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2 Bulletin of the Polish Academy of Sciences. Mathematics

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Multiplicative relations of points on algebraic groups

Flicker Y. Z., Krasoń P.

Bulletin of the Polish Academy of Sciences. Mathematics

2017

Vol. 65, no. 2

125--138

Our aim here is to restructure the area of multiplicative relations on points and congruences, by proposing a novel conjecture in the context of general reductive linear algebraic groups. To support our conjecture we check it in a few elementary but new cases, and claim this extends classical work in number theory on multiplicative relations on points and congruences, initiated by Skolem and Schinzel, which we rephrase group-theoretically as Hasse principles on commutative linear algebraic groups, or tori, so that a part of it becomes the abelian case of our conjecture. Our conjecture can then be viewed as an extension to general-not necessarily commutative-reductive linear algebraic groups of a part of Schinzel's result. We relate it to the Erdős support problem. To motivate our conjecture from another perspective we note that analogues have been extensively developed for abelian varieties. We give a short account of this, and state a question on the ʺdetecting linear dependence" problem.

Rational points on certain hyperelliptic curves over finite fields

Ulas M.

Bulletin of the Polish Academy of Sciences. Mathematics

2007

Vol. 55, no 2

97-104

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[...], 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 C1 : y2 = gi(x), (i = 1,2) defined over a finite field, in polynomial time.