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Rational solutions of certain Diophantine equations involving norms

Maciej Ulas — 2014

Acta Arithmetica

We present some results concerning the unirationality of the algebraic variety $_{f}$ given by the equation $N_{K / k} (X ₁ + α X ₂ + α ² X ₃) = f (t)$ , 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 $_{f}$ contains a k-rational point (x₀,y₀,z₀,t₀) with f(t₀)≠0, then $_{f}$ 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

Maciej Ulas — 2010

Acta Arithmetica

Rational points on certain quintic hypersurfaces

Maciej Ulas — 2009

Acta Arithmetica

Rational points on certain elliptic surfaces

Maciej Ulas — 2007

Acta Arithmetica

On the diophantine equation f(x)f(y) = f(z)²

Maciej Ulas — 2007

Colloquium Mathematicae

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

Maciej Ulas — 2009

Colloquium Mathematicae

We show that the system of equations $t_{x} + t_{y} = t_{p}, t_{y} + t_{z} = t_{q}, t_{x} + t_{z} = t_{r}$ , where $t_{x} = x (x + 1) / 2$ 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 $t_{x} + t_{y} = t_{p}, t_{y} + t_{z} = t_{q}, t_{x} + t_{z} = t_{r}, t_{x} + t_{y} + t_{z} = t_{s}$ has infinitely many rational two-parameter solutions.

Rational Points on Certain Hyperelliptic Curves over Finite Fields

Maciej Ulas — 2007

Bulletin of the Polish Academy of Sciences. Mathematics

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 $S_{k}^{i} : u ² = \prod_{j = 1}^{k} g_{i} (x_{j})$ , 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 $C_{i} : y ² = g_{i} (x)$ , (i=1,2) defined over a finite field, in polynomial time.

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Currently displaying 1 – 12 of 12

Rational solutions of certain Diophantine equations involving norms

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

A note on Sierpiński's problem related to triangular numbers

Rational Points on Certain Hyperelliptic Curves over Finite Fields

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

On torsion points on an elliptic curves via division polynomials.

A note on arithmetic progressions on quartic elliptic curves.