Maciej Ulas

On certain arithmetic properties of Stern polynomials

We prove several theorems concerning arithmetic properties of Stern polynomials defined in the following way:

ON ARITHMETIC PROGRESSIONS ON GENUS TWO CURVES

B_{0}(t)=0, B_{1}(t)=1, B_{2n}(t)=tB_{n}(t)

Rational points in geometric progressions on certain hyperelliptic curves

, and

RATIONAL POINTS ON CERTAIN DEL PEZZO SURFACES OF DEGREE ONE

B_{2n+1}(t)=B_{n}(t)+B_{n+1}(t)

On some Diophantine systems involving symmetric polynomials

. We study also the sequence

Some observations on the diophantine equation

e(n)=\op{deg}_{t}B_{n}(t)

y^2=x!+A

and give various of its properties.

and related results

We study arithmetic progression in the x-coordinate of rational points on genus two curves. As we know, there are two models for the curve C of genus two: C : y 2 = f5(x) or C : y 2 = f6(x), where f5, f6 2 Q(x), deg f5 = 5, deg f6 = 6 and the polynomials f5, f6 do not have multiple roots. First we prove that there exists an infinite family of curves of the form y 2 = f(x), where f 2 Q(x) and deg f = 5 each containing 11 points in arithmetic progression. We also present an example of F 2 Q(x) with deg F = 5 such that on the curve y 2 = F(x) twelve points lie in arithmetic progression. Next, we show that there exist infinitely many curves of the form y 2 = g(x) where g 2 Q(x) and deg g = 6, each containing 16 points in arithmetic progression. Moreover, we present two examples of curves in this form with 18 points in arithmetic progression.

A NOTE ON ERDŐS–STRAUS AND ERDŐS–GRAHAM DIVISIBILITY PROBLEMS (WITH AN APPENDIX BY ANDRZEJ SCHINZEL)

We pose a simple Diophantine problem which may be expressed in the language of geometry. Let C be a hyperelliptic curve given by the equation y2 = f(x), where f ∈ Z[x] is without multiple roots. We say that points Pi = (xi, yi) ∈ C(Q) for i = 1, 2, . . . , k, are in geometric progression if the numbers xi for i = 1, 2, . . . , k, are in geometric progression. Let n ≥ 3 be a given integer. In this paper we show that there exist polynomials a, b ∈ Z[t] such that on the curve y2 = a(t)xn + b(t) (defined over the field Q(t)) we can find four points in geometric progression. In particular this result generalizes earlier results of Berczes and Ziegler concerning the existence of geometric progressions on Pell type quadrics y2 = ax2 + b. We also investigate for fixed b ∈ Z, when there can exist rationals yi, i = 1, ..., 4, with {y2 i − b} forming a geometric progression, with particular attention to the case b = 1. Finally, we show that there exist infinitely many parabolas y2 = ax + b which contain five points in geometric progression.

ARITHMETIC PROPERTIES OF THE SEQUENCE OF DEGREES OF STERN POLYNOMIALS AND RELATED RESULTS

Let and let us consider a del Pezzo surface of degree one given by the equation . In this paper we prove that if the set of rational points on the curve E a,b : Y 2 = X 3 + 135(2 a −15) X −1350(5 a + 2 b − 26) is infinite then the set of rational points on the surface ϵ f is dense in the Zariski topology.