On torsion points on an elliptic curves via division polynomials.
A note on arithmetic progressions on quartic elliptic curves.
Rational solutions of certain Diophantine equations involving norms
We present some results concerning the unirationality of the algebraic variety given by the equation , where k is a number field, K=k(α), α is a root of an irreducible polynomial h(x) = x³ + ax + b ∈ k[x] and f ∈ k[t]. We are mainly interested in the case of pure cubic extensions, i.e. a = 0 and b ∈ k∖k³. We prove that if deg f = 4 and contains a k-rational point (x₀,y₀,z₀,t₀) with f(t₀)≠0, then is k-unirational. A similar result is proved for a broad family of quintic polynomials f satisfying...
On certain Diophantine systems with infinitely many parametric solutions and applications
Rational points on certain quintic hypersurfaces
Rational points on certain elliptic surfaces
On the diophantine equation f(x)f(y) = f(z)²
Let f ∈ ℚ [X] and deg f ≤ 3. We prove that if deg f = 2, then the diophantine equation f(x)f(y) = f(z)² has infinitely many nontrivial solutions in ℚ (t). In the case when deg f = 3 and f(X) = X(X²+aX+b) we show that for all but finitely many a,b ∈ ℤ satisfying ab ≠ 0 and additionally, if p|a, then p²∤b, the equation f(x)f(y) = f(z)² has infinitely many nontrivial solutions in rationals.
A note on Sierpiński's problem related to triangular numbers
We show that the system of equations , where is a triangular number, has infinitely many solutions in integers. Moreover, we show that this system has a rational three-parameter solution. Using this result we show that the system has infinitely many rational two-parameter solutions.
Rational Points on Certain Hyperelliptic Curves over Finite Fields
Let K be a field, a,b ∈ K and ab ≠ 0. Consider the polynomials g₁(x) = xⁿ+ax+b, g₂(x) = xⁿ+ax²+bx, where n is a fixed positive integer. We show that for each k≥ 2 the hypersurface given by the equation , i=1,2, contains a rational curve. Using the above and van de Woestijne’s recent results we show how to construct a rational point different from the point at infinity on the curves , (i=1,2) defined over a finite field, in polynomial time.
On the reducibility type of trinomials
Characterization of the torsion of the Jacobian of y² = x⁵+Ax and some applications
Tuples of hyperelliptic curves y² = xⁿ+a
Page 1