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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 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 = ∏ 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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