholarly journals Bounds for the size of integral solutions to Ym = f(X)

1999 ◽  
Vol 42 (1) ◽  
pp. 127-141
Author(s):  
Dimitrios Poulakis
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Diophantine Equations ◽  
Algebraic Number Field ◽  
Upper Bounds ◽  
Ring Of Integers ◽  
Odd Order ◽  
Integral Solutions

Let K be an algebraic number field with ring of integers OK and f(X) ∈ OK[X]. In this paper we establish improved explicit upper bounds for the size of solutions in OK, of diophantine equations Y2 = f(X), where f(X) has at least three roots of odd order, and Ym = f(X), where m is an integer ≥ 3 and f(X) has at least two roots of order prime to m.

scholarly journals CONCENTRATION OF POINTS ON MODULAR QUADRATIC FORMS

2011 ◽  
Vol 07 (07) ◽  
pp. 1835-1839 ◽  
Author(s):  
ANA ZUMALACÁRREGUI
Keyword(s):  
Quadratic Form ◽  
Upper Bound ◽  
Quadratic Forms ◽  
Irreducible Polynomial ◽  
Number Of Solutions ◽  
Modulo P ◽  
Absolutely Irreducible Polynomial ◽  
Mod P

Let Q(x, y) be a quadratic form with discriminant D ≠ 0. We obtain non-trivial upper bound estimates for the number of solutions of the congruence Q(x, y) ≡ λ ( mod p), where p is a prime and x, y lie in certain intervals of length M, under the assumption that Q(x, y) - λ is an absolutely irreducible polynomial modulo p. In particular, we prove that the number of solutions to this congruence is Mo(1) when M ≪ p1/4. These estimates generalize a previous result by Cilleruelo and Garaev on the particular congruence xy ≡ λ( mod p).

INTEGER POINTS CLOSE TO ALGEBRAIC CURVES

2002 ◽  
Vol 65 (01) ◽  
pp. 10-26
Author(s):  
FLORIN P. BOCA ◽  
MARIAN VÂJÂITU ◽  
ALEXANDRU ZAHARESCU
Keyword(s):  
Algebraic Curves ◽  
Integer Points
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