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1
Content available remote

Rational solutions of certain Diophantine equations involving norms

100%
Ulas M.
Acta Arithmetica
|
2014
|
tom 165
|
nr 1
47-56
EN
We present some results concerning the unirationality of the algebraic variety $𝓢_{f}$ given by the equation $N_{K/k} (X₁ + αX₂ + α²X₃) = f(t)$, 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 $𝓢_{f}$ contains a k-rational point (x₀,y₀,z₀,t₀) with f(t₀)≠0, then $𝓢_{f}$ is k-unirational. A similar result is proved for a broad family of quintic polynomials f satisfying some mild conditions (for example this family contains all irreducible polynomials). Moreover, the unirationality of $𝓢_{f}$ (with a non-trivial k-rational point) is proved for any polynomial f of degree 6 with f not equivalent to a polynomial h satisfying h(t) = h(ζ₃t), where ζ₃ is the primitive third root of unity. We are able to prove the same result for an extension of degree 3 generated by a root of the polynomial h(x) = x³ +ax + b ∈ k[x], provided that f(t) = t⁶ + a₄t⁴ + a₁t + a₀ ∈ k[t] with a₁a₄ ≠ 0.
2
Content available remote

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

100%
Ulas M.
Colloquium Mathematicum
|
2007
|
tom 107
|
nr 1
1-6
EN
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.
3
Content available remote

On certain Diophantine systems with infinitely many parametric solutions and applications

100%
Ulas M.
Acta Arithmetica
|
2010
|
tom 145
|
nr 3
305-313
4
Content available remote

Rational Points on Certain Hyperelliptic Curves over Finite Fields

100%
Ulas M.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2007
|
tom 55
|
nr 2
97-104
EN
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 $S_{k}^{i}: u² = ∏_{j=1}^{k} g_{i}(x_{j})$, 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 $C_{i}:y² = g_{i}(x)$, (i=1,2) defined over a finite field, in polynomial time.
5
Content available remote

A note on Sierpiński's problem related to triangular numbers

100%
Ulas M.
Colloquium Mathematicum
|
2009
|
tom 117
|
nr 2
165-173
EN
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.
6
Content available remote

Rational points on certain quintic hypersurfaces

100%
Ulas M.
Acta Arithmetica
|
2009
|
tom 138
|
nr 4
347-356
7
Content available remote

Rational points on certain elliptic surfaces

88%
Ulas M.
Acta Arithmetica
|
2007
|
tom 129
|
nr 2
167-185
8
Content available remote

Characterization of the torsion of the Jacobian of y² = x⁵+Ax and some applications

64%
Jędrzejak T., Ulas M.
Acta Arithmetica
|
2010
|
tom 144
|
nr 2
183-191
9
Content available remote

On the reducibility type of trinomials

64%
Bremner A., Ulas M.
Acta Arithmetica
|
2012
|
tom 153
|
nr 4
349-372
10
Content available remote

Tuples of hyperelliptic curves y² = xⁿ+a

51%
Jędrzejak T., Top J., Ulas M.
Acta Arithmetica
|
2011
|
tom 150
|
nr 2
105-113
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